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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|---|---|---|---|---|---|---|---|---|---|---|
self_trans_symm (e : α ≃+*o β) : e.trans e.symm = order_ring_iso.refl α | ext e.left_inv | lemma | order_ring_iso.self_trans_symm | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"order_ring_iso.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans_self (e : α ≃+*o β) : e.symm.trans e = order_ring_iso.refl β | ext e.right_inv | lemma | order_ring_iso.symm_trans_self | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"order_ring_iso.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_bijective : bijective (order_ring_iso.symm : (α ≃+*o β) → β ≃+*o α) | ⟨λ f g h, f.symm_symm.symm.trans $ (congr_arg order_ring_iso.symm h).trans g.symm_symm,
λ f, ⟨f.symm, f.symm_symm⟩⟩ | lemma | order_ring_iso.symm_bijective | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"order_ring_iso.symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_order_ring_hom (f : α ≃+*o β) : α →+*o β | ⟨f.to_ring_equiv.to_ring_hom, λ a b, (map_le_map_iff f).2⟩ | def | order_ring_iso.to_order_ring_hom | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | Reinterpret an ordered ring isomorphism as an ordered ring homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_order_ring_hom_eq_coe (f : α ≃+*o β) : f.to_order_ring_hom = f | rfl | lemma | order_ring_iso.to_order_ring_hom_eq_coe | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_order_ring_hom (f : α ≃+*o β) : ⇑(f : α →+*o β) = f | rfl | lemma | order_ring_iso.coe_to_order_ring_hom | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_order_ring_hom_refl : (order_ring_iso.refl α : α →+*o α) = order_ring_hom.id α | rfl | lemma | order_ring_iso.coe_to_order_ring_hom_refl | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"order_ring_hom.id",
"order_ring_iso.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_order_ring_hom_injective : injective (to_order_ring_hom : (α ≃+*o β) → α →+*o β) | λ f g h, fun_like.coe_injective $ by convert fun_like.ext'_iff.1 h | lemma | order_ring_iso.to_order_ring_hom_injective | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"fun_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_ring_hom.subsingleton [linear_ordered_field α] [linear_ordered_field β]
[archimedean β] :
subsingleton (α →+*o β) | ⟨λ f g, begin
ext x,
by_contra' h' : f x ≠ g x,
wlog h : f x < g x,
{ exact this g f x (ne.symm h') (h'.lt_or_lt.resolve_left h), },
obtain ⟨q, hf, hg⟩ := exists_rat_btwn h,
rw ←map_rat_cast f at hf,
rw ←map_rat_cast g at hg,
exact (lt_asymm ((order_hom_class.mono g).reflect_lt hg) $
(order_hom_clas... | instance | order_ring_hom.subsingleton | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"archimedean",
"exists_rat_btwn",
"linear_ordered_field",
"order_hom_class.mono"
] | There is at most one ordered ring homomorphism from a linear ordered field to an archimedean
linear ordered field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_ring_iso.subsingleton_right [linear_ordered_field α] [linear_ordered_field β]
[archimedean β] :
subsingleton (α ≃+*o β) | order_ring_iso.to_order_ring_hom_injective.subsingleton | instance | order_ring_iso.subsingleton_right | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"archimedean",
"linear_ordered_field"
] | There is at most one ordered ring isomorphism between a linear ordered field and an archimedean
linear ordered field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_ring_iso.subsingleton_left [linear_ordered_field α] [archimedean α]
[linear_ordered_field β] :
subsingleton (α ≃+*o β) | order_ring_iso.symm_bijective.injective.subsingleton | instance | order_ring_iso.subsingleton_left | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"archimedean",
"linear_ordered_field"
] | There is at most one ordered ring isomorphism between an archimedean linear ordered field and a
linear ordered field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.ordered_comm_monoid [ordered_comm_monoid α] {β : Type*}
[has_one β] [has_mul β] [has_pow β ℕ]
(f : β → α) (hf : function.injective f) (one : f 1 = 1)
(mul : ∀ x y, f (x * y) = f x * f y) (npow : ∀ x (n : ℕ), f (x ^ n) = f x ^ n) :
ordered_comm_monoid β | { mul_le_mul_left := λ a b ab c, show f (c * a) ≤ f (c * b), by
{ rw [mul, mul], apply mul_le_mul_left', exact ab },
..partial_order.lift f hf,
