statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
pos_iff_pos_of_mul_pos [pos_mul_reflect_lt α] [mul_pos_reflect_lt α] (hab : 0 < a * b) :
0 < a ↔ 0 < b | ⟨pos_of_mul_pos_right hab ∘ le_of_lt, pos_of_mul_pos_left hab ∘ le_of_lt⟩ | lemma | pos_iff_pos_of_mul_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_pos_reflect_lt",
"pos_mul_reflect_lt",
"pos_of_mul_pos_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_of_le_of_le [pos_mul_mono α] [mul_pos_mono α]
(h₁ : a ≤ b) (h₂ : c ≤ d) (a0 : 0 ≤ a) (d0 : 0 ≤ d) : a * c ≤ b * d | (mul_le_mul_of_nonneg_left h₂ a0).trans $ mul_le_mul_of_nonneg_right h₁ d0 | lemma | mul_le_mul_of_le_of_le | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"mul_pos_mono",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul [pos_mul_mono α] [mul_pos_mono α]
(h₁ : a ≤ b) (h₂ : c ≤ d) (c0 : 0 ≤ c) (b0 : 0 ≤ b) : a * c ≤ b * d | (mul_le_mul_of_nonneg_right h₁ c0).trans $ mul_le_mul_of_nonneg_left h₂ b0 | lemma | mul_le_mul | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"mul_pos_mono",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_self_le_mul_self [pos_mul_mono α] [mul_pos_mono α] (ha : 0 ≤ a) (hab : a ≤ b) :
a * a ≤ b * b | mul_le_mul hab hab ha $ ha.trans hab | lemma | mul_self_le_mul_self | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul",
"mul_pos_mono",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_of_mul_le_of_nonneg_left [pos_mul_mono α] (h : a * b ≤ c) (hle : d ≤ b) (a0 : 0 ≤ a) :
a * d ≤ c | (mul_le_mul_of_nonneg_left hle a0).trans h | lemma | mul_le_of_mul_le_of_nonneg_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_left",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_of_le_mul_of_nonneg_left [pos_mul_mono α] (h : a ≤ b * c) (hle : c ≤ d) (b0 : 0 ≤ b) :
a ≤ b * d | h.trans (mul_le_mul_of_nonneg_left hle b0) | lemma | le_mul_of_le_mul_of_nonneg_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_left",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_of_mul_le_of_nonneg_right [mul_pos_mono α] (h : a * b ≤ c) (hle : d ≤ a) (b0 : 0 ≤ b) :
d * b ≤ c | (mul_le_mul_of_nonneg_right hle b0).trans h | lemma | mul_le_of_mul_le_of_nonneg_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_right",
"mul_pos_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_of_le_mul_of_nonneg_right [mul_pos_mono α] (h : a ≤ b * c) (hle : b ≤ d) (c0 : 0 ≤ c) :
a ≤ d * c | h.trans (mul_le_mul_of_nonneg_right hle c0) | lemma | le_mul_of_le_mul_of_nonneg_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_right",
"mul_pos_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_mul_mono_iff_covariant_pos : pos_mul_mono α ↔ covariant_class α>0 α (λ x y, x * y) (≤) | ⟨@pos_mul_mono.to_covariant_class_pos_mul_le _ _ _ _, λ h, ⟨λ a b c h, begin
obtain ha | ha := a.prop.eq_or_gt,
{ simp only [ha, zero_mul] },
{ exactI @covariant_class.elim α>0 α (λ x y, x * y) (≤) _ ⟨_, ha⟩ _ _ h }
end⟩⟩ | lemma | pos_mul_mono_iff_covariant_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"covariant_class",
"pos_mul_mono",
"pos_mul_mono.to_covariant_class_pos_mul_le",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_pos_mono_iff_covariant_pos : mul_pos_mono α ↔ covariant_class α>0 α (λ x y, y * x) (≤) | ⟨@mul_pos_mono.to_covariant_class_pos_mul_le _ _ _ _, λ h, ⟨λ a b c h, begin
obtain ha | ha := a.prop.eq_or_gt,
{ simp only [ha, mul_zero] },
{ exactI @covariant_class.elim α>0 α (λ x y, y * x) (≤) _ ⟨_, ha⟩ _ _ h }
end⟩⟩ | lemma | mul_pos_mono_iff_covariant_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"covariant_class",
"mul_pos_mono",
"mul_pos_mono.to_covariant_class_pos_mul_le",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_mul_reflect_lt_iff_contravariant_pos :
pos_mul_reflect_lt α ↔ contravariant_class α>0 α (λ x y, x * y) (<) | ⟨@pos_mul_reflect_lt.to_contravariant_class_pos_mul_lt _ _ _ _, λ h, ⟨λ a b c h, begin
obtain ha | ha := a.prop.eq_or_gt,
{ simpa [ha] using h },
{ exactI (@contravariant_class.elim α>0 α (λ x y, x * y) (<) _ ⟨_, ha⟩ _ _ h) }
end⟩⟩ | lemma | pos_mul_reflect_lt_iff_contravariant_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"contravariant_class",
"pos_mul_reflect_lt",
"pos_mul_reflect_lt.to_contravariant_class_pos_mul_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_pos_reflect_lt_iff_contravariant_pos :
mul_pos_reflect_lt α ↔ contravariant_class α>0 α (λ x y, y * x) (<) | ⟨@mul_pos_reflect_lt.to_contravariant_class_pos_mul_lt _ _ _ _, λ h, ⟨λ a b c h, begin
obtain ha | ha := a.prop.eq_or_gt,
{ simpa [ha] using h },
{ exactI (@contravariant_class.elim α>0 α (λ x y, y * x) (<) _ ⟨_, ha⟩ _ _ h) }
