Lemmas about linear ordered (semi)fields #

Relating one division with another term. #

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@[deprecated lt_div_iff₀]

theorem lt_div_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hc : 0 < c) :

a < b / c ↔ a * c < b

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@[deprecated lt_div_iff₀']

theorem lt_div_iff' {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hc : 0 < c) :

a < b / c ↔ c * a < b

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@[deprecated div_lt_iff₀]

theorem div_lt_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hc : 0 < c) :

b / c < a ↔ b < a * c

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@[deprecated div_lt_iff₀']

theorem div_lt_iff' {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hc : 0 < c) :

b / c < a ↔ b < c * a

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@[deprecated inv_mul_le_iff₀]

theorem inv_mul_le_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (h : 0 < b) :

b⁻¹ * a ≤ c ↔ a ≤ b * c

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@[deprecated inv_mul_le_iff₀']

theorem inv_mul_le_iff' {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (h : 0 < b) :

b⁻¹ * a ≤ c ↔ a ≤ c * b

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@[deprecated mul_inv_le_iff₀']

theorem mul_inv_le_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (h : 0 < b) :

a * b⁻¹ ≤ c ↔ a ≤ b * c

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@[deprecated mul_inv_le_iff₀]

theorem mul_inv_le_iff' {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (h : 0 < b) :

a * b⁻¹ ≤ c ↔ a ≤ c * b

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@[deprecated inv_mul_lt_iff₀]

theorem inv_mul_lt_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (h : 0 < b) :

b⁻¹ * a < c ↔ a < b * c

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@[deprecated inv_mul_lt_iff₀']

theorem inv_mul_lt_iff' {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (h : 0 < b) :

b⁻¹ * a < c ↔ a < c * b

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@[deprecated mul_inv_lt_iff₀']

theorem mul_inv_lt_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (h : 0 < b) :

a * b⁻¹ < c ↔ a < b * c

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@[deprecated mul_inv_lt_iff₀]

theorem mul_inv_lt_iff' {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (h : 0 < b) :

a * b⁻¹ < c ↔ a < c * b

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@[deprecated inv_le_iff_one_le_mul₀]

theorem inv_pos_le_iff_one_le_mul {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) :

a⁻¹ ≤ b ↔ 1 ≤ b * a

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@[deprecated inv_le_iff_one_le_mul₀']

theorem inv_pos_le_iff_one_le_mul' {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) :

a⁻¹ ≤ b ↔ 1 ≤ a * b

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@[deprecated inv_lt_iff_one_lt_mul₀]

theorem inv_pos_lt_iff_one_lt_mul {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) :

a⁻¹ < b ↔ 1 < b * a

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@[deprecated inv_lt_iff_one_lt_mul₀']

theorem inv_pos_lt_iff_one_lt_mul' {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) :

a⁻¹ < b ↔ 1 < a * b

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@[deprecated div_le_of_le_mul₀]

theorem div_le_of_nonneg_of_le_mul {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ c * b) :

a / b ≤ c

One direction of div_le_iff where b is allowed to be 0 (but c must be nonnegative)

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@[deprecated mul_le_of_le_div₀]

theorem mul_le_of_nonneg_of_le_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ b / c) :

a * c ≤ b

One direction of div_le_iff where c is allowed to be 0 (but b must be nonnegative)

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@[deprecated div_le_one_of_le₀]

theorem div_le_one_of_le {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (h : a ≤ b) (hb : 0 ≤ b) :

a / b ≤ 1

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@[deprecated mul_inv_le_one_of_le₀]

theorem mul_inv_le_one_of_le {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (h : a ≤ b) (hb : 0 ≤ b) :

a * b⁻¹ ≤ 1

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@[deprecated inv_mul_le_one_of_le₀]

theorem inv_mul_le_one_of_le {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (h : a ≤ b) (hb : 0 ≤ b) :

b⁻¹ * a ≤ 1

Bi-implications of inequalities using inversions #

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@[deprecated inv_anti₀]

theorem inv_le_inv_of_le {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (h : a ≤ b) :

b⁻¹ ≤ a⁻¹

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@[deprecated inv_le_inv₀]

theorem inv_le_inv {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) :

a⁻¹ ≤ b⁻¹ ↔ b ≤ a

See inv_le_inv_of_le for the implication from right-to-left with one fewer assumption.

