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
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|---|---|---|---|---|---|---|---|---|---|---|
comm_group_with_zero_of_is_unit_or_eq_zero [hM : comm_monoid_with_zero M]
(h : ∀ (a : M), is_unit a ∨ a = 0) : comm_group_with_zero M | { .. (group_with_zero_of_is_unit_or_eq_zero h), .. hM } | def | comm_group_with_zero_of_is_unit_or_eq_zero | algebra.group_with_zero.units | src/algebra/group_with_zero/units/basic.lean | [
"algebra.group_with_zero.basic",
"algebra.group.units",
"tactic.nontriviality",
"tactic.assert_exists"
] | [
"comm_group_with_zero",
"comm_monoid_with_zero",
"group_with_zero_of_is_unit_or_eq_zero",
"is_unit"
] | Constructs a `comm_group_with_zero` structure on a `comm_monoid_with_zero`
consisting only of units and 0. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
div_self (h : a ≠ 0) : a / a = 1 | h.is_unit.div_self | lemma | div_self | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_mul_inv_iff_mul_eq₀ (hc : c ≠ 0) : a = b * c⁻¹ ↔ a * c = b | hc.is_unit.eq_mul_inv_iff_mul_eq | lemma | eq_mul_inv_iff_mul_eq₀ | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_inv_mul_iff_mul_eq₀ (hb : b ≠ 0) : a = b⁻¹ * c ↔ b * a = c | hb.is_unit.eq_inv_mul_iff_mul_eq | lemma | eq_inv_mul_iff_mul_eq₀ | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_eq_iff_eq_mul₀ (ha : a ≠ 0) : a⁻¹ * b = c ↔ b = a * c | ha.is_unit.inv_mul_eq_iff_eq_mul | lemma | inv_mul_eq_iff_eq_mul₀ | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_eq_iff_eq_mul₀ (hb : b ≠ 0) : a * b⁻¹ = c ↔ a = c * b | hb.is_unit.mul_inv_eq_iff_eq_mul | lemma | mul_inv_eq_iff_eq_mul₀ | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_eq_one₀ (hb : b ≠ 0) : a * b⁻¹ = 1 ↔ a = b | hb.is_unit.mul_inv_eq_one | lemma | mul_inv_eq_one₀ | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_eq_one₀ (ha : a ≠ 0) : a⁻¹ * b = 1 ↔ a = b | ha.is_unit.inv_mul_eq_one | lemma | inv_mul_eq_one₀ | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_one_iff_eq_inv₀ (hb : b ≠ 0) : a * b = 1 ↔ a = b⁻¹ | hb.is_unit.mul_eq_one_iff_eq_inv | lemma | mul_eq_one_iff_eq_inv₀ | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_one_iff_inv_eq₀ (ha : a ≠ 0) : a * b = 1 ↔ a⁻¹ = b | ha.is_unit.mul_eq_one_iff_inv_eq | lemma | mul_eq_one_iff_inv_eq₀ | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul_cancel (a : G₀) (h : b ≠ 0) : a / b * b = a | h.is_unit.div_mul_cancel _ | lemma | div_mul_cancel | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_cancel (a : G₀) (h : b ≠ 0) : a * b / b = a | h.is_unit.mul_div_cancel _ | lemma | mul_div_cancel | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_one_div_cancel (h : a ≠ 0) : a * (1 / a) = 1 | h.is_unit.mul_one_div_cancel | lemma | mul_one_div_cancel | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_mul_cancel (h : a ≠ 0) : (1 / a) * a = 1 | h.is_unit.one_div_mul_cancel | lemma | one_div_mul_cancel | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_left_inj' (hc : c ≠ 0) : a / c = b / c ↔ a = b | hc.is_unit.div_left_inj | lemma | div_left_inj' | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_iff (hb : b ≠ 0) : a / b = c ↔ a = c * b | hb.is_unit.div_eq_iff | lemma | div_eq_iff | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_div_iff (hb : b ≠ 0) : c = a / b ↔ c * b = a | hb.is_unit.eq_div_iff | lemma | eq_div_iff | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_iff_mul_eq (hb : b ≠ 0) : a / b = c ↔ c * b = a | hb.is_unit.div_eq_iff.trans eq_comm | lemma | div_eq_iff_mul_eq | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_div_iff_mul_eq (hc : c ≠ 0) : a = b / c ↔ a * c = b | hc.is_unit.eq_div_iff | lemma | eq_div_iff_mul_eq | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_of_eq_mul (hb : b ≠ 0) : a = c * b → a / b = c | hb.is_unit.div_eq_of_eq_mul | lemma | div_eq_of_eq_mul | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_div_of_mul_eq (hc : c ≠ 0) : a * c = b → a = b / c | hc.is_unit.eq_div_of_mul_eq | lemma | eq_div_of_mul_eq | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_one_iff_eq (hb : b ≠ 0) : a / b = 1 ↔ a = b | hb.is_unit.div_eq_one_iff_eq | lemma | div_eq_one_iff_eq | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul_left (hb : b ≠ 0) : b / (a * b) = 1 / a | hb.is_unit.div_mul_left | lemma | div_mul_left | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_mul_right (a b : G₀) (hc : c ≠ 0) : (a * c) / (b * c) = a / b | hc.is_unit.mul_div_mul_right _ _ | lemma | mul_div_mul_right | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mul_div (a : G₀) (hb : b ≠ 0) : a = a * b * (1 / b) | (hb.is_unit.mul_mul_div _).symm | lemma | mul_mul_div | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_div_div_cancel_right (a : G₀) (hc : c ≠ 0) : (a / c) / (b / c) = a / b | by rw [div_div_eq_mul_div, div_mul_cancel _ hc] | lemma | div_div_div_cancel_right | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [
"div_div_eq_mul_div",
"div_mul_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul_div_cancel (a : G₀) (hc : c ≠ 0) : (a / c) * (c / b) = a / b | by rw [← mul_div_assoc, div_mul_cancel _ hc] | lemma | div_mul_div_cancel | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [
"div_mul_cancel",
"mul_div_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul_cancel_of_imp {a b : G₀} (h : b = 0 → a = 0) : a / b * b = a | classical.by_cases (λ hb : b = 0, by simp [*]) (div_mul_cancel a) | lemma | div_mul_cancel_of_imp | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [
"div_mul_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_cancel_of_imp {a b : G₀} (h : b = 0 → a = 0) : a * b / b = a | classical.by_cases (λ hb : b = 0, by simp [*]) (mul_div_cancel a) | lemma | mul_div_cancel_of_imp | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [
"mul_div_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
divp_mk0 (a : G₀) {b : G₀} (hb : b ≠ 0) :
a /ₚ units.mk0 b hb = a / b | divp_eq_div _ _ | theorem | divp_mk0 | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [
"divp_eq_div",
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul_right (b : G₀) (ha : a ≠ 0) : a / (a * b) = 1 / b | ha.is_unit.div_mul_right _ | lemma | div_mul_right | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_cancel_left_of_imp {a b : G₀} (h : a = 0 → b = 0) : a * b / a = b | by rw [mul_comm, mul_div_cancel_of_imp h] | lemma | mul_div_cancel_left_of_imp | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [
"mul_comm",
"mul_div_cancel_of_imp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_cancel_left (b : G₀) (ha : a ≠ 0) : a * b / a = b | ha.is_unit.mul_div_cancel_left _ | lemma | mul_div_cancel_left | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_cancel_of_imp' {a b : G₀} (h : b = 0 → a = 0) : b * (a / b) = a | by rw [mul_comm, div_mul_cancel_of_imp h] | lemma | mul_div_cancel_of_imp' | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [
"div_mul_cancel_of_imp",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_cancel' (a : G₀) (hb : b ≠ 0) : b * (a / b) = a | hb.is_unit.mul_div_cancel' _ | lemma | mul_div_cancel' | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_mul_left (a b : G₀) (hc : c ≠ 0) : (c * a) / (c * b) = a / b | hc.is_unit.mul_div_mul_left _ _ | lemma | mul_div_mul_left | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_mul_of_div_eq_div (a : G₀) {b : G₀} (c : G₀) {d : G₀} (hb : b ≠ 0) (hd : d ≠ 0)
(h : a / b = c / d) : a * d = c * b | by rw [←mul_one a, ←div_self hb, ←mul_comm_div, h, div_mul_eq_mul_div, div_mul_cancel _ hd] | lemma | mul_eq_mul_of_div_eq_div | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [
"div_mul_cancel",
"div_mul_eq_mul_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_div_iff (hb : b ≠ 0) (hd : d ≠ 0) : a / b = c / d ↔ a * d = c * b | hb.is_unit.div_eq_div_iff hd.is_unit | lemma | div_eq_div_iff | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_div_cancel' (ha : a ≠ 0) : a / (a / b) = b | ha.is_unit.div_div_cancel | lemma | div_div_cancel' | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_div_cancel_left' (ha : a ≠ 0) : a / b / a = b⁻¹ | ha.is_unit.div_div_cancel_left | lemma | div_div_cancel_left' | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_helper (b : G₀) (h : a ≠ 0) : 1 / (a * b) * a = 1 / b | by rw [div_mul_eq_mul_div, one_mul, div_mul_right _ h] | lemma | div_helper | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [
"div_mul_eq_mul_div",
"div_mul_right",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_ne_zero : f a ≠ 0 ↔ a ≠ 0 | ⟨λ hfa ha, hfa $ ha.symm ▸ map_zero f, λ ha, ((is_unit.mk0 a ha).map f).ne_zero⟩ | lemma | map_ne_zero | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [
"is_unit.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_zero : f a = 0 ↔ a = 0 | not_iff_not.1 (map_ne_zero f) | lemma | map_eq_zero | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [
"map_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_on_inv₀ (f g : F') (h : f a = g a) : f a⁻¹ = g a⁻¹ | begin
rcases eq_or_ne a 0 with rfl|ha,
{ rw [inv_zero, map_zero, map_zero] },
{ exact (is_unit.mk0 a ha).eq_on_inv f g h }
end | lemma | eq_on_inv₀ | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [
"eq_on_inv",
"eq_or_ne",
"inv_zero",
"is_unit.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_inv₀ : f a⁻¹ = (f a)⁻¹ | begin
by_cases h : a = 0, by simp [h],
apply eq_inv_of_mul_eq_one_left,
rw [← map_mul, inv_mul_cancel h, map_one]
end | lemma | map_inv₀ | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [
"eq_inv_of_mul_eq_one_left",
"inv_mul_cancel",
"map_mul",
"map_one"
] | A monoid homomorphism between groups with zeros sending `0` to `0` sends `a⁻¹` to `(f a)⁻¹`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_div₀ : f (a / b) = f a / f b | map_div' f (map_inv₀ f) a b | lemma | map_div₀ | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [
"map_div'",
"map_inv₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_with_zero.inverse {M : Type*} [comm_monoid_with_zero M] :
M →*₀ M | { to_fun := ring.inverse,
map_zero' := ring.inverse_zero _,
map_one' := ring.inverse_one _,
map_mul' := λ x y, (ring.mul_inverse_rev x y).trans (mul_comm _ _) } | def | monoid_with_zero.inverse | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [
"comm_monoid_with_zero",
"mul_comm",
"ring.inverse",
"ring.inverse_one",
"ring.inverse_zero",
"ring.mul_inverse_rev"
] | We define the inverse as a `monoid_with_zero_hom` by extending the inverse map by zero
on non-units. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_with_zero.coe_inverse {M : Type*} [comm_monoid_with_zero M] :
(monoid_with_zero.inverse : M → M) = ring.inverse | rfl | lemma | monoid_with_zero.coe_inverse | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [
"comm_monoid_with_zero",
"monoid_with_zero.inverse",
"ring.inverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_with_zero.inverse_apply {M : Type*} [comm_monoid_with_zero M] (a : M) :
monoid_with_zero.inverse a = ring.inverse a | rfl | lemma | monoid_with_zero.inverse_apply | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [
"comm_monoid_with_zero",
"monoid_with_zero.inverse",
"ring.inverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_monoid_with_zero_hom {G₀ : Type*} [comm_group_with_zero G₀] : G₀ →*₀ G₀ | { map_zero' := inv_zero,
..inv_monoid_hom } | def | inv_monoid_with_zero_hom | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [
"comm_group_with_zero",
"inv_monoid_hom",
"inv_zero"
] | Inversion on a commutative group with zero, considered as a monoid with zero homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_mk0 {α : Type*} [has_smul G₀ α] {g : G₀} (hg : g ≠ 0) (a : α) :
(mk0 g hg) • a = g • a | rfl | lemma | units.smul_mk0 | algebra.group_with_zero.units | src/algebra/group_with_zero/units/lemmas.lean | [
"algebra.group_with_zero.commute",
"algebra.hom.units",