end⟩⟩ | lemma | mul_pos_reflect_lt_iff_contravariant_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"contravariant_class",
"mul_pos_reflect_lt",
"mul_pos_reflect_lt.to_contravariant_class_pos_mul_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_mul_strict_mono.to_pos_mul_mono [pos_mul_strict_mono α] : pos_mul_mono α | pos_mul_mono_iff_covariant_pos.2 $ ⟨λ a, strict_mono.monotone $ @covariant_class.elim _ _ _ _ _ _⟩ | instance | pos_mul_strict_mono.to_pos_mul_mono | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"pos_mul_mono",
"pos_mul_strict_mono",
"strict_mono.monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_pos_strict_mono.to_mul_pos_mono [mul_pos_strict_mono α] : mul_pos_mono α | mul_pos_mono_iff_covariant_pos.2 $ ⟨λ a, strict_mono.monotone $ @covariant_class.elim _ _ _ _ _ _⟩ | instance | mul_pos_strict_mono.to_mul_pos_mono | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_pos_mono",
"mul_pos_strict_mono",
"strict_mono.monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_mul_mono_rev.to_pos_mul_reflect_lt [pos_mul_mono_rev α] : pos_mul_reflect_lt α | pos_mul_reflect_lt_iff_contravariant_pos.2
⟨λ a b c h, (le_of_mul_le_mul_of_pos_left h.le a.2).lt_of_ne $ by { rintro rfl, simpa using h }⟩ | instance | pos_mul_mono_rev.to_pos_mul_reflect_lt | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"pos_mul_mono_rev",
"pos_mul_reflect_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_pos_mono_rev.to_mul_pos_reflect_lt [mul_pos_mono_rev α] : mul_pos_reflect_lt α | mul_pos_reflect_lt_iff_contravariant_pos.2
⟨λ a b c h, (le_of_mul_le_mul_of_pos_right h.le a.2).lt_of_ne $ by { rintro rfl, simpa using h }⟩ | instance | mul_pos_mono_rev.to_mul_pos_reflect_lt | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_pos_mono_rev",
"mul_pos_reflect_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_cancel_iff_of_pos [pos_mul_mono_rev α] (a0 : 0 < a) : a * b = a * c ↔ b = c | ⟨λ h, (le_of_mul_le_mul_of_pos_left h.le a0).antisymm $ le_of_mul_le_mul_of_pos_left h.ge a0,
congr_arg _⟩ | lemma | mul_left_cancel_iff_of_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"pos_mul_mono_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_cancel_iff_of_pos [mul_pos_mono_rev α] (b0 : 0 < b) : a * b = c * b ↔ a = c | ⟨λ h, (le_of_mul_le_mul_of_pos_right h.le b0).antisymm $ le_of_mul_le_mul_of_pos_right h.ge b0,
congr_arg _⟩ | lemma | mul_right_cancel_iff_of_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_pos_mono_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_mul_iff_eq_and_eq_of_pos [pos_mul_strict_mono α] [mul_pos_strict_mono α]
[pos_mul_mono_rev α] [mul_pos_mono_rev α]
(hac : a ≤ b) (hbd : c ≤ d) (a0 : 0 < a) (d0 : 0 < d) :
a * c = b * d ↔ a = b ∧ c = d | begin
refine ⟨λ h, _, λ h, congr_arg2 (*) h.1 h.2⟩,
rcases hac.eq_or_lt with rfl | hac,
{ exact ⟨rfl, (mul_left_cancel_iff_of_pos a0).mp h⟩ },
rcases eq_or_lt_of_le hbd with rfl | hbd,
{ exact ⟨(mul_right_cancel_iff_of_pos d0).mp h, rfl⟩ },
exact ((mul_lt_mul_of_pos_of_pos hac hbd a0 d0).ne h).elim,
end | lemma | mul_eq_mul_iff_eq_and_eq_of_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"congr_arg2",
"eq_or_lt_of_le",
"mul_left_cancel_iff_of_pos",
"mul_lt_mul_of_pos_of_pos",
"mul_pos_mono_rev",
"mul_pos_strict_mono",
"mul_right_cancel_iff_of_pos",
"pos_mul_mono_rev",
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_mul_iff_eq_and_eq_of_pos' [pos_mul_strict_mono α] [mul_pos_strict_mono α]
[pos_mul_mono_rev α] [mul_pos_mono_rev α]
(hac : a ≤ b) (hbd : c ≤ d) (b0 : 0 < b) (c0 : 0 < c) :
a * c = b * d ↔ a = b ∧ c = d | begin
refine ⟨λ h, _, λ h, congr_arg2 (*) h.1 h.2⟩,
rcases hac.eq_or_lt with rfl | hac,
{ exact ⟨rfl, (mul_left_cancel_iff_of_pos b0).mp h⟩ },
rcases eq_or_lt_of_le hbd with rfl | hbd,
{ exact ⟨(mul_right_cancel_iff_of_pos c0).mp h, rfl⟩ },
exact ((mul_lt_mul_of_lt_of_lt' hac hbd b0 c0).ne h).elim,
end | lemma | mul_eq_mul_iff_eq_and_eq_of_pos' | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"congr_arg2",
"eq_or_lt_of_le",
"mul_left_cancel_iff_of_pos",
"mul_lt_mul_of_lt_of_lt'",
"mul_pos_mono_rev",
"mul_pos_strict_mono",
"mul_right_cancel_iff_of_pos",
"pos_mul_mono_rev",
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_and_pos_or_neg_and_neg_of_mul_pos [pos_mul_mono α] [mul_pos_mono α]
(hab : 0 < a * b) :
(0 < a ∧ 0 < b) ∨ (a < 0 ∧ b < 0) | begin
rcases lt_trichotomy 0 a with ha | rfl | ha,
{ refine or.inl ⟨ha, lt_imp_lt_of_le_imp_le (λ hb, _) hab⟩,
exact mul_nonpos_of_nonneg_of_nonpos ha.le hb },
{ rw [zero_mul] at hab, exact hab.false.elim },
{ refine or.inr ⟨ha, lt_imp_lt_of_le_imp_le (λ hb, _) hab⟩,
exact mul_nonpos_of_nonpos_of_nonneg... | lemma | pos_and_pos_or_neg_and_neg_of_mul_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"lt_imp_lt_of_le_imp_le",