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@[deprecated inv_le_comm₀]

theorem inv_le {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) :

a⁻¹ ≤ b ↔ b⁻¹ ≤ a

In a linear ordered field, for positive a and b we have a⁻¹ ≤ b ↔ b⁻¹ ≤ a. See also inv_le_of_inv_le for a one-sided implication with one fewer assumption.

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@[deprecated inv_le_of_inv_le₀]

theorem inv_le_of_inv_le {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (h : a⁻¹ ≤ b) :

b⁻¹ ≤ a

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@[deprecated le_inv_comm₀]

theorem le_inv {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) :

a ≤ b⁻¹ ↔ b ≤ a⁻¹

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@[deprecated inv_lt_inv₀]

theorem inv_lt_inv {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) :

a⁻¹ < b⁻¹ ↔ b < a

See inv_lt_inv_of_lt for the implication from right-to-left with one fewer assumption.

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@[deprecated inv_strictAnti₀]

theorem inv_lt_inv_of_lt {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (hb : 0 < b) (h : b < a) :

a⁻¹ < b⁻¹

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@[deprecated inv_lt_comm₀]

theorem inv_lt {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) :

a⁻¹ < b ↔ b⁻¹ < a

In a linear ordered field, for positive a and b we have a⁻¹ < b ↔ b⁻¹ < a. See also inv_lt_of_inv_lt for a one-sided implication with one fewer assumption.

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@[deprecated inv_lt_of_inv_lt₀]

theorem inv_lt_of_inv_lt {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (h : a⁻¹ < b) :

b⁻¹ < a

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@[deprecated lt_inv_comm₀]

theorem lt_inv {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) :

a < b⁻¹ ↔ b < a⁻¹

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@[deprecated inv_lt_one_of_one_lt₀]

theorem inv_lt_one {α : Type u_2} [LinearOrderedSemifield α] {a : α} (ha : 1 < a) :

a⁻¹ < 1

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@[deprecated one_lt_inv₀]

theorem one_lt_inv {α : Type u_2} [LinearOrderedSemifield α] {a : α} (h₁ : 0 < a) (h₂ : a < 1) :

1 < a⁻¹

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@[deprecated inv_le_one_of_one_le₀]

theorem inv_le_one {α : Type u_2} [LinearOrderedSemifield α] {a : α} (ha : 1 ≤ a) :

a⁻¹ ≤ 1

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@[deprecated one_le_inv₀]

theorem one_le_inv {α : Type u_2} [LinearOrderedSemifield α] {a : α} (h₁ : 0 < a) (h₂ : a ≤ 1) :

1 ≤ a⁻¹

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@[deprecated inv_lt_one₀]

theorem inv_lt_one_iff_of_pos {α : Type u_2} [LinearOrderedSemifield α] {a : α} (h₀ : 0 < a) :

a⁻¹ < 1 ↔ 1 < a

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@[deprecated inv_lt_one_iff₀]

theorem inv_lt_one_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} :

a⁻¹ < 1 ↔ a ≤ 0 ∨ 1 < a

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@[deprecated one_lt_inv_iff₀]

theorem one_lt_inv_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} :

1 < a⁻¹ ↔ 0 < a ∧ a < 1

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@[deprecated inv_le_one_iff₀]

theorem inv_le_one_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} :

a⁻¹ ≤ 1 ↔ a ≤ 0 ∨ 1 ≤ a

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@[deprecated one_le_inv_iff₀]

theorem one_le_inv_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} :