..hf.comm_monoid f one mul npow } | def | function.injective.ordered_comm_monoid | algebra.order.monoid | src/algebra/order/monoid/basic.lean | [
"algebra.order.monoid.defs",
"algebra.group.inj_surj",
"order.hom.basic"
] | [
"mul_le_mul_left",
"mul_le_mul_left'",
"ordered_comm_monoid",
"partial_order.lift"
] | Pullback an `ordered_comm_monoid` under an injective map.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.linear_ordered_comm_monoid [linear_ordered_comm_monoid α] {β : Type*}
[has_one β] [has_mul β] [has_pow β ℕ] [has_sup β] [has_inf β]
(f : β → α) (hf : function.injective f) (one : f 1 = 1)
(mul : ∀ x y, f (x * y) = f x * f y) (npow : ∀ x (n : ℕ), f (x ^ n) = f x ^ n)
(hsup : ∀ x y, f (x ⊔ y) =... | { .. hf.ordered_comm_monoid f one mul npow,
.. linear_order.lift f hf hsup hinf } | def | function.injective.linear_ordered_comm_monoid | algebra.order.monoid | src/algebra/order/monoid/basic.lean | [
"algebra.order.monoid.defs",
"algebra.group.inj_surj",
"order.hom.basic"
] | [
"has_inf",
"has_sup",
"linear_order.lift",
"linear_ordered_comm_monoid"
] | Pullback a `linear_ordered_comm_monoid` under an injective map.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_embedding.mul_left
{α : Type*} [has_mul α] [linear_order α] [covariant_class α α (*) (<)] (m : α) : α ↪o α | order_embedding.of_strict_mono (λ n, m * n) (λ a b w, mul_lt_mul_left' w m) | def | order_embedding.mul_left | algebra.order.monoid | src/algebra/order/monoid/basic.lean | [
"algebra.order.monoid.defs",
"algebra.group.inj_surj",
"order.hom.basic"
] | [
"covariant_class",
"mul_lt_mul_left'",
"order_embedding.of_strict_mono"
] | The order embedding sending `b` to `a * b`, for some fixed `a`.
See also `order_iso.mul_left` when working in an ordered group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_embedding.mul_right
{α : Type*} [has_mul α] [linear_order α] [covariant_class α α (swap (*)) (<)] (m : α) :
α ↪o α | order_embedding.of_strict_mono (λ n, n * m) (λ a b w, mul_lt_mul_right' w m) | def | order_embedding.mul_right | algebra.order.monoid | src/algebra/order/monoid/basic.lean | [
"algebra.order.monoid.defs",
"algebra.group.inj_surj",
"order.hom.basic"
] | [
"covariant_class",
"mul_lt_mul_right'",
"order_embedding.of_strict_mono"
] | The order embedding sending `b` to `b * a`, for some fixed `a`.
See also `order_iso.mul_right` when working in an ordered group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ordered_comm_monoid (α : Type*) extends comm_monoid α, partial_order α | (mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b) | class | ordered_comm_monoid | algebra.order.monoid | src/algebra/order/monoid/defs.lean | [
"algebra.order.monoid.lemmas",
"order.bounded_order"
] | [
"comm_monoid",
"mul_le_mul_left"
] | An ordered commutative monoid is a commutative monoid
with a partial order such that `a ≤ b → c * a ≤ c * b` (multiplication is monotone) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ordered_add_comm_monoid (α : Type*) extends add_comm_monoid α, partial_order α | (add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b) | class | ordered_add_comm_monoid | algebra.order.monoid | src/algebra/order/monoid/defs.lean | [
"algebra.order.monoid.lemmas",
"order.bounded_order"
] | [
"add_comm_monoid"
] | An ordered (additive) commutative monoid is a commutative monoid
with a partial order such that `a ≤ b → c + a ≤ c + b` (addition is monotone) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ordered_comm_monoid.to_covariant_class_left (M : Type*) [ordered_comm_monoid M] :
covariant_class M M (*) (≤) | { elim := λ a b c bc, ordered_comm_monoid.mul_le_mul_left _ _ bc a } | instance | ordered_comm_monoid.to_covariant_class_left | algebra.order.monoid | src/algebra/order/monoid/defs.lean | [
"algebra.order.monoid.lemmas",
"order.bounded_order"
] | [
"covariant_class",
"ordered_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_comm_monoid.to_covariant_class_right (M : Type*) [ordered_comm_monoid M] :
covariant_class M M (swap (*)) (≤) | covariant_swap_mul_le_of_covariant_mul_le M | instance | ordered_comm_monoid.to_covariant_class_right | algebra.order.monoid | src/algebra/order/monoid/defs.lean | [
"algebra.order.monoid.lemmas",
"order.bounded_order"
] | [
"covariant_class",
"covariant_swap_mul_le_of_covariant_mul_le",
"ordered_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_mul.to_covariant_class_left
(M : Type*) [has_mul M] [partial_order M] [covariant_class M M (*) (<)] :