"mul_nonpos_of_nonneg_of_nonpos",
"mul_nonpos_of_nonpos_of_nonneg",
"mul_pos_mono",
"pos_mul_mono",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_of_mul_pos_right [pos_mul_mono α] [mul_pos_mono α]
(h : 0 < a * b) (ha : a ≤ 0) :
b < 0 | ((pos_and_pos_or_neg_and_neg_of_mul_pos h).resolve_left $ λ h, h.1.not_le ha).2 | lemma | neg_of_mul_pos_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_pos_mono",
"pos_and_pos_or_neg_and_neg_of_mul_pos",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_of_mul_pos_left [pos_mul_mono α] [mul_pos_mono α]
(h : 0 < a * b) (ha : b ≤ 0) :
a < 0 | ((pos_and_pos_or_neg_and_neg_of_mul_pos h).resolve_left $ λ h, h.2.not_le ha).1 | lemma | neg_of_mul_pos_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_pos_mono",
"pos_and_pos_or_neg_and_neg_of_mul_pos",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_iff_neg_of_mul_pos [pos_mul_mono α] [mul_pos_mono α]
(hab : 0 < a * b) :
a < 0 ↔ b < 0 | ⟨neg_of_mul_pos_right hab ∘ le_of_lt, neg_of_mul_pos_left hab ∘ le_of_lt⟩ | lemma | neg_iff_neg_of_mul_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_pos_mono",
"neg_of_mul_pos_left",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left.neg_of_mul_neg_left [pos_mul_mono α]
(h : a * b < 0) (h1 : 0 ≤ a) :
b < 0 | lt_of_not_ge (assume h2 : b ≥ 0, (left.mul_nonneg h1 h2).not_lt h) | lemma | left.neg_of_mul_neg_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"left.mul_nonneg",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right.neg_of_mul_neg_left [mul_pos_mono α]
(h : a * b < 0) (h1 : 0 ≤ a) :
b < 0 | lt_of_not_ge (assume h2 : b ≥ 0, (right.mul_nonneg h1 h2).not_lt h) | lemma | right.neg_of_mul_neg_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_pos_mono",
"right.mul_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left.neg_of_mul_neg_right [pos_mul_mono α]
(h : a * b < 0) (h1 : 0 ≤ b) : a < 0 | lt_of_not_ge (assume h2 : a ≥ 0, (left.mul_nonneg h2 h1).not_lt h) | lemma | left.neg_of_mul_neg_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"left.mul_nonneg",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right.neg_of_mul_neg_right [mul_pos_mono α]
(h : a * b < 0) (h1 : 0 ≤ b) : a < 0 | lt_of_not_ge (assume h2 : a ≥ 0, (right.mul_nonneg h2 h1).not_lt h) | lemma | right.neg_of_mul_neg_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_pos_mono",
"right.mul_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_iff_one_le_right
[pos_mul_mono α] [pos_mul_mono_rev α]
(a0 : 0 < a) :
a ≤ a * b ↔ 1 ≤ b | iff.trans (by rw [mul_one]) (mul_le_mul_left a0) | lemma | le_mul_iff_one_le_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_left",
"mul_one",
"pos_mul_mono",
"pos_mul_mono_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_iff_one_lt_right
[pos_mul_strict_mono α] [pos_mul_reflect_lt α]
(a0 : 0 < a) :
a < a * b ↔ 1 < b | iff.trans (by rw [mul_one]) (mul_lt_mul_left a0) | lemma | lt_mul_iff_one_lt_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_mul_left",
"mul_one",
"pos_mul_reflect_lt",
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_iff_le_one_right
[pos_mul_mono α] [pos_mul_mono_rev α]
(a0 : 0 < a) :
a * b ≤ a ↔ b ≤ 1 | iff.trans (by rw [mul_one]) (mul_le_mul_left a0) | lemma | mul_le_iff_le_one_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_left",
"mul_one",
"pos_mul_mono",
"pos_mul_mono_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_iff_lt_one_right
[pos_mul_strict_mono α] [pos_mul_reflect_lt α]
(a0 : 0 < a) :
a * b < a ↔ b < 1 | iff.trans (by rw [mul_one]) (mul_lt_mul_left a0) | lemma | mul_lt_iff_lt_one_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_mul_left",
"mul_one",
"pos_mul_reflect_lt",
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_iff_one_le_left
[mul_pos_mono α] [mul_pos_mono_rev α]
(a0 : 0 < a) :
a ≤ b * a ↔ 1 ≤ b | iff.trans (by rw [one_mul]) (mul_le_mul_right a0) | lemma | le_mul_iff_one_le_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_right",
"mul_pos_mono",
"mul_pos_mono_rev",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_iff_one_lt_left
[mul_pos_strict_mono α] [mul_pos_reflect_lt α]
(a0 : 0 < a) :
a < b * a ↔ 1 < b | iff.trans (by rw [one_mul]) (mul_lt_mul_right a0) | lemma | lt_mul_iff_one_lt_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_mul_right",
"mul_pos_reflect_lt",
"mul_pos_strict_mono",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_iff_le_one_left
[mul_pos_mono α] [mul_pos_mono_rev α]
(b0 : 0 < b) :
a * b ≤ b ↔ a ≤ 1 | iff.trans (by rw [one_mul]) (mul_le_mul_right b0) | lemma | mul_le_iff_le_one_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_right",