1 ≤ a⁻¹ ↔ 0 < a ∧ a ≤ 1

Relating two divisions. #

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theorem div_le_div_of_nonneg_right {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hab : a ≤ b) (hc : 0 ≤ c) :

a / c ≤ b / c

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theorem div_lt_div_of_pos_right {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (h : a < b) (hc : 0 < c) :

a / c < b / c

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theorem div_le_div_of_nonneg_left {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (ha : 0 ≤ a) (hc : 0 < c) (h : c ≤ b) :

a / b ≤ a / c

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theorem div_lt_div_of_pos_left {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (ha : 0 < a) (hc : 0 < c) (h : c < b) :

a / b < a / c

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@[deprecated div_le_div_of_nonneg_right]

theorem div_le_div_of_le_of_nonneg {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hab : a ≤ b) (hc : 0 ≤ c) :

a / c ≤ b / c

Alias of div_le_div_of_nonneg_right.

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@[deprecated div_lt_div_of_pos_right]

theorem div_lt_div_of_lt {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (h : a < b) (hc : 0 < c) :

a / c < b / c

Alias of div_lt_div_of_pos_right.

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@[deprecated div_le_div_of_nonneg_left]

theorem div_le_div_of_le_left {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (ha : 0 ≤ a) (hc : 0 < c) (h : c ≤ b) :

a / b ≤ a / c

Alias of div_le_div_of_nonneg_left.

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@[deprecated div_lt_div_of_pos_left]

theorem div_lt_div_of_lt_left {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (ha : 0 < a) (hc : 0 < c) (h : c < b) :

a / b < a / c

Alias of div_lt_div_of_pos_left.

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@[deprecated div_le_div_of_nonneg_right]

theorem div_le_div_of_le {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hc : 0 ≤ c) (hab : a ≤ b) :

a / c ≤ b / c

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theorem div_le_div_right {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hc : 0 < c) :

a / c ≤ b / c ↔ a ≤ b

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theorem div_lt_div_right {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (hc : 0 < c) :

a / c < b / c ↔ a < b

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theorem div_lt_div_left {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :

a / b < a / c ↔ c < b

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theorem div_le_div_left {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :

a / b ≤ a / c ↔ c ≤ b

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theorem div_lt_div_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} {d : α} (b0 : 0 < b) (d0 : 0 < d) :

a / b < c / d ↔ a * d < c * b

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theorem div_le_div_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} {d : α} (b0 : 0 < b) (d0 : 0 < d) :

a / b ≤ c / d ↔ a * d ≤ c * b

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theorem div_le_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} {d : α} (hc : 0 ≤ c) (hac : a ≤ c) (hd : 0 < d) (hbd : d ≤ b) :

a / b ≤ c / d

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theorem div_lt_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} {d : α} (hac : a < c) (hbd : d ≤ b) (c0 : 0 ≤ c) (d0 : 0 < d) :

a / b < c / d

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theorem div_lt_div' {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} {d : α} (hac : a ≤ c) (hbd : d < b) (c0 : 0 < c) (d0 : 0 < d) :

a / b < c / d

Relating one division and involving `1` #

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theorem div_le_self {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 ≤ a) (hb : 1 ≤ b) :

a / b ≤ a

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theorem div_lt_self {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 1 < b) :

a / b < a

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theorem le_div_self {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 ≤ a) (hb₀ : 0 < b) (hb₁ : b ≤ 1) :

a ≤ a / b

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theorem one_le_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (hb : 0 < b) :

1 ≤ a / b ↔ b ≤ a

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theorem div_le_one {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (hb : 0 < b) :

a / b ≤ 1 ↔ a ≤ b

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theorem one_lt_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (hb : 0 < b) :

1 < a / b ↔ b < a

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theorem div_lt_one {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (hb : 0 < b) :

a / b < 1 ↔ a < b

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theorem one_div_le {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) :

1 / a ≤ b ↔ 1 / b ≤ a

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theorem one_div_lt {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) :