covariant_class M M (*) (≤) | ⟨covariant_le_of_covariant_lt _ _ _ covariant_class.elim⟩ | lemma | has_mul.to_covariant_class_left | algebra.order.monoid | src/algebra/order/monoid/defs.lean | [
"algebra.order.monoid.lemmas",
"order.bounded_order"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_mul.to_covariant_class_right
(M : Type*) [has_mul M] [partial_order M] [covariant_class M M (swap (*)) (<)] :
covariant_class M M (swap (*)) (≤) | ⟨covariant_le_of_covariant_lt _ _ _ covariant_class.elim⟩ | lemma | has_mul.to_covariant_class_right | algebra.order.monoid | src/algebra/order/monoid/defs.lean | [
"algebra.order.monoid.lemmas",
"order.bounded_order"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_pos [ordered_add_comm_monoid α] {a : α} (h : 0 < a) : 0 < bit0 a | add_pos' h h | lemma | bit0_pos | algebra.order.monoid | src/algebra/order/monoid/defs.lean | [
"algebra.order.monoid.lemmas",
"order.bounded_order"
] | [
"ordered_add_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_add_comm_monoid (α : Type*)
extends linear_order α, ordered_add_comm_monoid α. | class | linear_ordered_add_comm_monoid | algebra.order.monoid | src/algebra/order/monoid/defs.lean | [
"algebra.order.monoid.lemmas",
"order.bounded_order"
] | [
"ordered_add_comm_monoid"
] | A linearly ordered additive commutative monoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_comm_monoid (α : Type*)
extends linear_order α, ordered_comm_monoid α. | class | linear_ordered_comm_monoid | algebra.order.monoid | src/algebra/order/monoid/defs.lean | [
"algebra.order.monoid.lemmas",
"order.bounded_order"
] | [
"ordered_comm_monoid"
] | A linearly ordered commutative monoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_add_comm_monoid_with_top (α : Type*)
extends linear_ordered_add_comm_monoid α, has_top α | (le_top : ∀ x : α, x ≤ ⊤)
(top_add' : ∀ x : α, ⊤ + x = ⊤) | class | linear_ordered_add_comm_monoid_with_top | algebra.order.monoid | src/algebra/order/monoid/defs.lean | [
"algebra.order.monoid.lemmas",
"order.bounded_order"
] | [
"has_top",
"le_top",
"linear_ordered_add_comm_monoid"
] | A linearly ordered commutative monoid with an additively absorbing `⊤` element.
Instances should include number systems with an infinite element adjoined.` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_ordered_add_comm_monoid_with_top.to_order_top (α : Type u)
[h : linear_ordered_add_comm_monoid_with_top α] : order_top α | { ..h } | instance | linear_ordered_add_comm_monoid_with_top.to_order_top | algebra.order.monoid | src/algebra/order/monoid/defs.lean | [
"algebra.order.monoid.lemmas",
"order.bounded_order"
] | [
"linear_ordered_add_comm_monoid_with_top",
"order_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_add (a : α) : ⊤ + a = ⊤ | linear_ordered_add_comm_monoid_with_top.top_add' a | lemma | top_add | algebra.order.monoid | src/algebra/order/monoid/defs.lean | [
"algebra.order.monoid.lemmas",
"order.bounded_order"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_top (a : α) : a + ⊤ = ⊤ | trans (add_comm _ _) (top_add _) | lemma | add_top | algebra.order.monoid | src/algebra/order/monoid/defs.lean | [
"algebra.order.monoid.lemmas",
"order.bounded_order"
] | [
"top_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_left' [covariant_class α α (*) (≤)]
{b c : α} (bc : b ≤ c) (a : α) :
a * b ≤ a * c | covariant_class.elim _ bc | lemma | mul_le_mul_left' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_mul_le_mul_left' [contravariant_class α α (*) (≤)]
{a b c : α} (bc : a * b ≤ a * c) :
b ≤ c | contravariant_class.elim _ bc | lemma | le_of_mul_le_mul_left' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_right' [covariant_class α α (swap (*)) (≤)]
{b c : α} (bc : b ≤ c) (a : α) :
b * a ≤ c * a | covariant_class.elim a bc | lemma | mul_le_mul_right' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_mul_le_mul_right' [contravariant_class α α (swap (*)) (≤)]
{a b c : α} (bc : b * a ≤ c * a) :
b ≤ c | contravariant_class.elim a bc | lemma | le_of_mul_le_mul_right' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_iff_left [covariant_class α α (*) (≤)] [contravariant_class α α (*) (≤)]
(a : α) {b c : α} :
a * b ≤ a * c ↔ b ≤ c | rel_iff_cov α α (*) (≤) a | lemma | mul_le_mul_iff_left | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"covariant_class",
"rel_iff_cov"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_iff_right
[covariant_class α α (swap (*)) (≤)] [contravariant_class α α (swap (*)) (≤)]
(a : α) {b c : α} :