"mul_pos_mono",
"mul_pos_mono_rev",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_iff_lt_one_left
[mul_pos_strict_mono α] [mul_pos_reflect_lt α]
(b0 : 0 < b) :
a * b < b ↔ a < 1 | iff.trans (by rw [one_mul]) (mul_lt_mul_right b0) | lemma | mul_lt_iff_lt_one_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_mul_right",
"mul_pos_reflect_lt",
"mul_pos_strict_mono",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_of_le_one_left [mul_pos_mono α] (hb : 0 ≤ b) (h : a ≤ 1) : a * b ≤ b | by simpa only [one_mul] using mul_le_mul_of_nonneg_right h hb | lemma | mul_le_of_le_one_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_right",
"mul_pos_mono",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_of_one_le_left [mul_pos_mono α] (hb : 0 ≤ b) (h : 1 ≤ a) : b ≤ a * b | by simpa only [one_mul] using mul_le_mul_of_nonneg_right h hb | lemma | le_mul_of_one_le_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_right",
"mul_pos_mono",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_of_le_one_right [pos_mul_mono α] (ha : 0 ≤ a) (h : b ≤ 1) : a * b ≤ a | by simpa only [mul_one] using mul_le_mul_of_nonneg_left h ha | lemma | mul_le_of_le_one_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_left",
"mul_one",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_of_one_le_right [pos_mul_mono α] (ha : 0 ≤ a) (h : 1 ≤ b) : a ≤ a * b | by simpa only [mul_one] using mul_le_mul_of_nonneg_left h ha | lemma | le_mul_of_one_le_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_left",
"mul_one",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_lt_one_left [mul_pos_strict_mono α] (hb : 0 < b) (h : a < 1) : a * b < b | by simpa only [one_mul] using mul_lt_mul_of_pos_right h hb | lemma | mul_lt_of_lt_one_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_mul_of_pos_right",
"mul_pos_strict_mono",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_one_lt_left [mul_pos_strict_mono α] (hb : 0 < b) (h : 1 < a) : b < a * b | by simpa only [one_mul] using mul_lt_mul_of_pos_right h hb | lemma | lt_mul_of_one_lt_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_mul_of_pos_right",
"mul_pos_strict_mono",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_lt_one_right [pos_mul_strict_mono α] (ha : 0 < a) (h : b < 1) : a * b < a | by simpa only [mul_one] using mul_lt_mul_of_pos_left h ha | lemma | mul_lt_of_lt_one_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_mul_of_pos_left",
"mul_one",
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_one_lt_right [pos_mul_strict_mono α] (ha : 0 < a) (h : 1 < b) : a < a * b | by simpa only [mul_one] using mul_lt_mul_of_pos_left h ha | lemma | lt_mul_of_one_lt_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_mul_of_pos_left",
"mul_one",
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_of_le_of_le_one_of_nonneg [pos_mul_mono α] (h : b ≤ c) (ha : a ≤ 1) (hb : 0 ≤ b) :
b * a ≤ c | (mul_le_of_le_one_right hb ha).trans h | lemma | mul_le_of_le_of_le_one_of_nonneg | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_of_le_one_right",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_le_of_lt_one_of_pos [pos_mul_strict_mono α] (bc : b ≤ c) (ha : a < 1) (b0 : 0 < b) :
b * a < c | (mul_lt_of_lt_one_right b0 ha).trans_le bc | lemma | mul_lt_of_le_of_lt_one_of_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_of_lt_one_right",
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_lt_of_le_one_of_nonneg [pos_mul_mono α] (h : b < c) (ha : a ≤ 1) (hb : 0 ≤ b) :
b * a < c | (mul_le_of_le_one_right hb ha).trans_lt h | lemma | mul_lt_of_lt_of_le_one_of_nonneg | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_of_le_one_right",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left.mul_le_one_of_le_of_le [pos_mul_mono α] (ha : a ≤ 1) (hb : b ≤ 1) (a0 : 0 ≤ a) :
a * b ≤ 1 | mul_le_of_le_of_le_one_of_nonneg ha hb a0 | lemma | left.mul_le_one_of_le_of_le | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_of_le_of_le_one_of_nonneg",
"pos_mul_mono"
] | Assumes left covariance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left.mul_lt_of_le_of_lt_one_of_pos [pos_mul_strict_mono α]
(ha : a ≤ 1) (hb : b < 1) (a0 : 0 < a) : a * b < 1 | mul_lt_of_le_of_lt_one_of_pos ha hb a0 | lemma | left.mul_lt_of_le_of_lt_one_of_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_of_le_of_lt_one_of_pos",
"pos_mul_strict_mono"
] | Assumes left covariance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left.mul_lt_of_lt_of_le_one_of_nonneg [pos_mul_mono α]
(ha : a < 1) (hb : b ≤ 1) (a0 : 0 ≤ a) : a * b < 1 | mul_lt_of_lt_of_le_one_of_nonneg ha hb a0 | lemma | left.mul_lt_of_lt_of_le_one_of_nonneg | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_of_lt_of_le_one_of_nonneg",
