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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mul_right_eq_self₀ : a * b = a ↔ b = 1 ∨ a = 0 | calc a * b = a ↔ a * b = a * 1 : by rw mul_one
... ↔ b = 1 ∨ a = 0 : mul_eq_mul_left_iff | lemma | mul_right_eq_self₀ | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"mul_eq_mul_left_iff",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_eq_self₀ : a * b = b ↔ a = 1 ∨ b = 0 | calc a * b = b ↔ a * b = 1 * b : by rw one_mul
... ↔ a = 1 ∨ b = 0 : mul_eq_mul_right_iff | lemma | mul_left_eq_self₀ | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"mul_eq_mul_right_iff",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_left₀ (ha : a ≠ 0) : a * b = a ↔ b = 1 | by rw [iff.comm, ←mul_right_inj' ha, mul_one] | lemma | mul_eq_left₀ | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_right₀ (hb : b ≠ 0) : a * b = b ↔ a = 1 | by rw [iff.comm, ←mul_left_inj' hb, one_mul] | lemma | mul_eq_right₀ | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_eq_mul₀ (ha : a ≠ 0) : a = a * b ↔ b = 1 | by rw [eq_comm, mul_eq_left₀ ha] | lemma | left_eq_mul₀ | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"mul_eq_left₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_eq_mul₀ (hb : b ≠ 0) : b = a * b ↔ a = 1 | by rw [eq_comm, mul_eq_right₀ hb] | lemma | right_eq_mul₀ | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"mul_eq_right₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_of_mul_eq_self_right (h₁ : b ≠ 1) (h₂ : a * b = a) : a = 0 | classical.by_contradiction $ λ ha, h₁ $ mul_left_cancel₀ ha $ h₂.symm ▸ (mul_one a).symm | theorem | eq_zero_of_mul_eq_self_right | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"mul_left_cancel₀",
"mul_one"
] | An element of a `cancel_monoid_with_zero` fixed by right multiplication by an element other
than one must be zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_zero_of_mul_eq_self_left (h₁ : b ≠ 1) (h₂ : b * a = a) : a = 0 | classical.by_contradiction $ λ ha, h₁ $ mul_right_cancel₀ ha $ h₂.symm ▸ (one_mul a).symm | theorem | eq_zero_of_mul_eq_self_left | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"mul_right_cancel₀",
"one_mul"
] | An element of a `cancel_monoid_with_zero` fixed by left multiplication by an element other
than one must be zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_inv_cancel_right₀ (h : b ≠ 0) (a : G₀) :
(a * b) * b⁻¹ = a | calc (a * b) * b⁻¹ = a * (b * b⁻¹) : mul_assoc _ _ _
... = a : by simp [h] | lemma | mul_inv_cancel_right₀ | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_cancel_left₀ (h : a ≠ 0) (b : G₀) :
a * (a⁻¹ * b) = b | calc a * (a⁻¹ * b) = (a * a⁻¹) * b : (mul_assoc _ _ _).symm
... = b : by simp [h] | lemma | mul_inv_cancel_left₀ | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_ne_zero (h : a ≠ 0) : a⁻¹ ≠ 0 | assume a_eq_0, by simpa [a_eq_0] using mul_inv_cancel h | lemma | inv_ne_zero | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"mul_inv_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_cancel (h : a ≠ 0) : a⁻¹ * a = 1 | calc a⁻¹ * a = (a⁻¹ * a) * a⁻¹ * a⁻¹⁻¹ : by simp [inv_ne_zero h]
... = a⁻¹ * a⁻¹⁻¹ : by simp [h]
... = 1 : by simp [inv_ne_zero h] | lemma | inv_mul_cancel | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"inv_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group_with_zero.mul_left_injective (h : x ≠ 0) :
function.injective (λ y, x * y) | λ y y' w, by simpa only [←mul_assoc, inv_mul_cancel h, one_mul] using congr_arg (λ y, x⁻¹ * y) w | lemma | group_with_zero.mul_left_injective | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"inv_mul_cancel",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group_with_zero.mul_right_injective (h : x ≠ 0) :
function.injective (λ y, y * x) | λ y y' w, by simpa only [mul_assoc, mul_inv_cancel h, mul_one] using congr_arg (λ y, y * x⁻¹) w | lemma | group_with_zero.mul_right_injective | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"mul_assoc",
"mul_inv_cancel",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_cancel_right₀ (h : b ≠ 0) (a : G₀) :
(a * b⁻¹) * b = a | calc (a * b⁻¹) * b = a * (b⁻¹ * b) : mul_assoc _ _ _
... = a : by simp [h] | lemma | inv_mul_cancel_right₀ | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_cancel_left₀ (h : a ≠ 0) (b : G₀) :