1 / a < b ↔ 1 / b < a

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theorem le_one_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) :

a ≤ 1 / b ↔ b ≤ 1 / a

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theorem lt_one_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) :

a < 1 / b ↔ b < 1 / a

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theorem Bound.one_lt_div_of_pos_of_lt {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (b0 : 0 < b) :

b < a → 1 < a / b

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theorem Bound.div_lt_one_of_pos_of_lt {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (b0 : 0 < b) :

a < b → a / b < 1

Relating two divisions, involving `1` #

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theorem one_div_le_one_div_of_le {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (h : a ≤ b) :

1 / b ≤ 1 / a

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theorem one_div_lt_one_div_of_lt {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (h : a < b) :

1 / b < 1 / a

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theorem le_of_one_div_le_one_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (h : 1 / a ≤ 1 / b) :

b ≤ a

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theorem lt_of_one_div_lt_one_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (h : 1 / a < 1 / b) :

b < a

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theorem one_div_le_one_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) :

1 / a ≤ 1 / b ↔ b ≤ a

For the single implications with fewer assumptions, see one_div_le_one_div_of_le and le_of_one_div_le_one_div

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theorem one_div_lt_one_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) (hb : 0 < b) :

1 / a < 1 / b ↔ b < a

For the single implications with fewer assumptions, see one_div_lt_one_div_of_lt and lt_of_one_div_lt_one_div

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theorem one_lt_one_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} (h1 : 0 < a) (h2 : a < 1) :

1 < 1 / a

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theorem one_le_one_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} (h1 : 0 < a) (h2 : a ≤ 1) :

1 ≤ 1 / a

Results about halving. #

The equalities also hold in semifields of characteristic 0.

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theorem half_pos {α : Type u_2} [LinearOrderedSemifield α] {a : α} (h : 0 < a) :

0 < a / 2

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theorem one_half_pos {α : Type u_2} [LinearOrderedSemifield α] :

0 < 1 / 2

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@[simp]

theorem half_le_self_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} :

a / 2 ≤ a ↔ 0 ≤ a

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@[simp]

theorem half_lt_self_iff {α : Type u_2} [LinearOrderedSemifield α] {a : α} :

a / 2 < a ↔ 0 < a

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theorem half_le_self {α : Type u_2} [LinearOrderedSemifield α] {a : α} :

0 ≤ a → a / 2 ≤ a

Alias of the reverse direction of half_le_self_iff.

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theorem half_lt_self {α : Type u_2} [LinearOrderedSemifield α] {a : α} :

0 < a → a / 2 < a

Alias of the reverse direction of half_lt_self_iff.

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theorem div_two_lt_of_pos {α : Type u_2} [LinearOrderedSemifield α] {a : α} :

0 < a → a / 2 < a

Alias of the reverse direction of half_lt_self_iff.

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theorem one_half_lt_one {α : Type u_2} [LinearOrderedSemifield α] :

1 / 2 < 1

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theorem two_inv_lt_one {α : Type u_2} [LinearOrderedSemifield α] :

2⁻¹ < 1

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theorem left_lt_add_div_two {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} :

a < (a + b) / 2 ↔ a < b

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theorem add_div_two_lt_right {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} :

(a + b) / 2 < b ↔ a < b

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theorem add_thirds {α : Type u_2} [LinearOrderedSemifield α] (a : α) :

a / 3 + a / 3 + a / 3 = a

Miscellaneous lemmas #

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@[simp]

theorem div_pos_iff_of_pos_left {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 < a) :

0 < a / b ↔ 0 < b

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@[simp]

theorem div_pos_iff_of_pos_right {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} (hb : 0 < b) :

0 < a / b ↔ 0 < a

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theorem mul_le_mul_of_mul_div_le {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} {d : α} (h : a * (b / c) ≤ d) (hc : 0 < c) :

b * a ≤ d * c

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theorem div_mul_le_div_mul_of_div_le_div {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {c : α} {d : α} {e : α} (h : a / b ≤ c / d) (he : 0 ≤ e) :

a / (b * e) ≤ c / (d * e)

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theorem exists_pos_mul_lt {α : Type u_2} [LinearOrderedSemifield α] {a : α} (h : 0 < a) (b : α) :