"pos_mul_mono"
] | Assumes left covariance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_le_of_le_of_le_one' [pos_mul_mono α] [mul_pos_mono α]
(bc : b ≤ c) (ha : a ≤ 1) (a0 : 0 ≤ a) (c0 : 0 ≤ c) : b * a ≤ c | (mul_le_mul_of_nonneg_right bc a0).trans $ mul_le_of_le_one_right c0 ha | lemma | mul_le_of_le_of_le_one' | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_right",
"mul_le_of_le_one_right",
"mul_pos_mono",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_lt_of_le_one' [pos_mul_mono α] [mul_pos_strict_mono α]
(bc : b < c) (ha : a ≤ 1) (a0 : 0 < a) (c0 : 0 ≤ c) : b * a < c | (mul_lt_mul_of_pos_right bc a0).trans_le $ mul_le_of_le_one_right c0 ha | lemma | mul_lt_of_lt_of_le_one' | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_of_le_one_right",
"mul_lt_mul_of_pos_right",
"mul_pos_strict_mono",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_le_of_lt_one' [pos_mul_strict_mono α] [mul_pos_mono α]
(bc : b ≤ c) (ha : a < 1) (a0 : 0 ≤ a) (c0 : 0 < c) : b * a < c | (mul_le_mul_of_nonneg_right bc a0).trans_lt $ mul_lt_of_lt_one_right c0 ha | lemma | mul_lt_of_le_of_lt_one' | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_right",
"mul_lt_of_lt_one_right",
"mul_pos_mono",
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_lt_of_lt_one_of_pos [pos_mul_mono α] [mul_pos_strict_mono α]
(bc : b < c) (ha : a ≤ 1) (a0 : 0 < a) (c0 : 0 ≤ c) : b * a < c | (mul_lt_mul_of_pos_right bc a0).trans_le $ mul_le_of_le_one_right c0 ha | lemma | mul_lt_of_lt_of_lt_one_of_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_of_le_one_right",
"mul_lt_mul_of_pos_right",
"mul_pos_strict_mono",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_of_le_of_one_le_of_nonneg [pos_mul_mono α] (h : b ≤ c) (ha : 1 ≤ a) (hc : 0 ≤ c) :
b ≤ c * a | h.trans $ le_mul_of_one_le_right hc ha | lemma | le_mul_of_le_of_one_le_of_nonneg | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"le_mul_of_one_le_right",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_le_of_one_lt_of_pos [pos_mul_strict_mono α] (bc : b ≤ c) (ha : 1 < a) (c0 : 0 < c) :
b < c * a | bc.trans_lt $ lt_mul_of_one_lt_right c0 ha | lemma | lt_mul_of_le_of_one_lt_of_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"lt_mul_of_one_lt_right",
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_lt_of_one_le_of_nonneg [pos_mul_mono α] (h : b < c) (ha : 1 ≤ a) (hc : 0 ≤ c) :
b < c * a | h.trans_le $ le_mul_of_one_le_right hc ha | lemma | lt_mul_of_lt_of_one_le_of_nonneg | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"le_mul_of_one_le_right",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left.one_le_mul_of_le_of_le [pos_mul_mono α] (ha : 1 ≤ a) (hb : 1 ≤ b) (a0 : 0 ≤ a) :
1 ≤ a * b | le_mul_of_le_of_one_le_of_nonneg ha hb a0 | lemma | left.one_le_mul_of_le_of_le | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"le_mul_of_le_of_one_le_of_nonneg",
"pos_mul_mono"
] | Assumes left covariance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left.one_lt_mul_of_le_of_lt_of_pos [pos_mul_strict_mono α]
(ha : 1 ≤ a) (hb : 1 < b) (a0 : 0 < a) : 1 < a * b | lt_mul_of_le_of_one_lt_of_pos ha hb a0 | lemma | left.one_lt_mul_of_le_of_lt_of_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"lt_mul_of_le_of_one_lt_of_pos",
"pos_mul_strict_mono"
] | Assumes left covariance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left.lt_mul_of_lt_of_one_le_of_nonneg [pos_mul_mono α]
(ha : 1 < a) (hb : 1 ≤ b) (a0 : 0 ≤ a) : 1 < a * b | lt_mul_of_lt_of_one_le_of_nonneg ha hb a0 | lemma | left.lt_mul_of_lt_of_one_le_of_nonneg | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"lt_mul_of_lt_of_one_le_of_nonneg",
"pos_mul_mono"
] | Assumes left covariance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_mul_of_le_of_one_le' [pos_mul_mono α] [mul_pos_mono α]
(bc : b ≤ c) (ha : 1 ≤ a) (a0 : 0 ≤ a) (b0 : 0 ≤ b) : b ≤ c * a | (le_mul_of_one_le_right b0 ha).trans $ mul_le_mul_of_nonneg_right bc a0 | lemma | le_mul_of_le_of_one_le' | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"le_mul_of_one_le_right",
"mul_le_mul_of_nonneg_right",
"mul_pos_mono",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_le_of_one_lt' [pos_mul_strict_mono α] [mul_pos_mono α]
(bc : b ≤ c) (ha : 1 < a) (a0 : 0 ≤ a) (b0 : 0 < b) : b < c * a | (lt_mul_of_one_lt_right b0 ha).trans_le $ mul_le_mul_of_nonneg_right bc a0 | lemma | lt_mul_of_le_of_one_lt' | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"lt_mul_of_one_lt_right",
"mul_le_mul_of_nonneg_right",
"mul_pos_mono",