a⁻¹ * (a * b) = b | calc a⁻¹ * (a * b) = (a⁻¹ * a) * b : (mul_assoc _ _ _).symm
... = b : by simp [h] | lemma | inv_mul_cancel_left₀ | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_of_mul (h : a * b = 1) : a⁻¹ = b | by rw [← inv_mul_cancel_left₀ (left_ne_zero_of_mul_eq_one h) b, h, mul_one] | lemma | inv_eq_of_mul | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"inv_mul_cancel_left₀",
"left_ne_zero_of_mul_eq_one",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group_with_zero.to_division_monoid : division_monoid G₀ | { inv := has_inv.inv,
inv_inv := λ a, begin
by_cases h : a = 0,
{ simp [h] },
{ exact left_inv_eq_right_inv (inv_mul_cancel $ inv_ne_zero h) (inv_mul_cancel h) }
end,
mul_inv_rev := λ a b, begin
by_cases ha : a = 0, { simp [ha] },
by_cases hb : b = 0, { simp [hb] },
refine inv_eq_of_mul _,... | instance | group_with_zero.to_division_monoid | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"division_monoid",
"inv_eq_of_mul",
"inv_inv",
"inv_mul_cancel",
"inv_ne_zero",
"left_inv_eq_right_inv",
"mul_assoc",
"mul_inv_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group_with_zero.to_cancel_monoid_with_zero : cancel_monoid_with_zero G₀ | { mul_left_cancel_of_ne_zero := λ x y z hx h,
by rw [← inv_mul_cancel_left₀ hx y, h, inv_mul_cancel_left₀ hx z],
mul_right_cancel_of_ne_zero := λ x y z hy h,
by rw [← mul_inv_cancel_right₀ hy x, h, mul_inv_cancel_right₀ hy z],
..‹group_with_zero G₀› } | instance | group_with_zero.to_cancel_monoid_with_zero | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"cancel_monoid_with_zero",
"inv_mul_cancel_left₀",
"mul_inv_cancel_right₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_div (a : G₀) : 0 / a = 0 | by rw [div_eq_mul_inv, zero_mul] | lemma | zero_div | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"div_eq_mul_inv",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_zero (a : G₀) : a / 0 = 0 | by rw [div_eq_mul_inv, inv_zero, mul_zero] | lemma | div_zero | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"div_eq_mul_inv",
"inv_zero",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_self_mul_inv (a : G₀) : a * a * a⁻¹ = a | begin
by_cases h : a = 0,
{ rw [h, inv_zero, mul_zero] },
{ rw [mul_assoc, mul_inv_cancel h, mul_one] }
end | lemma | mul_self_mul_inv | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"inv_zero",
"mul_assoc",
"mul_inv_cancel",
"mul_one",
"mul_zero"
] | Multiplying `a` by itself and then by its inverse results in `a`
(whether or not `a` is zero). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_inv_mul_self (a : G₀) : a * a⁻¹ * a = a | begin
by_cases h : a = 0,
{ rw [h, inv_zero, mul_zero] },
{ rw [mul_inv_cancel h, one_mul] }
end | lemma | mul_inv_mul_self | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"inv_zero",
"mul_inv_cancel",
"mul_zero",
"one_mul"
] | Multiplying `a` by its inverse and then by itself results in `a`
(whether or not `a` is zero). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_mul_mul_self (a : G₀) : a⁻¹ * a * a = a | begin
by_cases h : a = 0,
{ rw [h, inv_zero, mul_zero] },
{ rw [inv_mul_cancel h, one_mul] }
end | lemma | inv_mul_mul_self | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"inv_mul_cancel",
"inv_zero",
"mul_zero",
"one_mul"
] | Multiplying `a⁻¹` by `a` twice results in `a` (whether or not `a`
is zero). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_self_div_self (a : G₀) : a * a / a = a | by rw [div_eq_mul_inv, mul_self_mul_inv a] | lemma | mul_self_div_self | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"div_eq_mul_inv",
"mul_self_mul_inv"
] | Multiplying `a` by itself and then dividing by itself results in `a`, whether or not `a` is
zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
div_self_mul_self (a : G₀) : a / a * a = a | by rw [div_eq_mul_inv, mul_inv_mul_self a] | lemma | div_self_mul_self | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"div_eq_mul_inv",
"mul_inv_mul_self"
] | Dividing `a` by itself and then multiplying by itself results in `a`, whether or not `a` is
zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
div_self_mul_self' (a : G₀) : a / (a * a) = a⁻¹ | calc a / (a * a) = a⁻¹⁻¹ * a⁻¹ * a⁻¹ : by simp [mul_inv_rev]