∃ (c : α), 0 < c ∧ b * c < a

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theorem exists_pos_lt_mul {α : Type u_2} [LinearOrderedSemifield α] {a : α} (h : 0 < a) (b : α) :

∃ (c : α), 0 < c ∧ b < c * a

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theorem monotone_div_right_of_nonneg {α : Type u_2} [LinearOrderedSemifield α] {a : α} (ha : 0 ≤ a) :

Monotone fun (x : α) => x / a

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theorem strictMono_div_right_of_pos {α : Type u_2} [LinearOrderedSemifield α] {a : α} (ha : 0 < a) :

StrictMono fun (x : α) => x / a

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theorem Monotone.div_const {α : Type u_2} [LinearOrderedSemifield α] {β : Type u_4} [Preorder β] {f : β → α} (hf : Monotone f) {c : α} (hc : 0 ≤ c) :

Monotone fun (x : β) => f x / c

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theorem StrictMono.div_const {α : Type u_2} [LinearOrderedSemifield α] {β : Type u_4} [Preorder β] {f : β → α} (hf : StrictMono f) {c : α} (hc : 0 < c) :

StrictMono fun (x : β) => f x / c

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@[instance 100]

instance LinearOrderedSemiField.toDenselyOrdered {α : Type u_2} [LinearOrderedSemifield α] :

DenselyOrdered α

Equations

⋯ = ⋯

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theorem min_div_div_right {α : Type u_2} [LinearOrderedSemifield α] {c : α} (hc : 0 ≤ c) (a : α) (b : α) :

a / c ⊓ b / c = (a ⊓ b) / c

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theorem max_div_div_right {α : Type u_2} [LinearOrderedSemifield α] {c : α} (hc : 0 ≤ c) (a : α) (b : α) :

a / c ⊔ b / c = (a ⊔ b) / c

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theorem one_div_strictAntiOn {α : Type u_2} [LinearOrderedSemifield α] :

StrictAntiOn (fun (x : α) => 1 / x) (Set.Ioi 0)

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theorem one_div_pow_le_one_div_pow_of_le {α : Type u_2} [LinearOrderedSemifield α] {a : α} (a1 : 1 ≤ a) {m : ℕ} {n : ℕ} (mn : m ≤ n) :

1 / a ^ n ≤ 1 / a ^ m

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theorem one_div_pow_lt_one_div_pow_of_lt {α : Type u_2} [LinearOrderedSemifield α] {a : α} (a1 : 1 < a) {m : ℕ} {n : ℕ} (mn : m < n) :

1 / a ^ n < 1 / a ^ m

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theorem one_div_pow_anti {α : Type u_2} [LinearOrderedSemifield α] {a : α} (a1 : 1 ≤ a) :

Antitone fun (n : ℕ) => 1 / a ^ n

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theorem one_div_pow_strictAnti {α : Type u_2} [LinearOrderedSemifield α] {a : α} (a1 : 1 < a) :

StrictAnti fun (n : ℕ) => 1 / a ^ n

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theorem inv_strictAntiOn {α : Type u_2} [LinearOrderedSemifield α] :

StrictAntiOn (fun (x : α) => x⁻¹) (Set.Ioi 0)

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theorem inv_pow_le_inv_pow_of_le {α : Type u_2} [LinearOrderedSemifield α] {a : α} (a1 : 1 ≤ a) {m : ℕ} {n : ℕ} (mn : m ≤ n) :

(a ^ n)⁻¹ ≤ (a ^ m)⁻¹

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theorem inv_pow_lt_inv_pow_of_lt {α : Type u_2} [LinearOrderedSemifield α] {a : α} (a1 : 1 < a) {m : ℕ} {n : ℕ} (mn : m < n) :

(a ^ n)⁻¹ < (a ^ m)⁻¹

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theorem inv_pow_anti {α : Type u_2} [LinearOrderedSemifield α] {a : α} (a1 : 1 ≤ a) :