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_lt_of_one_le' [pos_mul_mono α] [mul_pos_strict_mono α]
(bc : b < c) (ha : 1 ≤ a) (a0 : 0 < a) (b0 : 0 ≤ b) : b < c * a | (le_mul_of_one_le_right b0 ha).trans_lt $ mul_lt_mul_of_pos_right bc a0 | lemma | lt_mul_of_lt_of_one_le' | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"le_mul_of_one_le_right",
"mul_lt_mul_of_pos_right",
"mul_pos_strict_mono",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_lt_of_one_lt_of_pos [pos_mul_strict_mono α] [mul_pos_strict_mono α]
(bc : b < c) (ha : 1 < a) (a0 : 0 < a) (b0 : 0 < b) : b < c * a | (lt_mul_of_one_lt_right b0 ha).trans $ mul_lt_mul_of_pos_right bc a0 | lemma | lt_mul_of_lt_of_one_lt_of_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"lt_mul_of_one_lt_right",
"mul_lt_mul_of_pos_right",
"mul_pos_strict_mono",
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_of_le_one_of_le_of_nonneg [mul_pos_mono α] (ha : a ≤ 1) (h : b ≤ c) (hb : 0 ≤ b) :
a * b ≤ c | (mul_le_of_le_one_left hb ha).trans h | lemma | mul_le_of_le_one_of_le_of_nonneg | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_of_le_one_left",
"mul_pos_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_lt_one_of_le_of_pos [mul_pos_strict_mono α] (ha : a < 1) (h : b ≤ c) (hb : 0 < b) :
a * b < c | (mul_lt_of_lt_one_left hb ha).trans_le h | lemma | mul_lt_of_lt_one_of_le_of_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_of_lt_one_left",
"mul_pos_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_le_one_of_lt_of_nonneg [mul_pos_mono α] (ha : a ≤ 1) (h : b < c) (hb : 0 ≤ b) :
a * b < c | (mul_le_of_le_one_left hb ha).trans_lt h | lemma | mul_lt_of_le_one_of_lt_of_nonneg | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_of_le_one_left",
"mul_pos_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right.mul_lt_one_of_lt_of_le_of_pos [mul_pos_strict_mono α]
(ha : a < 1) (hb : b ≤ 1) (b0 : 0 < b) : a * b < 1 | mul_lt_of_lt_one_of_le_of_pos ha hb b0 | lemma | right.mul_lt_one_of_lt_of_le_of_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_of_lt_one_of_le_of_pos",
"mul_pos_strict_mono"
] | Assumes right covariance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right.mul_lt_one_of_le_of_lt_of_nonneg [mul_pos_mono α]
(ha : a ≤ 1) (hb : b < 1) (b0 : 0 ≤ b) : a * b < 1 | mul_lt_of_le_one_of_lt_of_nonneg ha hb b0 | lemma | right.mul_lt_one_of_le_of_lt_of_nonneg | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_of_le_one_of_lt_of_nonneg",
"mul_pos_mono"
] | Assumes right covariance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_lt_of_lt_one_of_lt_of_pos [pos_mul_strict_mono α] [mul_pos_strict_mono α]
(ha : a < 1) (bc : b < c) (a0 : 0 < a) (c0 : 0 < c) : a * b < c | (mul_lt_mul_of_pos_left bc a0).trans $ mul_lt_of_lt_one_left c0 ha | lemma | mul_lt_of_lt_one_of_lt_of_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_mul_of_pos_left",
"mul_lt_of_lt_one_left",
"mul_pos_strict_mono",
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right.mul_le_one_of_le_of_le [mul_pos_mono α]
(ha : a ≤ 1) (hb : b ≤ 1) (b0 : 0 ≤ b) : a * b ≤ 1 | mul_le_of_le_one_of_le_of_nonneg ha hb b0 | lemma | right.mul_le_one_of_le_of_le | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_of_le_one_of_le_of_nonneg",
"mul_pos_mono"
] | Assumes right covariance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_le_of_le_one_of_le' [pos_mul_mono α] [mul_pos_mono α]
(ha : a ≤ 1) (bc : b ≤ c) (a0 : 0 ≤ a) (c0 : 0 ≤ c) : a * b ≤ c | (mul_le_mul_of_nonneg_left bc a0).trans $ mul_le_of_le_one_left c0 ha | lemma | mul_le_of_le_one_of_le' | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_left",
"mul_le_of_le_one_left",
"mul_pos_mono",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_lt_one_of_le' [pos_mul_mono α] [mul_pos_strict_mono α]
(ha : a < 1) (bc : b ≤ c) (a0 : 0 ≤ a) (c0 : 0 < c) : a * b < c | (mul_le_mul_of_nonneg_left bc a0).trans_lt $ mul_lt_of_lt_one_left c0 ha | lemma | mul_lt_of_lt_one_of_le' | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_left",
"mul_lt_of_lt_one_left",
"mul_pos_strict_mono",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_le_one_of_lt' [pos_mul_strict_mono α] [mul_pos_mono α]
(ha : a ≤ 1) (bc : b < c) (a0 : 0 < a) (c0 : 0 ≤ c) : a * b < c | (mul_lt_mul_of_pos_left bc a0).trans_le $ mul_le_of_le_one_left c0 ha | lemma | mul_lt_of_le_one_of_lt' | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_of_le_one_left",
"mul_lt_mul_of_pos_left",
"mul_pos_mono",
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_one_lt_of_le_of_pos [mul_pos_strict_mono α] (ha : 1 < a) (h : b ≤ c) (hc : 0 < c) :
b < a * c | h.trans_lt $ lt_mul_of_one_lt_left hc ha | lemma | lt_mul_of_one_lt_of_le_of_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"lt_mul_of_one_lt_left",