... = a⁻¹ : inv_mul_mul_self _ | lemma | div_self_mul_self' | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"inv_mul_mul_self",
"mul_inv_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_ne_zero {a : G₀} (h : a ≠ 0) : 1 / a ≠ 0 | by simpa only [one_div] using inv_ne_zero h | lemma | one_div_ne_zero | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"inv_ne_zero",
"one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_zero {a : G₀} : a⁻¹ = 0 ↔ a = 0 | by rw [inv_eq_iff_eq_inv, inv_zero] | lemma | inv_eq_zero | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"inv_eq_iff_eq_inv",
"inv_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_eq_inv {a : G₀} : 0 = a⁻¹ ↔ 0 = a | eq_comm.trans $ inv_eq_zero.trans eq_comm | lemma | zero_eq_inv | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_div_self (a : G₀) : a / (a / a) = a | begin
rw div_div_eq_mul_div,
exact mul_self_div_self a
end | lemma | div_div_self | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"div_div_eq_mul_div",
"mul_self_div_self"
] | Dividing `a` by the result of dividing `a` by itself results in
`a` (whether or not `a` is zero). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ne_zero_of_one_div_ne_zero {a : G₀} (h : 1 / a ≠ 0) : a ≠ 0 | assume ha : a = 0, begin rw [ha, div_zero] at h, contradiction end | lemma | ne_zero_of_one_div_ne_zero | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"div_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_of_one_div_eq_zero {a : G₀} (h : 1 / a = 0) : a = 0 | classical.by_cases
(assume ha, ha)
(assume ha, ((one_div_ne_zero ha) h).elim) | lemma | eq_zero_of_one_div_eq_zero | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"one_div_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_surjective₀ {a : G₀} (h : a ≠ 0) : surjective (λ g, a * g) | λ g, ⟨a⁻¹ * g, by simp [← mul_assoc, mul_inv_cancel h]⟩ | lemma | mul_left_surjective₀ | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"mul_assoc",
"mul_inv_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_surjective₀ {a : G₀} (h : a ≠ 0) : surjective (λ g, g * a) | λ g, ⟨g * a⁻¹, by simp [mul_assoc, inv_mul_cancel h]⟩ | lemma | mul_right_surjective₀ | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"inv_mul_cancel",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul_eq_mul_div₀ (a b c : G₀) : (a / c) * b = a * b / c | by simp_rw [div_eq_mul_inv, mul_assoc, mul_comm c⁻¹] | lemma | div_mul_eq_mul_div₀ | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"div_eq_mul_inv",
"mul_assoc",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inverse_rev' {a b : M₀} (h : commute a b) : inverse (a * b) = inverse b * inverse a | begin
by_cases hab : is_unit (a * b),
{ obtain ⟨⟨a, rfl⟩, b, rfl⟩ := h.is_unit_mul_iff.mp hab,
rw [←units.coe_mul, inverse_unit, inverse_unit, inverse_unit, ←units.coe_mul,
mul_inv_rev], },
obtain ha | hb := not_and_distrib.mp (mt h.is_unit_mul_iff.mpr hab),
{ rw [inverse_non_unit _ hab, inverse_non_u... | lemma | ring.mul_inverse_rev' | algebra.group_with_zero | src/algebra/group_with_zero/commute.lean | [
"algebra.group_with_zero.semiconj",
"algebra.group.commute",
"tactic.nontriviality"
] | [
"commute",
"is_unit",
"mul_inv_rev",
"mul_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inverse_rev {M₀} [comm_monoid_with_zero M₀] (a b : M₀) :
ring.inverse (a * b) = inverse b * inverse a | mul_inverse_rev' (commute.all _ _) | lemma | ring.mul_inverse_rev | algebra.group_with_zero | src/algebra/group_with_zero/commute.lean | [
"algebra.group_with_zero.semiconj",
"algebra.group.commute",
"tactic.nontriviality"
] | [
"comm_monoid_with_zero",
"commute.all",
"ring.inverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.ring_inverse_ring_inverse {a b : M₀} (h : commute a b) :
commute (ring.inverse a) (ring.inverse b) | (ring.mul_inverse_rev' h.symm).symm.trans $ (congr_arg _ h.symm.eq).trans $ ring.mul_inverse_rev' h | lemma | commute.ring_inverse_ring_inverse | algebra.group_with_zero | src/algebra/group_with_zero/commute.lean | [
"algebra.group_with_zero.semiconj",
"algebra.group.commute",
"tactic.nontriviality"
] | [
"commute",
"ring.inverse",