Antitone fun (n : ℕ) => (a ^ n)⁻¹

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theorem inv_pow_strictAnti {α : Type u_2} [LinearOrderedSemifield α] {a : α} (a1 : 1 < a) :

StrictAnti fun (n : ℕ) => (a ^ n)⁻¹

Results about `IsGLB` #

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theorem IsGLB.mul_left {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {s : Set α} (ha : 0 ≤ a) (hs : IsGLB s b) :

IsGLB ((fun (b : α) => a * b) '' s) (a * b)

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theorem IsGLB.mul_right {α : Type u_2} [LinearOrderedSemifield α] {a : α} {b : α} {s : Set α} (ha : 0 ≤ a) (hs : IsGLB s b) :

IsGLB ((fun (b : α) => b * a) '' s) (b * a)

Lemmas about pos, nonneg, nonpos, neg #

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theorem div_pos_iff {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} :

0 < a / b ↔ 0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0

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theorem div_neg_iff {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} :

a / b < 0 ↔ 0 < a ∧ b < 0 ∨ a < 0 ∧ 0 < b

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theorem div_nonneg_iff {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} :

0 ≤ a / b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0

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theorem div_nonpos_iff {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} :

a / b ≤ 0 ↔ 0 ≤ a ∧ b ≤ 0 ∨ a ≤ 0 ∧ 0 ≤ b

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theorem div_nonneg_of_nonpos {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a ≤ 0) (hb : b ≤ 0) :

0 ≤ a / b

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theorem div_pos_of_neg_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) :

0 < a / b

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theorem div_neg_of_neg_of_pos {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : 0 < b) :

a / b < 0

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theorem div_neg_of_pos_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : 0 < a) (hb : b < 0) :

a / b < 0

Relating one division with another term #

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theorem div_le_iff_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (hc : c < 0) :

b / c ≤ a ↔ a * c ≤ b

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theorem div_le_iff_of_neg' {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (hc : c < 0) :

b / c ≤ a ↔ c * a ≤ b

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theorem le_div_iff_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (hc : c < 0) :

a ≤ b / c ↔ b ≤ a * c

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theorem le_div_iff_of_neg' {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (hc : c < 0) :

a ≤ b / c ↔ b ≤ c * a

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theorem div_lt_iff_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (hc : c < 0) :

b / c < a ↔ a * c < b

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theorem div_lt_iff_of_neg' {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (hc : c < 0) :

b / c < a ↔ c * a < b

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theorem lt_div_iff_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (hc : c < 0) :

a < b / c ↔ b < a * c

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theorem lt_div_iff_of_neg' {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (hc : c < 0) :

a < b / c ↔ b < c * a

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theorem div_le_one_of_ge {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (h : b ≤ a) (hb : b ≤ 0) :

a / b ≤ 1

Bi-implications of inequalities using inversions #

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theorem inv_le_inv_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) :

a⁻¹ ≤ b⁻¹ ↔ b ≤ a

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theorem inv_le_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) :

a⁻¹ ≤ b ↔ b⁻¹ ≤ a

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theorem le_inv_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) :

a ≤ b⁻¹ ↔ b ≤ a⁻¹

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theorem inv_lt_inv_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) :

a⁻¹ < b⁻¹ ↔ b < a

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theorem inv_lt_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) :

a⁻¹ < b ↔ b⁻¹ < a

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theorem lt_inv_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) :

a < b⁻¹ ↔ b < a⁻¹

Monotonicity results involving inversion #

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theorem sub_inv_antitoneOn_Ioi {α : Type u_2} [LinearOrderedField α] {c : α} :

AntitoneOn (fun (x : α) => (x - c)⁻¹) (Set.Ioi c)

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theorem sub_inv_antitoneOn_Iio {α : Type u_2} [LinearOrderedField α] {c : α} :

AntitoneOn (fun (x : α) => (x - c)⁻¹) (Set.Iio c)

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theorem sub_inv_antitoneOn_Icc_right {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (ha : c < a) :