"mul_pos_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_one_le_of_lt_of_nonneg [mul_pos_mono α] (ha : 1 ≤ a) (h : b < c) (hc : 0 ≤ c) :
b < a * c | h.trans_le $ le_mul_of_one_le_left hc ha | lemma | lt_mul_of_one_le_of_lt_of_nonneg | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"le_mul_of_one_le_left",
"mul_pos_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_one_lt_of_lt_of_pos [mul_pos_strict_mono α] (ha : 1 < a) (h : b < c) (hc : 0 < c) :
b < a * c | h.trans $ lt_mul_of_one_lt_left hc ha | lemma | lt_mul_of_one_lt_of_lt_of_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"lt_mul_of_one_lt_left",
"mul_pos_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right.one_lt_mul_of_lt_of_le_of_pos [mul_pos_strict_mono α]
(ha : 1 < a) (hb : 1 ≤ b) (b0 : 0 < b) : 1 < a * b | lt_mul_of_one_lt_of_le_of_pos ha hb b0 | lemma | right.one_lt_mul_of_lt_of_le_of_pos | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"lt_mul_of_one_lt_of_le_of_pos",
"mul_pos_strict_mono"
] | Assumes right covariance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right.one_lt_mul_of_le_of_lt_of_nonneg [mul_pos_mono α]
(ha : 1 ≤ a) (hb : 1 < b) (b0 : 0 ≤ b) : 1 < a * b | lt_mul_of_one_le_of_lt_of_nonneg ha hb b0 | lemma | right.one_lt_mul_of_le_of_lt_of_nonneg | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"lt_mul_of_one_le_of_lt_of_nonneg",
"mul_pos_mono"
] | Assumes right covariance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right.one_lt_mul_of_lt_of_lt [mul_pos_strict_mono α]
(ha : 1 < a) (hb : 1 < b) (b0 : 0 < b) : 1 < a * b | lt_mul_of_one_lt_of_lt_of_pos ha hb b0 | lemma | right.one_lt_mul_of_lt_of_lt | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"lt_mul_of_one_lt_of_lt_of_pos",
"mul_pos_strict_mono"
] | Assumes right covariance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lt_mul_of_one_lt_of_lt_of_nonneg [mul_pos_mono α] (ha : 1 ≤ a) (h : b < c) (hc : 0 ≤ c) :
b < a * c | h.trans_le $ le_mul_of_one_le_left hc ha | lemma | lt_mul_of_one_lt_of_lt_of_nonneg | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"le_mul_of_one_le_left",
"mul_pos_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_mul_lt_of_one_le_of_nonneg_left [pos_mul_mono α] (h : a * b < c) (hle : 1 ≤ b)
(ha : 0 ≤ a) :
a < c | (le_mul_of_one_le_right ha hle).trans_lt h | lemma | lt_of_mul_lt_of_one_le_of_nonneg_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"le_mul_of_one_le_right",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_lt_mul_of_le_one_of_nonneg_left [pos_mul_mono α] (h : a < b * c) (hc : c ≤ 1)
(hb : 0 ≤ b) :
a < b | h.trans_le $ mul_le_of_le_one_right hb hc | lemma | lt_of_lt_mul_of_le_one_of_nonneg_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_of_le_one_right",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_lt_mul_of_le_one_of_nonneg_right [mul_pos_mono α] (h : a < b * c) (hb : b ≤ 1)
(hc : 0 ≤ c) :
a < c | h.trans_le $ mul_le_of_le_one_left hc hb | lemma | lt_of_lt_mul_of_le_one_of_nonneg_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_of_le_one_left",
"mul_pos_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_of_one_le_of_le_of_nonneg [mul_pos_mono α] (ha : 1 ≤ a) (bc : b ≤ c) (c0 : 0 ≤ c) :
b ≤ a * c | bc.trans $ le_mul_of_one_le_left c0 ha | lemma | le_mul_of_one_le_of_le_of_nonneg | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"le_mul_of_one_le_left",
"mul_pos_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right.one_le_mul_of_le_of_le [mul_pos_mono α] (ha : 1 ≤ a) (hb : 1 ≤ b) (b0 : 0 ≤ b) :
1 ≤ a * b | le_mul_of_one_le_of_le_of_nonneg ha hb b0 | lemma | right.one_le_mul_of_le_of_le | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"le_mul_of_one_le_of_le_of_nonneg",
"mul_pos_mono"
] | Assumes right covariance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_of_mul_le_of_one_le_of_nonneg_left [pos_mul_mono α] (h : a * b ≤ c) (hb : 1 ≤ b)
(ha : 0 ≤ a) :
a ≤ c | (le_mul_of_one_le_right ha hb).trans h | lemma | le_of_mul_le_of_one_le_of_nonneg_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"le_mul_of_one_le_right",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_le_mul_of_le_one_of_nonneg_left [pos_mul_mono α] (h : a ≤ b * c) (hc : c ≤ 1)
(hb : 0 ≤ b) :
a ≤ b | h.trans $ mul_le_of_le_one_right hb hc | lemma | le_of_le_mul_of_le_one_of_nonneg_left | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_of_le_one_right",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_mul_le_of_one_le_nonneg_right [mul_pos_mono α] (h : a * b ≤ c) (ha : 1 ≤ a)
(hb : 0 ≤ b) :