b * a ≤ c * a ↔ b ≤ c | rel_iff_cov α α (swap (*)) (≤) a | lemma | mul_le_mul_iff_right | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"covariant_class",
"rel_iff_cov"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_iff_left [covariant_class α α (*) (<)] [contravariant_class α α (*) (<)]
(a : α) {b c : α} :
a * b < a * c ↔ b < c | rel_iff_cov α α (*) (<) a | lemma | mul_lt_mul_iff_left | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"covariant_class",
"rel_iff_cov"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_iff_right
[covariant_class α α (swap (*)) (<)] [contravariant_class α α (swap (*)) (<)]
(a : α) {b c : α} :
b * a < c * a ↔ b < c | rel_iff_cov α α (swap (*)) (<) a | lemma | mul_lt_mul_iff_right | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"covariant_class",
"rel_iff_cov"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_left' [covariant_class α α (*) (<)]
{b c : α} (bc : b < c) (a : α) :
a * b < a * c | covariant_class.elim _ bc | lemma | mul_lt_mul_left' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_mul_lt_mul_left' [contravariant_class α α (*) (<)]
{a b c : α} (bc : a * b < a * c) :
b < c | contravariant_class.elim _ bc | lemma | lt_of_mul_lt_mul_left' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_right' [covariant_class α α (swap (*)) (<)]
{b c : α} (bc : b < c) (a : α) :
b * a < c * a | covariant_class.elim a bc | lemma | mul_lt_mul_right' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_mul_lt_mul_right' [contravariant_class α α (swap (*)) (<)]
{a b c : α} (bc : b * a < c * a) :
b < c | contravariant_class.elim a bc | lemma | lt_of_mul_lt_mul_right' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_of_lt_of_lt [covariant_class α α (*) (<)] [covariant_class α α (swap (*)) (<)]
{a b c d : α} (h₁ : a < b) (h₂ : c < d) : a * c < b * d | calc a * c < a * d : mul_lt_mul_left' h₂ a
... < b * d : mul_lt_mul_right' h₁ d | lemma | mul_lt_mul_of_lt_of_lt | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_mul_left'",
"mul_lt_mul_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_of_le_of_lt [covariant_class α α (*) (<)] [covariant_class α α (swap (*)) (≤)]
{a b c d : α} (h₁ : a ≤ b) (h₂ : c < d) : a * c < b * d | (mul_le_mul_right' h₁ _).trans_lt (mul_lt_mul_left' h₂ b) | lemma | mul_lt_mul_of_le_of_lt | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_mul_right'",
"mul_lt_mul_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_of_lt_of_le [covariant_class α α (*) (≤)] [covariant_class α α (swap (*)) (<)]
{a b c d : α} (h₁ : a < b) (h₂ : c ≤ d) : a * c < b * d | (mul_le_mul_left' h₂ _).trans_lt (mul_lt_mul_right' h₁ d) | lemma | mul_lt_mul_of_lt_of_le | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_mul_left'",
"mul_lt_mul_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left.mul_lt_mul [covariant_class α α (*) (<)] [covariant_class α α (swap (*)) (≤)]
{a b c d : α} (h₁ : a < b) (h₂ : c < d) : a * c < b * d | mul_lt_mul_of_le_of_lt h₁.le h₂ | lemma | left.mul_lt_mul | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_mul_of_le_of_lt"
] | Only assumes left strict covariance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right.mul_lt_mul [covariant_class α α (*) (≤)] [covariant_class α α (swap (*)) (<)]
{a b c d : α} (h₁ : a < b) (h₂ : c < d) : a * c < b * d | mul_lt_mul_of_lt_of_le h₁ h₂.le | lemma | right.mul_lt_mul | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_mul_of_lt_of_le"
] | Only assumes right strict covariance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_le_mul' [covariant_class α α (*) (≤)] [covariant_class α α (swap (*)) (≤)]
{a b c d : α} (h₁ : a ≤ b) (h₂ : c ≤ d) : a * c ≤ b * d | (mul_le_mul_left' h₂ _).trans (mul_le_mul_right' h₁ d) | lemma | mul_le_mul' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_mul_left'",
"mul_le_mul_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_three [covariant_class α α (*) (≤)] [covariant_class α α (swap (*)) (≤)]
{a b c d e f : α} (h₁ : a ≤ d) (h₂ : b ≤ e) (h₃ : c ≤ f) :
a * b * c ≤ d * e * f | mul_le_mul' (mul_le_mul' h₁ h₂) h₃ | lemma | mul_le_mul_three | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_mul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_mul_lt_left [covariant_class α α (*) (≤)]
{a b c d : α} (h : a * b < c) (hle : d ≤ b) :
a * d < c | (mul_le_mul_left' hle a).trans_lt h | lemma | mul_lt_of_mul_lt_left | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_mul_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_of_mul_le_left [covariant_class α α (*) (≤)]