AntitoneOn (fun (x : α) => (x - c)⁻¹) (Set.Icc a b)

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theorem sub_inv_antitoneOn_Icc_left {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (ha : b < c) :

AntitoneOn (fun (x : α) => (x - c)⁻¹) (Set.Icc a b)

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theorem inv_antitoneOn_Ioi {α : Type u_2} [LinearOrderedField α] :

AntitoneOn (fun (x : α) => x⁻¹) (Set.Ioi 0)

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theorem inv_antitoneOn_Iio {α : Type u_2} [LinearOrderedField α] :

AntitoneOn (fun (x : α) => x⁻¹) (Set.Iio 0)

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theorem inv_antitoneOn_Icc_right {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : 0 < a) :

AntitoneOn (fun (x : α) => x⁻¹) (Set.Icc a b)

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theorem inv_antitoneOn_Icc_left {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (hb : b < 0) :

AntitoneOn (fun (x : α) => x⁻¹) (Set.Icc a b)

Relating two divisions #

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theorem div_le_div_of_nonpos_of_le {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (hc : c ≤ 0) (h : b ≤ a) :

a / c ≤ b / c

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theorem div_lt_div_of_neg_of_lt {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (hc : c < 0) (h : b < a) :

a / c < b / c

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theorem div_le_div_right_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (hc : c < 0) :

a / c ≤ b / c ↔ b ≤ a

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theorem div_lt_div_right_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (hc : c < 0) :

a / c < b / c ↔ b < a

Relating one division and involving `1` #

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theorem one_le_div_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (hb : b < 0) :

1 ≤ a / b ↔ a ≤ b

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theorem div_le_one_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (hb : b < 0) :

a / b ≤ 1 ↔ b ≤ a

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theorem one_lt_div_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (hb : b < 0) :

1 < a / b ↔ a < b

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theorem div_lt_one_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (hb : b < 0) :

a / b < 1 ↔ b < a

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theorem one_div_le_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) :

1 / a ≤ b ↔ 1 / b ≤ a

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theorem one_div_lt_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) :

1 / a < b ↔ 1 / b < a

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theorem le_one_div_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) :

a ≤ 1 / b ↔ b ≤ 1 / a

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theorem lt_one_div_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) :

a < 1 / b ↔ b < 1 / a

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theorem one_lt_div_iff {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} :

1 < a / b ↔ 0 < b ∧ b < a ∨ b < 0 ∧ a < b

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theorem one_le_div_iff {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} :

1 ≤ a / b ↔ 0 < b ∧ b ≤ a ∨ b < 0 ∧ a ≤ b

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theorem div_lt_one_iff {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} :

a / b < 1 ↔ 0 < b ∧ a < b ∨ b = 0 ∨ b < 0 ∧ b < a

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theorem div_le_one_iff {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} :

a / b ≤ 1 ↔ 0 < b ∧ a ≤ b ∨ b = 0 ∨ b < 0 ∧ b ≤ a

Relating two divisions, involving `1` #

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theorem one_div_le_one_div_of_neg_of_le {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (hb : b < 0) (h : a ≤ b) :

1 / b ≤ 1 / a

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theorem one_div_lt_one_div_of_neg_of_lt {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (hb : b < 0) (h : a < b) :

1 / b < 1 / a

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theorem le_of_neg_of_one_div_le_one_div {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (hb : b < 0) (h : 1 / a ≤ 1 / b) :

b ≤ a

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theorem lt_of_neg_of_one_div_lt_one_div {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (hb : b < 0) (h : 1 / a < 1 / b) :

b < a

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theorem one_div_le_one_div_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) :

1 / a ≤ 1 / b ↔ b ≤ a

For the single implications with fewer assumptions, see one_div_lt_one_div_of_neg_of_lt and lt_of_one_div_lt_one_div

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theorem one_div_lt_one_div_of_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (ha : a < 0) (hb : b < 0) :

1 / a < 1 / b ↔ b < a

For the single implications with fewer assumptions, see one_div_lt_one_div_of_lt and lt_of_one_div_lt_one_div