"ring.mul_inverse_rev'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_right [mul_zero_class G₀] (a : G₀) : commute a 0 | semiconj_by.zero_right a | theorem | commute.zero_right | algebra.group_with_zero | src/algebra/group_with_zero/commute.lean | [
"algebra.group_with_zero.semiconj",
"algebra.group.commute",
"tactic.nontriviality"
] | [
"commute",
"mul_zero_class",
"semiconj_by.zero_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_left [mul_zero_class G₀] (a : G₀) : commute 0 a | semiconj_by.zero_left a a | theorem | commute.zero_left | algebra.group_with_zero | src/algebra/group_with_zero/commute.lean | [
"algebra.group_with_zero.semiconj",
"algebra.group.commute",
"tactic.nontriviality"
] | [
"commute",
"mul_zero_class",
"semiconj_by.zero_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_left_iff₀ : commute a⁻¹ b ↔ commute a b | semiconj_by.inv_symm_left_iff₀ | theorem | commute.inv_left_iff₀ | algebra.group_with_zero | src/algebra/group_with_zero/commute.lean | [
"algebra.group_with_zero.semiconj",
"algebra.group.commute",
"tactic.nontriviality"
] | [
"commute",
"semiconj_by.inv_symm_left_iff₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_left₀ (h : commute a b) : commute a⁻¹ b | inv_left_iff₀.2 h | theorem | commute.inv_left₀ | algebra.group_with_zero | src/algebra/group_with_zero/commute.lean | [
"algebra.group_with_zero.semiconj",
"algebra.group.commute",
"tactic.nontriviality"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_right_iff₀ : commute a b⁻¹ ↔ commute a b | semiconj_by.inv_right_iff₀ | theorem | commute.inv_right_iff₀ | algebra.group_with_zero | src/algebra/group_with_zero/commute.lean | [
"algebra.group_with_zero.semiconj",
"algebra.group.commute",
"tactic.nontriviality"
] | [
"commute",
"semiconj_by.inv_right_iff₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_right₀ (h : commute a b) : commute a b⁻¹ | inv_right_iff₀.2 h | theorem | commute.inv_right₀ | algebra.group_with_zero | src/algebra/group_with_zero/commute.lean | [
"algebra.group_with_zero.semiconj",
"algebra.group.commute",
"tactic.nontriviality"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_right (hab : commute a b) (hac : commute a c) :
commute a (b / c) | hab.div_right hac | theorem | commute.div_right | algebra.group_with_zero | src/algebra/group_with_zero/commute.lean | [
"algebra.group_with_zero.semiconj",
"algebra.group.commute",
"tactic.nontriviality"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_left (hac : commute a c) (hbc : commute b c) :
commute (a / b) c | by { rw div_eq_mul_inv, exact hac.mul_left hbc.inv_left₀ } | theorem | commute.div_left | algebra.group_with_zero | src/algebra/group_with_zero/commute.lean | [
"algebra.group_with_zero.semiconj",
"algebra.group.commute",
"tactic.nontriviality"
] | [
"commute",
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_zero_class (M₀ : Type*) extends has_mul M₀, has_zero M₀ | (zero_mul : ∀ a : M₀, 0 * a = 0)
(mul_zero : ∀ a : M₀, a * 0 = 0) | class | mul_zero_class | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"mul_zero",
"zero_mul"
] | Typeclass for expressing that a type `M₀` with multiplication and a zero satisfies
`0 * a = 0` and `a * 0 = 0` for all `a : M₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_mul (a : M₀) : 0 * a = 0 | mul_zero_class.zero_mul a | lemma | zero_mul | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_zero (a : M₀) : a * 0 = 0 | mul_zero_class.mul_zero a | lemma | mul_zero | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_left_cancel_mul_zero (M₀ : Type u) [has_mul M₀] [has_zero M₀] : Prop | (mul_left_cancel_of_ne_zero : ∀ {a b c : M₀}, a ≠ 0 → a * b = a * c → b = c) | class | is_left_cancel_mul_zero | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [] | A mixin for left cancellative multiplication by nonzero elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_left_cancel₀ (ha : a ≠ 0) (h : a * b = a * c) : b = c | is_left_cancel_mul_zero.mul_left_cancel_of_ne_zero ha h | lemma | mul_left_cancel₀ | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_injective₀ (ha : a ≠ 0) : function.injective ((*) a) | λ b c, mul_left_cancel₀ ha | lemma | mul_right_injective₀ | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"mul_left_cancel₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_right_cancel_mul_zero (M₀ : Type u) [has_mul M₀] [has_zero M₀] : Prop | (mul_right_cancel_of_ne_zero : ∀ {a b c : M₀}, b ≠ 0 → a * b = c * b → a = c) | class | is_right_cancel_mul_zero | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [] | A mixin for right cancellative multiplication by nonzero elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_right_cancel₀ (hb : b ≠ 0) (h : a * b = c * b) : a = c | is_right_cancel_mul_zero.mul_right_cancel_of_ne_zero hb h | lemma | mul_right_cancel₀ | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_injective₀ (hb : b ≠ 0) : function.injective (λ a, a * b) | λ a c, mul_right_cancel₀ hb | lemma | mul_left_injective₀ | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"mul_right_cancel₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_cancel_mul_zero (M₀ : Type u) [has_mul M₀] [has_zero M₀]
extends is_left_cancel_mul_zero M₀, is_right_cancel_mul_zero M₀ : Prop | class | is_cancel_mul_zero | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"is_left_cancel_mul_zero",
"is_right_cancel_mul_zero"
] | A mixin for cancellative multiplication by nonzero elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_left_cancel_mul_zero.to_is_right_cancel_mul_zero [is_left_cancel_mul_zero M₀] :
is_right_cancel_mul_zero M₀ | ⟨λ a b c ha h, mul_left_cancel₀ ha $ (mul_comm _ _).trans $ (h.trans (mul_comm _ _))⟩ | lemma | is_left_cancel_mul_zero.to_is_right_cancel_mul_zero | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"is_left_cancel_mul_zero",
"is_right_cancel_mul_zero",
"mul_comm",
"mul_left_cancel₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_right_cancel_mul_zero.to_is_left_cancel_mul_zero [is_right_cancel_mul_zero M₀] :
is_left_cancel_mul_zero M₀ | ⟨λ a b c ha h, mul_right_cancel₀ ha $ (mul_comm _ _).trans $ (h.trans (mul_comm _ _))⟩ | lemma | is_right_cancel_mul_zero.to_is_left_cancel_mul_zero | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"is_left_cancel_mul_zero",
"is_right_cancel_mul_zero",
"mul_comm",
"mul_right_cancel₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_left_cancel_mul_zero.to_is_cancel_mul_zero [is_left_cancel_mul_zero M₀] :
is_cancel_mul_zero M₀ | { .. ‹is_left_cancel_mul_zero M₀›, .. is_left_cancel_mul_zero.to_is_right_cancel_mul_zero } | lemma | is_left_cancel_mul_zero.to_is_cancel_mul_zero | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"is_cancel_mul_zero",
"is_left_cancel_mul_zero",
"is_left_cancel_mul_zero.to_is_right_cancel_mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_right_cancel_mul_zero.to_is_cancel_mul_zero [is_right_cancel_mul_zero M₀] :
is_cancel_mul_zero M₀ | { .. ‹is_right_cancel_mul_zero M₀›, .. is_right_cancel_mul_zero.to_is_left_cancel_mul_zero } | lemma | is_right_cancel_mul_zero.to_is_cancel_mul_zero | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"is_cancel_mul_zero",
"is_right_cancel_mul_zero",
"is_right_cancel_mul_zero.to_is_left_cancel_mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
no_zero_divisors (M₀ : Type*) [has_mul M₀] [has_zero M₀] : Prop | (eq_zero_or_eq_zero_of_mul_eq_zero : ∀ {a b : M₀}, a * b = 0 → a = 0 ∨ b = 0) | class | no_zero_divisors | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [] | Predicate typeclass for expressing that `a * b = 0` implies `a = 0` or `b = 0`
for all `a` and `b` of type `G₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
semigroup_with_zero (S₀ : Type*) extends semigroup S₀, mul_zero_class S₀. | class | semigroup_with_zero | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"mul_zero_class",
"semigroup"
] | A type `S₀` is a "semigroup with zero” if it is a semigroup with zero element, and `0` is left
and right absorbing. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_zero_one_class (M₀ : Type*) extends mul_one_class M₀, mul_zero_class M₀. | class | mul_zero_one_class | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"mul_one_class",
"mul_zero_class"
] | A typeclass for non-associative monoids with zero elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_with_zero (M₀ : Type*) extends monoid M₀, mul_zero_one_class M₀. | class | monoid_with_zero | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"monoid",