{a b c d : α} (h : a * b ≤ c) (hle : d ≤ b) :
a * d ≤ c | @act_rel_of_rel_of_act_rel _ _ _ (≤) _ ⟨λ _ _ _, le_trans⟩ a _ _ _ hle h | lemma | mul_le_of_mul_le_left | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"act_rel_of_rel_of_act_rel",
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_mul_lt_right [covariant_class α α (swap (*)) (≤)]
{a b c d : α} (h : a * b < c) (hle : d ≤ a) :
d * b < c | (mul_le_mul_right' hle b).trans_lt h | lemma | mul_lt_of_mul_lt_right | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_mul_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_of_mul_le_right [covariant_class α α (swap (*)) (≤)]
{a b c d : α} (h : a * b ≤ c) (hle : d ≤ a) :
d * b ≤ c | (mul_le_mul_right' hle b).trans h | lemma | mul_le_of_mul_le_right | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_mul_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_lt_mul_left [covariant_class α α (*) (≤)]
{a b c d : α} (h : a < b * c) (hle : c ≤ d) :
a < b * d | h.trans_le (mul_le_mul_left' hle b) | lemma | lt_mul_of_lt_mul_left | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_mul_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_of_le_mul_left [covariant_class α α (*) (≤)]
{a b c d : α} (h : a ≤ b * c) (hle : c ≤ d) :
a ≤ b * d | @rel_act_of_rel_of_rel_act _ _ _ (≤) _ ⟨λ _ _ _, le_trans⟩ b _ _ _ hle h | lemma | le_mul_of_le_mul_left | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"rel_act_of_rel_of_rel_act"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_lt_mul_right [covariant_class α α (swap (*)) (≤)]
{a b c d : α} (h : a < b * c) (hle : b ≤ d) :
a < d * c | h.trans_le (mul_le_mul_right' hle c) | lemma | lt_mul_of_lt_mul_right | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_mul_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_of_le_mul_right [covariant_class α α (swap (*)) (≤)]
{a b c d : α} (h : a ≤ b * c) (hle : b ≤ d) :
a ≤ d * c | h.trans (mul_le_mul_right' hle c) | lemma | le_mul_of_le_mul_right | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_mul_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_cancel'' [contravariant_class α α (*) (≤)]
{a b c : α} (h : a * b = a * c) :
b = c | (le_of_mul_le_mul_left' h.le).antisymm (le_of_mul_le_mul_left' h.ge) | lemma | mul_left_cancel'' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"le_of_mul_le_mul_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_cancel'' [contravariant_class α α (swap (*)) (≤)]
{a b c : α} (h : a * b = c * b) :
a = c | le_antisymm (le_of_mul_le_mul_right' h.le) (le_of_mul_le_mul_right' h.ge) | lemma | mul_right_cancel'' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"le_of_mul_le_mul_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_le_max_of_mul_le_mul (h : a * b ≤ c * d) : min a b ≤ max c d | by { simp_rw [min_le_iff, le_max_iff], contrapose! h, exact mul_lt_mul_of_lt_of_lt h.1.1 h.2.2 } | lemma | min_le_max_of_mul_le_mul | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"le_max_iff",
"min_le_iff",
"mul_lt_mul_of_lt_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_mul_mul_le_max_mul_max' :
max (a * b) (c * d) ≤ max a c * max b d | max_le (mul_le_mul' (le_max_left _ _) $ le_max_left _ _) $
mul_le_mul' (le_max_right _ _) $ le_max_right _ _ | lemma | max_mul_mul_le_max_mul_max' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"mul_le_mul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_mul_min_le_min_mul_mul' :
min a c * min b d ≤ min (a * b) (c * d) | le_min (mul_le_mul' (min_le_left _ _) $ min_le_left _ _) $
mul_le_mul' (min_le_right _ _) $ min_le_right _ _ | lemma | min_mul_min_le_min_mul_mul' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"mul_le_mul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_of_one_le_right' [covariant_class α α (*) (≤)]
{a b : α} (h : 1 ≤ b) :
a ≤ a * b | calc a = a * 1 : (mul_one a).symm
... ≤ a * b : mul_le_mul_left' h a | lemma | le_mul_of_one_le_right' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_mul_left'",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_of_le_one_right' [covariant_class α α (*) (≤)]
{a b : α} (h : b ≤ 1) :
a * b ≤ a | calc a * b ≤ a * 1 : mul_le_mul_left' h a
... = a : mul_one a | lemma | mul_le_of_le_one_right' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_mul_left'",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_of_one_le_left' [covariant_class α α (swap (*)) (≤)]
{a b : α} (h : 1 ≤ b) :