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theorem one_div_lt_neg_one {α : Type u_2} [LinearOrderedField α] {a : α} (h1 : a < 0) (h2 : -1 < a) :

1 / a < -1

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theorem one_div_le_neg_one {α : Type u_2} [LinearOrderedField α] {a : α} (h1 : a < 0) (h2 : -1 ≤ a) :

1 / a ≤ -1

Results about halving #

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theorem sub_self_div_two {α : Type u_2} [LinearOrderedField α] (a : α) :

a - a / 2 = a / 2

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theorem div_two_sub_self {α : Type u_2} [LinearOrderedField α] (a : α) :

a / 2 - a = -(a / 2)

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theorem add_sub_div_two_lt {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (h : a < b) :

a + (b - a) / 2 < b

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theorem sub_one_div_inv_le_two {α : Type u_2} [LinearOrderedField α] {a : α} (a2 : 2 ≤ a) :

(1 - 1 / a)⁻¹ ≤ 2

An inequality involving 2.

Results about `IsLUB` #

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theorem IsLUB.mul_left {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {s : Set α} (ha : 0 ≤ a) (hs : IsLUB s b) :

IsLUB ((fun (b : α) => a * b) '' s) (a * b)

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theorem IsLUB.mul_right {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {s : Set α} (ha : 0 ≤ a) (hs : IsLUB s b) :

IsLUB ((fun (b : α) => b * a) '' s) (b * a)

Miscellaneous lemmas #

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theorem mul_sub_mul_div_mul_neg_iff {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} {d : α} (hc : c ≠ 0) (hd : d ≠ 0) :

(a * d - b * c) / (c * d) < 0 ↔ a / c < b / d

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theorem mul_sub_mul_div_mul_nonpos_iff {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} {d : α} (hc : c ≠ 0) (hd : d ≠ 0) :

(a * d - b * c) / (c * d) ≤ 0 ↔ a / c ≤ b / d

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theorem mul_sub_mul_div_mul_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} {d : α} (hc : c ≠ 0) (hd : d ≠ 0) :

a / c < b / d → (a * d - b * c) / (c * d) < 0

Alias of the reverse direction of mul_sub_mul_div_mul_neg_iff.

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theorem div_lt_div_of_mul_sub_mul_div_neg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} {d : α} (hc : c ≠ 0) (hd : d ≠ 0) :

(a * d - b * c) / (c * d) < 0 → a / c < b / d

Alias of the forward direction of mul_sub_mul_div_mul_neg_iff.

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theorem div_le_div_of_mul_sub_mul_div_nonpos {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} {d : α} (hc : c ≠ 0) (hd : d ≠ 0) :

(a * d - b * c) / (c * d) ≤ 0 → a / c ≤ b / d

Alias of the forward direction of mul_sub_mul_div_mul_nonpos_iff.

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theorem mul_sub_mul_div_mul_nonpos {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} {d : α} (hc : c ≠ 0) (hd : d ≠ 0) :

a / c ≤ b / d → (a * d - b * c) / (c * d) ≤ 0

Alias of the reverse direction of mul_sub_mul_div_mul_nonpos_iff.

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theorem exists_add_lt_and_pos_of_lt {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (h : b < a) :

∃ (c : α), b + c < a ∧ 0 < c

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theorem le_of_forall_sub_le {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} (h : ∀ ε > 0, b - ε ≤ a) :

b ≤ a

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theorem le_mul_of_forall_lt₀ {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (h : ∀ a' > a, ∀ b' > b, c ≤ a' * b') :

c ≤ a * b

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theorem mul_le_of_forall_lt_of_nonneg {α : Type u_2} [LinearOrderedField α] {a : α} {b : α} {c : α} (ha : 0 ≤ a) (hc : 0 ≤ c) (h : ∀ a' ≥ 0, a' < a → ∀ b' ≥ 0, b' < b → a' * b' ≤ c) :

a * b ≤ c

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