"mul_zero_one_class"
] | A type `M₀` is a “monoid with zero” if it is a monoid with zero element, and `0` is left
and right absorbing. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_with_zero.to_semigroup_with_zero (M₀ : Type*) [monoid_with_zero M₀] :
semigroup_with_zero M₀ | { ..‹monoid_with_zero M₀› } | instance | monoid_with_zero.to_semigroup_with_zero | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"monoid_with_zero",
"semigroup_with_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_monoid_with_zero (M₀ : Type*) extends monoid_with_zero M₀ | (mul_left_cancel_of_ne_zero : ∀ {a b c : M₀}, a ≠ 0 → a * b = a * c → b = c)
(mul_right_cancel_of_ne_zero : ∀ {a b c : M₀}, b ≠ 0 → a * b = c * b → a = c) | class | cancel_monoid_with_zero | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"monoid_with_zero"
] | A type `M` is a `cancel_monoid_with_zero` if it is a monoid with zero element, `0` is left
and right absorbing, and left/right multiplication by a non-zero element is injective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cancel_monoid_with_zero.to_is_cancel_mul_zero [cancel_monoid_with_zero M₀] :
is_cancel_mul_zero M₀ | { .. ‹cancel_monoid_with_zero M₀› } | instance | cancel_monoid_with_zero.to_is_cancel_mul_zero | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"cancel_monoid_with_zero",
"is_cancel_mul_zero"
] | A `cancel_monoid_with_zero` satisfies `is_cancel_mul_zero`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comm_monoid_with_zero (M₀ : Type*) extends comm_monoid M₀, monoid_with_zero M₀. | class | comm_monoid_with_zero | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"comm_monoid",
"monoid_with_zero"
] | A type `M` is a commutative “monoid with zero” if it is a commutative monoid with zero
element, and `0` is left and right absorbing. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_comm_monoid_with_zero (M₀ : Type*) extends comm_monoid_with_zero M₀ | (mul_left_cancel_of_ne_zero : ∀ {a b c : M₀}, a ≠ 0 → a * b = a * c → b = c) | class | cancel_comm_monoid_with_zero | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"comm_monoid_with_zero"
] | A type `M` is a `cancel_comm_monoid_with_zero` if it is a commutative monoid with zero element,
`0` is left and right absorbing,
and left/right multiplication by a non-zero element is injective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cancel_comm_monoid_with_zero.to_cancel_monoid_with_zero
[h : cancel_comm_monoid_with_zero M₀] : cancel_monoid_with_zero M₀ | { .. h, .. @is_left_cancel_mul_zero.to_is_right_cancel_mul_zero M₀ _ _ { .. h } } | instance | cancel_comm_monoid_with_zero.to_cancel_monoid_with_zero | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"cancel_comm_monoid_with_zero",
"cancel_monoid_with_zero",
"is_left_cancel_mul_zero.to_is_right_cancel_mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group_with_zero (G₀ : Type u) extends
monoid_with_zero G₀, div_inv_monoid G₀, nontrivial G₀ | (inv_zero : (0 : G₀)⁻¹ = 0)
(mul_inv_cancel : ∀ a:G₀, a ≠ 0 → a * a⁻¹ = 1) | class | group_with_zero | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"div_inv_monoid",
"inv_zero",
"monoid_with_zero",
"mul_inv_cancel",
"nontrivial"
] | A type `G₀` is a “group with zero” if it is a monoid with zero element (distinct from `1`)
such that every nonzero element is invertible.
The type is required to come with an “inverse” function, and the inverse of `0` must be `0`.
Examples include division rings and the ordered monoids that are the
target of valuation... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_zero : (0 : G₀)⁻¹ = 0 | group_with_zero.inv_zero | lemma | inv_zero | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_cancel {a : G₀} (h : a ≠ 0) : a * a⁻¹ = 1 | group_with_zero.mul_inv_cancel a h | lemma | mul_inv_cancel | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_group_with_zero (G₀ : Type*) extends comm_monoid_with_zero G₀, group_with_zero G₀. | class | comm_group_with_zero | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"comm_monoid_with_zero",
"group_with_zero"
] | A type `G₀` is a commutative “group with zero”
if it is a commutative monoid with zero element (distinct from `1`)
such that every nonzero element is invertible.