a ≤ b * a | calc a = 1 * a : (one_mul a).symm
... ≤ b * a : mul_le_mul_right' h a | lemma | le_mul_of_one_le_left' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_mul_right'",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_of_le_one_left' [covariant_class α α (swap (*)) (≤)]
{a b : α} (h : b ≤ 1) :
b * a ≤ a | calc b * a ≤ 1 * a : mul_le_mul_right' h a
... = a : one_mul a | lemma | mul_le_of_le_one_left' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_mul_right'",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_of_le_mul_right [contravariant_class α α (*) (≤)] {a b : α} (h : a ≤ a * b) : 1 ≤ b | le_of_mul_le_mul_left' $ by simpa only [mul_one] | lemma | one_le_of_le_mul_right | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"le_of_mul_le_mul_left'",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_one_of_mul_le_right [contravariant_class α α (*) (≤)] {a b : α} (h : a * b ≤ a) : b ≤ 1 | le_of_mul_le_mul_left' $ by simpa only [mul_one] | lemma | le_one_of_mul_le_right | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"le_of_mul_le_mul_left'",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_of_le_mul_left [contravariant_class α α (swap (*)) (≤)] {a b : α} (h : b ≤ a * b) :
1 ≤ a | le_of_mul_le_mul_right' $ by simpa only [one_mul] | lemma | one_le_of_le_mul_left | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"le_of_mul_le_mul_right'",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_one_of_mul_le_left [contravariant_class α α (swap (*)) (≤)] {a b : α} (h : a * b ≤ b) :
a ≤ 1 | le_of_mul_le_mul_right' $ by simpa only [one_mul] | lemma | le_one_of_mul_le_left | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"le_of_mul_le_mul_right'",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_iff_one_le_right'
[covariant_class α α (*) (≤)] [contravariant_class α α (*) (≤)]
(a : α) {b : α} :
a ≤ a * b ↔ 1 ≤ b | iff.trans (by rw [mul_one]) (mul_le_mul_iff_left a) | lemma | le_mul_iff_one_le_right' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"covariant_class",
"mul_le_mul_iff_left",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_iff_one_le_left'
[covariant_class α α (swap (*)) (≤)] [contravariant_class α α (swap (*)) (≤)]
(a : α) {b : α} :
a ≤ b * a ↔ 1 ≤ b | iff.trans (by rw one_mul) (mul_le_mul_iff_right a) | lemma | le_mul_iff_one_le_left' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"covariant_class",
"mul_le_mul_iff_right",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_iff_le_one_right'
[covariant_class α α (*) (≤)] [contravariant_class α α (*) (≤)]
(a : α) {b : α} :
a * b ≤ a ↔ b ≤ 1 | iff.trans (by rw [mul_one]) (mul_le_mul_iff_left a) | lemma | mul_le_iff_le_one_right' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"covariant_class",
"mul_le_mul_iff_left",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_iff_le_one_left'
[covariant_class α α (swap (*)) (≤)] [contravariant_class α α (swap (*)) (≤)]
{a b : α} :
a * b ≤ b ↔ a ≤ 1 | iff.trans (by rw one_mul) (mul_le_mul_iff_right b) | lemma | mul_le_iff_le_one_left' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"covariant_class",
"mul_le_mul_iff_right",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_one_lt_right' [covariant_class α α (*) (<)]
(a : α) {b : α} (h : 1 < b) :
a < a * b | calc a = a * 1 : (mul_one a).symm
... < a * b : mul_lt_mul_left' h a | lemma | lt_mul_of_one_lt_right' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_mul_left'",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_lt_one_right' [covariant_class α α (*) (<)]
(a : α) {b : α} (h : b < 1) :
a * b < a | calc a * b < a * 1 : mul_lt_mul_left' h a
... = a : mul_one a | lemma | mul_lt_of_lt_one_right' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_mul_left'",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_one_lt_left' [covariant_class α α (swap (*)) (<)]
(a : α) {b : α} (h : 1 < b) :
a < b * a | calc a = 1 * a : (one_mul a).symm
... < b * a : mul_lt_mul_right' h a | lemma | lt_mul_of_one_lt_left' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_mul_right'",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_lt_one_left' [covariant_class α α (swap (*)) (<)]
(a : α) {b : α} (h : b < 1) :
b * a < a | calc b * a < 1 * a : mul_lt_mul_right' h a