b ≤ c | (le_mul_of_one_le_left hb ha).trans h | lemma | le_of_mul_le_of_one_le_nonneg_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"le_mul_of_one_le_left",
"mul_pos_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_le_mul_of_le_one_of_nonneg_right [mul_pos_mono α] (h : a ≤ b * c) (hb : b ≤ 1)
(hc : 0 ≤ c) :
a ≤ c | h.trans $ mul_le_of_le_one_left hc hb | lemma | le_of_le_mul_of_le_one_of_nonneg_right | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_of_le_one_left",
"mul_pos_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_square_le' [pos_mul_strict_mono α] (a0 : 0 < a) : ∃ (b : α), b * b ≤ a | begin
obtain ha | ha := lt_or_le a 1,
{ exact ⟨a, (mul_lt_of_lt_one_right a0 ha).le⟩ },
{ exact ⟨1, by rwa mul_one⟩ }
end | lemma | exists_square_le' | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_lt_of_lt_one_right",
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_mul_mono.to_pos_mul_strict_mono [pos_mul_mono α] : pos_mul_strict_mono α | ⟨λ x a b h, (mul_le_mul_of_nonneg_left h.le x.2.le).lt_of_ne (h.ne ∘ mul_left_cancel₀ x.2.ne')⟩ | lemma | pos_mul_mono.to_pos_mul_strict_mono | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_left",
"mul_left_cancel₀",
"pos_mul_mono",
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_mul_mono_iff_pos_mul_strict_mono : pos_mul_mono α ↔ pos_mul_strict_mono α | ⟨@pos_mul_mono.to_pos_mul_strict_mono α _ _, @pos_mul_strict_mono.to_pos_mul_mono α _ _⟩ | lemma | pos_mul_mono_iff_pos_mul_strict_mono | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"pos_mul_mono",
"pos_mul_mono.to_pos_mul_strict_mono",
"pos_mul_strict_mono",
"pos_mul_strict_mono.to_pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_pos_mono.to_mul_pos_strict_mono [mul_pos_mono α] : mul_pos_strict_mono α | ⟨λ x a b h, (mul_le_mul_of_nonneg_right h.le x.2.le).lt_of_ne (h.ne ∘ mul_right_cancel₀ x.2.ne')⟩ | lemma | mul_pos_mono.to_mul_pos_strict_mono | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_le_mul_of_nonneg_right",
"mul_pos_mono",
"mul_pos_strict_mono",
"mul_right_cancel₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_pos_mono_iff_mul_pos_strict_mono : mul_pos_mono α ↔ mul_pos_strict_mono α | ⟨@mul_pos_mono.to_mul_pos_strict_mono α _ _, @mul_pos_strict_mono.to_mul_pos_mono α _ _⟩ | lemma | mul_pos_mono_iff_mul_pos_strict_mono | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_pos_mono",
"mul_pos_mono.to_mul_pos_strict_mono",
"mul_pos_strict_mono",
"mul_pos_strict_mono.to_mul_pos_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_mul_reflect_lt.to_pos_mul_mono_rev [pos_mul_reflect_lt α] : pos_mul_mono_rev α | ⟨λ x a b h, h.eq_or_lt.elim (le_of_eq ∘ mul_left_cancel₀ x.2.ne.symm)
(λ h', (lt_of_mul_lt_mul_left h' x.2.le).le)⟩ | lemma | pos_mul_reflect_lt.to_pos_mul_mono_rev | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"lt_of_mul_lt_mul_left",
"mul_left_cancel₀",
"pos_mul_mono_rev",
"pos_mul_reflect_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_mul_mono_rev_iff_pos_mul_reflect_lt : pos_mul_mono_rev α ↔ pos_mul_reflect_lt α | ⟨@pos_mul_mono_rev.to_pos_mul_reflect_lt α _ _, @pos_mul_reflect_lt.to_pos_mul_mono_rev α _ _⟩ | lemma | pos_mul_mono_rev_iff_pos_mul_reflect_lt | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"pos_mul_mono_rev",
"pos_mul_mono_rev.to_pos_mul_reflect_lt",
"pos_mul_reflect_lt",
"pos_mul_reflect_lt.to_pos_mul_mono_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_pos_reflect_lt.to_mul_pos_mono_rev [mul_pos_reflect_lt α] : mul_pos_mono_rev α | ⟨λ x a b h, h.eq_or_lt.elim (le_of_eq ∘ mul_right_cancel₀ x.2.ne.symm)
(λ h', (lt_of_mul_lt_mul_right h' x.2.le).le)⟩ | lemma | mul_pos_reflect_lt.to_mul_pos_mono_rev | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"lt_of_mul_lt_mul_right",
"mul_pos_mono_rev",
"mul_pos_reflect_lt",
"mul_right_cancel₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_pos_mono_rev_iff_mul_pos_reflect_lt : mul_pos_mono_rev α ↔ mul_pos_reflect_lt α | ⟨@mul_pos_mono_rev.to_mul_pos_reflect_lt α _ _, @mul_pos_reflect_lt.to_mul_pos_mono_rev α _ _⟩ | lemma | mul_pos_mono_rev_iff_mul_pos_reflect_lt | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_pos_mono_rev",
"mul_pos_mono_rev.to_mul_pos_reflect_lt",
"mul_pos_reflect_lt",
"mul_pos_reflect_lt.to_mul_pos_mono_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_mul_strict_mono_iff_mul_pos_strict_mono :
pos_mul_strict_mono α ↔ mul_pos_strict_mono α | by simp ! only [mul_comm] | lemma | pos_mul_strict_mono_iff_mul_pos_strict_mono | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_comm",
"mul_pos_strict_mono",
"pos_mul_strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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