The type is required to come with an “inverse” function, and the inverse of `0` must be `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero.one : ne_zero (1 : M₀) | ⟨begin
assume h,
rcases exists_pair_ne M₀ with ⟨x, y, hx⟩,
apply hx,
calc x = 1 * x : by rw [one_mul]
... = 0 : by rw [h, zero_mul]
... = 1 * y : by rw [h, zero_mul]
... = y : by rw [one_mul]
end⟩ | instance | ne_zero.one | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"exists_pair_ne",
"ne_zero",
"one_mul",
"zero_mul"
] | In a nontrivial monoid with zero, zero and one are different. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_nonzero [has_zero M₀'] [has_one M₀']
(f : M₀' → M₀) (zero : f 0 = 0) (one : f 1 = 1) : nontrivial M₀' | ⟨⟨0, 1, mt (congr_arg f) $ by { rw [zero, one], exact zero_ne_one }⟩⟩ | lemma | pullback_nonzero | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"nontrivial",
"zero_ne_one"
] | Pullback a `nontrivial` instance along a function sending `0` to `0` and `1` to `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_eq_zero_of_left {a : M₀} (h : a = 0) (b : M₀) : a * b = 0 | h.symm ▸ zero_mul b | lemma | mul_eq_zero_of_left | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_zero_of_right (a : M₀) {b : M₀} (h : b = 0) : a * b = 0 | h.symm ▸ mul_zero a | lemma | mul_eq_zero_of_right | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_zero : a * b = 0 ↔ a = 0 ∨ b = 0 | ⟨eq_zero_or_eq_zero_of_mul_eq_zero,
λo, o.elim (λ h, mul_eq_zero_of_left h b) (mul_eq_zero_of_right a)⟩ | theorem | mul_eq_zero | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"mul_eq_zero_of_left",
"mul_eq_zero_of_right"
] | If `α` has no zero divisors, then the product of two elements equals zero iff one of them
equals zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_eq_mul : 0 = a * b ↔ a = 0 ∨ b = 0 | by rw [eq_comm, mul_eq_zero] | theorem | zero_eq_mul | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"mul_eq_zero"
] | If `α` has no zero divisors, then the product of two elements equals zero iff one of them
equals zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_ne_zero_iff : a * b ≠ 0 ↔ a ≠ 0 ∧ b ≠ 0 | mul_eq_zero.not.trans not_or_distrib | theorem | mul_ne_zero_iff | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [
"not_or_distrib"
] | If `α` has no zero divisors, then the product of two elements is nonzero iff both of them
are nonzero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_eq_zero_comm : a * b = 0 ↔ b * a = 0 | mul_eq_zero.trans $ (or_comm _ _).trans mul_eq_zero.symm | theorem | mul_eq_zero_comm | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [] | If `α` has no zero divisors, then for elements `a, b : α`, `a * b` equals zero iff so is
`b * a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_ne_zero_comm : a * b ≠ 0 ↔ b * a ≠ 0 | mul_eq_zero_comm.not | theorem | mul_ne_zero_comm | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [] | If `α` has no zero divisors, then for elements `a, b : α`, `a * b` is nonzero iff so is
`b * a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_self_eq_zero : a * a = 0 ↔ a = 0 | by simp | lemma | mul_self_eq_zero | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_eq_mul_self : 0 = a * a ↔ a = 0 | by simp | lemma | zero_eq_mul_self | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_self_ne_zero : a * a ≠ 0 ↔ a ≠ 0 | mul_self_eq_zero.not | lemma | mul_self_ne_zero | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_ne_mul_self : 0 ≠ a * a ↔ a ≠ 0 | zero_eq_mul_self.not | lemma | zero_ne_mul_self | algebra.group_with_zero | src/algebra/group_with_zero/defs.lean | [
"algebra.group.defs",
"logic.nontrivial",
"algebra.ne_zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_of_zero_dvd (h : 0 ∣ a) : a = 0 | dvd.elim h (λ c H', H'.trans (zero_mul c)) | theorem | eq_zero_of_zero_dvd | algebra.group_with_zero | src/algebra/group_with_zero/divisibility.lean | [
"algebra.group_with_zero.basic",
"algebra.divisibility.units"
] | [
"dvd.elim",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_dvd_iff : 0 ∣ a ↔ a = 0 | ⟨eq_zero_of_zero_dvd, λ h, by { rw h, use 0, simp }⟩ | lemma | zero_dvd_iff | algebra.group_with_zero | src/algebra/group_with_zero/divisibility.lean | [
"algebra.group_with_zero.basic",
"algebra.divisibility.units"
] | [] | Given an element `a` of a commutative semigroup with zero, there exists another element whose
product with zero equals `a` iff `a` equals zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_zero (a : α) : a ∣ 0 | dvd.intro 0 (by simp) | theorem | dvd_zero | algebra.group_with_zero | src/algebra/group_with_zero/divisibility.lean | [