... = a : one_mul a | lemma | mul_lt_of_lt_one_left' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_mul_right'",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_of_lt_mul_right [contravariant_class α α (*) (<)] {a b : α} (h : a < a * b) : 1 < b | lt_of_mul_lt_mul_left' $ by simpa only [mul_one] | lemma | one_lt_of_lt_mul_right | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"lt_of_mul_lt_mul_left'",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_one_of_mul_lt_right [contravariant_class α α (*) (<)] {a b : α} (h : a * b < a) : b < 1 | lt_of_mul_lt_mul_left' $ by simpa only [mul_one] | lemma | lt_one_of_mul_lt_right | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"lt_of_mul_lt_mul_left'",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_of_lt_mul_left [contravariant_class α α (swap (*)) (<)] {a b : α} (h : b < a * b) :
1 < a | lt_of_mul_lt_mul_right' $ by simpa only [one_mul] | lemma | one_lt_of_lt_mul_left | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"lt_of_mul_lt_mul_right'",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_one_of_mul_lt_left [contravariant_class α α (swap (*)) (<)] {a b : α} (h : a * b < b) :
a < 1 | lt_of_mul_lt_mul_right' $ by simpa only [one_mul] | lemma | lt_one_of_mul_lt_left | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"lt_of_mul_lt_mul_right'",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_iff_one_lt_right'
[covariant_class α α (*) (<)] [contravariant_class α α (*) (<)]
(a : α) {b : α} :
a < a * b ↔ 1 < b | iff.trans (by rw mul_one) (mul_lt_mul_iff_left a) | lemma | lt_mul_iff_one_lt_right' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"covariant_class",
"mul_lt_mul_iff_left",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_iff_one_lt_left'
[covariant_class α α (swap (*)) (<)] [contravariant_class α α (swap (*)) (<)]
(a : α) {b : α} :
a < b * a ↔ 1 < b | iff.trans (by rw one_mul) (mul_lt_mul_iff_right a) | lemma | lt_mul_iff_one_lt_left' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"covariant_class",
"mul_lt_mul_iff_right",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_iff_lt_one_left'
[covariant_class α α (*) (<)] [contravariant_class α α (*) (<)]
{a b : α} :
a * b < a ↔ b < 1 | iff.trans (by rw mul_one) (mul_lt_mul_iff_left a) | lemma | mul_lt_iff_lt_one_left' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"covariant_class",
"mul_lt_mul_iff_left",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_iff_lt_one_right'
[covariant_class α α (swap (*)) (<)] [contravariant_class α α (swap (*)) (<)]
{a : α} (b : α) :
a * b < b ↔ a < 1 | iff.trans (by rw one_mul) (mul_lt_mul_iff_right b) | lemma | mul_lt_iff_lt_one_right' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"contravariant_class",
"covariant_class",
"mul_lt_mul_iff_right",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_of_le_of_le_one [covariant_class α α (*) (≤)]
{a b c : α} (hbc : b ≤ c) (ha : a ≤ 1) : b * a ≤ c | calc b * a ≤ b * 1 : mul_le_mul_left' ha b
... = b : mul_one b
... ≤ c : hbc | lemma | mul_le_of_le_of_le_one | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_mul_left'",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_le_of_lt_one [covariant_class α α (*) (<)]
{a b c : α} (hbc : b ≤ c) (ha : a < 1) : b * a < c | calc b * a < b * 1 : mul_lt_mul_left' ha b
... = b : mul_one b
... ≤ c : hbc | lemma | mul_lt_of_le_of_lt_one | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_mul_left'",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_lt_of_le_one [covariant_class α α (*) (≤)]
{a b c : α} (hbc : b < c) (ha : a ≤ 1) : b * a < c | calc b * a ≤ b * 1 : mul_le_mul_left' ha b
... = b : mul_one b
... < c : hbc | lemma | mul_lt_of_lt_of_le_one | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_mul_left'",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_lt_of_lt_one [covariant_class α α (*) (<)]
{a b c : α} (hbc : b < c) (ha : a < 1) : b * a < c | calc b * a < b * 1 : mul_lt_mul_left' ha b
... = b : mul_one b
... < c : hbc | lemma | mul_lt_of_lt_of_lt_one | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_mul_left'",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_lt_of_lt_one' [covariant_class α α (*) (≤)]
{a b c : α} (hbc : b < c) (ha : a < 1) : b * a < c | mul_lt_of_lt_of_le_one hbc ha.le | lemma | mul_lt_of_lt_of_lt_one' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_of_lt_of_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left.mul_le_one [covariant_class α α (*) (≤)]
{a b : α} (ha : a ≤ 1) (hb : b ≤ 1) : a * b ≤ 1 | mul_le_of_le_of_le_one ha hb | lemma | left.mul_le_one | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_of_le_of_le_one"
] | Assumes left covariance.