"algebra.group_with_zero.basic",
"algebra.divisibility.units"
] | [
"dvd.intro"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_dvd_mul_iff_left [cancel_monoid_with_zero α] {a b c : α}
(ha : a ≠ 0) : a * b ∣ a * c ↔ b ∣ c | exists_congr $ λ d, by rw [mul_assoc, mul_right_inj' ha] | theorem | mul_dvd_mul_iff_left | algebra.group_with_zero | src/algebra/group_with_zero/divisibility.lean | [
"algebra.group_with_zero.basic",
"algebra.divisibility.units"
] | [
"cancel_monoid_with_zero",
"mul_assoc",
"mul_right_inj'"
] | Given two elements `b`, `c` of a `cancel_monoid_with_zero` and a nonzero element `a`,
`a*b` divides `a*c` iff `b` divides `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_dvd_mul_iff_right [cancel_comm_monoid_with_zero α] {a b c : α} (hc : c ≠ 0) :
a * c ∣ b * c ↔ a ∣ b | exists_congr $ λ d, by rw [mul_right_comm, mul_left_inj' hc] | theorem | mul_dvd_mul_iff_right | algebra.group_with_zero | src/algebra/group_with_zero/divisibility.lean | [
"algebra.group_with_zero.basic",
"algebra.divisibility.units"
] | [
"cancel_comm_monoid_with_zero",
"mul_left_inj'",
"mul_right_comm"
] | Given two elements `a`, `b` of a commutative `cancel_monoid_with_zero` and a nonzero
element `c`, `a*c` divides `b*c` iff `a` divides `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_not_unit (a b : α) : Prop | a ≠ 0 ∧ ∃ x, ¬is_unit x ∧ b = a * x | def | dvd_not_unit | algebra.group_with_zero | src/algebra/group_with_zero/divisibility.lean | [
"algebra.group_with_zero.basic",
"algebra.divisibility.units"
] | [
"is_unit"
] | `dvd_not_unit a b` expresses that `a` divides `b` "strictly", i.e. that `b` divided by `a`
is not a unit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_not_unit_of_dvd_of_not_dvd {a b : α} (hd : a ∣ b) (hnd : ¬ b ∣ a) :
dvd_not_unit a b | begin
split,
{ rintro rfl, exact hnd (dvd_zero _) },
{ rcases hd with ⟨c, rfl⟩,
refine ⟨c, _, rfl⟩,
rintro ⟨u, rfl⟩,
simpa using hnd }
end | lemma | dvd_not_unit_of_dvd_of_not_dvd | algebra.group_with_zero | src/algebra/group_with_zero/divisibility.lean | [
"algebra.group_with_zero.basic",
"algebra.divisibility.units"
] | [
"dvd_not_unit",
"dvd_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_and_not_dvd_iff [cancel_comm_monoid_with_zero α] {x y : α} :
x ∣ y ∧ ¬y ∣ x ↔ dvd_not_unit x y | ⟨λ ⟨⟨d, hd⟩, hyx⟩, ⟨λ hx0, by simpa [hx0] using hyx, ⟨d,
mt is_unit_iff_dvd_one.1 (λ ⟨e, he⟩, hyx ⟨e, by rw [hd, mul_assoc, ← he, mul_one]⟩), hd⟩⟩,
λ ⟨hx0, d, hdu, hdx⟩, ⟨⟨d, hdx⟩, λ ⟨e, he⟩, hdu (is_unit_of_dvd_one _
⟨e, mul_left_cancel₀ hx0 $ by conv {to_lhs, rw [he, hdx]};simp [mul_assoc]⟩)⟩⟩ | lemma | dvd_and_not_dvd_iff | algebra.group_with_zero | src/algebra/group_with_zero/divisibility.lean | [
"algebra.group_with_zero.basic",
"algebra.divisibility.units"
] | [
"cancel_comm_monoid_with_zero",
"dvd_not_unit",
"is_unit_of_dvd_one",
"mul_assoc",
"mul_left_cancel₀",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero_of_dvd_ne_zero {p q : α} (h₁ : q ≠ 0)
(h₂ : p ∣ q) : p ≠ 0 | begin
rcases h₂ with ⟨u, rfl⟩,
exact left_ne_zero_of_mul h₁,
end | theorem | ne_zero_of_dvd_ne_zero | algebra.group_with_zero | src/algebra/group_with_zero/divisibility.lean | [
"algebra.group_with_zero.basic",
"algebra.divisibility.units"
] | [
"left_ne_zero_of_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_antisymm : a ∣ b → b ∣ a → a = b | begin
rintro ⟨c, rfl⟩ ⟨d, hcd⟩,
rw [mul_assoc, eq_comm, mul_right_eq_self₀, mul_eq_one] at hcd,
obtain ⟨rfl, -⟩ | rfl := hcd; simp,
end | lemma | dvd_antisymm | algebra.group_with_zero | src/algebra/group_with_zero/divisibility.lean | [
"algebra.group_with_zero.basic",
"algebra.divisibility.units"
] | [
"mul_assoc",
"mul_eq_one",
"mul_right_eq_self₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_antisymm' : a ∣ b → b ∣ a → b = a | flip dvd_antisymm | lemma | dvd_antisymm' | algebra.group_with_zero | src/algebra/group_with_zero/divisibility.lean | [
"algebra.group_with_zero.basic",
"algebra.divisibility.units"
] | [
"dvd_antisymm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_forall_dvd (h : ∀ c, a ∣ c ↔ b ∣ c) : a = b | ((h _).2 dvd_rfl).antisymm $ (h _).1 dvd_rfl | lemma | eq_of_forall_dvd | algebra.group_with_zero | src/algebra/group_with_zero/divisibility.lean | [
"algebra.group_with_zero.basic",
"algebra.divisibility.units"
] | [
"dvd_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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