The lemma assuming right covariance is `right.mul_le_one`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left.mul_lt_one_of_le_of_lt [covariant_class α α (*) (<)]
{a b : α} (ha : a ≤ 1) (hb : b < 1) : a * b < 1 | mul_lt_of_le_of_lt_one ha hb | lemma | left.mul_lt_one_of_le_of_lt | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_of_le_of_lt_one"
] | Assumes left covariance.
The lemma assuming right covariance is `right.mul_lt_one_of_le_of_lt`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left.mul_lt_one_of_lt_of_le [covariant_class α α (*) (≤)]
{a b : α} (ha : a < 1) (hb : b ≤ 1) : a * b < 1 | mul_lt_of_lt_of_le_one ha hb | lemma | left.mul_lt_one_of_lt_of_le | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_of_lt_of_le_one"
] | Assumes left covariance.
The lemma assuming right covariance is `right.mul_lt_one_of_lt_of_le`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left.mul_lt_one [covariant_class α α (*) (<)]
{a b : α} (ha : a < 1) (hb : b < 1) : a * b < 1 | mul_lt_of_lt_of_lt_one ha hb | lemma | left.mul_lt_one | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_of_lt_of_lt_one"
] | Assumes left covariance.
The lemma assuming right covariance is `right.mul_lt_one`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left.mul_lt_one' [covariant_class α α (*) (≤)]
{a b : α} (ha : a < 1) (hb : b < 1) : a * b < 1 | mul_lt_of_lt_of_lt_one' ha hb | lemma | left.mul_lt_one' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_of_lt_of_lt_one'"
] | Assumes left covariance.
The lemma assuming right covariance is `right.mul_lt_one'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_mul_of_le_of_one_le [covariant_class α α (*) (≤)]
{a b c : α} (hbc : b ≤ c) (ha : 1 ≤ a) : b ≤ c * a | calc b ≤ c : hbc
... = c * 1 : (mul_one c).symm
... ≤ c * a : mul_le_mul_left' ha c | lemma | le_mul_of_le_of_one_le | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_mul_left'",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_le_of_one_lt [covariant_class α α (*) (<)]
{a b c : α} (hbc : b ≤ c) (ha : 1 < a) : b < c * a | calc b ≤ c : hbc
... = c * 1 : (mul_one c).symm
... < c * a : mul_lt_mul_left' ha c | lemma | lt_mul_of_le_of_one_lt | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_mul_left'",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_lt_of_one_le [covariant_class α α (*) (≤)]
{a b c : α} (hbc : b < c) (ha : 1 ≤ a) : b < c * a | calc b < c : hbc
... = c * 1 : (mul_one c).symm
... ≤ c * a : mul_le_mul_left' ha c | lemma | lt_mul_of_lt_of_one_le | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_le_mul_left'",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_lt_of_one_lt [covariant_class α α (*) (<)]
{a b c : α} (hbc : b < c) (ha : 1 < a) : b < c * a | calc b < c : hbc
... = c * 1 : (mul_one c).symm
... < c * a : mul_lt_mul_left' ha c | lemma | lt_mul_of_lt_of_one_lt | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"mul_lt_mul_left'",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_of_lt_of_one_lt' [covariant_class α α (*) (≤)]
{a b c : α} (hbc : b < c) (ha : 1 < a) : b < c * a | lt_mul_of_lt_of_one_le hbc ha.le | lemma | lt_mul_of_lt_of_one_lt' | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"lt_mul_of_lt_of_one_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left.one_le_mul [covariant_class α α (*) (≤)]
{a b : α} (ha : 1 ≤ a) (hb : 1 ≤ b) : 1 ≤ a * b | le_mul_of_le_of_one_le ha hb | lemma | left.one_le_mul | algebra.order.monoid | src/algebra/order/monoid/lemmas.lean | [
"algebra.covariant_and_contravariant",
"order.min_max"
] | [
"covariant_class",
"le_mul_of_le_of_one_le"
] | Assumes left covariance.
The lemma assuming right covariance is `right.one_le_mul`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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