"group_theory.group_action.units"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_aut (M : Type*) [has_mul M] | M ≃* M | def | mul_aut | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [] | The group of multiplicative automorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mul (e₁ e₂ : mul_aut M) : ⇑(e₁ * e₂) = e₁ ∘ e₂ | rfl | lemma | mul_aut.coe_mul | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"mul_aut"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ⇑(1 : mul_aut M) = id | rfl | lemma | mul_aut.coe_one | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"mul_aut"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_def (e₁ e₂ : mul_aut M) : e₁ * e₂ = e₂.trans e₁ | rfl | lemma | mul_aut.mul_def | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"mul_aut"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_def : (1 : mul_aut M) = mul_equiv.refl _ | rfl | lemma | mul_aut.one_def | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"mul_aut",
"mul_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_def (e₁ : mul_aut M) : e₁⁻¹ = e₁.symm | rfl | lemma | mul_aut.inv_def | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"mul_aut"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_apply (e₁ e₂ : mul_aut M) (m : M) : (e₁ * e₂) m = e₁ (e₂ m) | rfl | lemma | mul_aut.mul_apply | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"mul_aut"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_apply (m : M) : (1 : mul_aut M) m = m | rfl | lemma | mul_aut.one_apply | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"mul_aut"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_inv_self (e : mul_aut M) (m : M) : e (e⁻¹ m) = m | mul_equiv.apply_symm_apply _ _ | lemma | mul_aut.apply_inv_self | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"mul_aut",
"mul_equiv.apply_symm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_apply_self (e : mul_aut M) (m : M) : e⁻¹ (e m) = m | mul_equiv.apply_symm_apply _ _ | lemma | mul_aut.inv_apply_self | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"mul_aut",
"mul_equiv.apply_symm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_perm : mul_aut M →* equiv.perm M | by refine_struct { to_fun := mul_equiv.to_equiv }; intros; refl | def | mul_aut.to_perm | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"equiv.perm",
"mul_aut"
] | Monoid hom from the group of multiplicative automorphisms to the group of permutations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_mul_distrib_mul_action {M} [monoid M] : mul_distrib_mul_action (mul_aut M) M | { smul := ($),
one_smul := λ _, rfl,
mul_smul := λ _ _ _, rfl,
smul_one := mul_equiv.map_one,
smul_mul := mul_equiv.map_mul } | instance | mul_aut.apply_mul_distrib_mul_action | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"monoid",
"mul_aut",
"mul_distrib_mul_action",
"mul_equiv.map_mul",
"mul_equiv.map_one",
"one_smul"
] | The tautological action by `mul_aut M` on `M`.
This generalizes `function.End.apply_mul_action`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_def {M} [monoid M] (f : mul_aut M) (a : M) : f • a = f a | rfl | lemma | mul_aut.smul_def | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"monoid",
"mul_aut"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_has_faithful_smul {M} [monoid M] : has_faithful_smul (mul_aut M) M | ⟨λ _ _, mul_equiv.ext⟩ | instance | mul_aut.apply_has_faithful_smul | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"has_faithful_smul",
"monoid",
"mul_aut"
] | `mul_aut.apply_mul_action` is faithful. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conj [group G] : G →* mul_aut G | { to_fun := λ g,
{ to_fun := λ h, g * h * g⁻¹,
inv_fun := λ h, g⁻¹ * h * g,
left_inv := λ _, by simp [mul_assoc],
right_inv := λ _, by simp [mul_assoc],
map_mul' := by simp [mul_assoc] },
map_mul' := λ _ _, by ext; simp [mul_assoc],
map_one' := by ext; simp [mul_assoc] } | def | mul_aut.conj | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"group",
"inv_fun",
"mul_assoc",
"mul_aut"
] | Group conjugation, `mul_aut.conj g h = g * h * g⁻¹`, as a monoid homomorphism
mapping multiplication in `G` into multiplication in the automorphism group `mul_aut G`.
See also the type `conj_act G` for any group `G`, which has a `mul_action (conj_act G) G` instance
where `conj G` acts on `G` by conjugation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conj_apply [group G] (g h : G) : conj g h = g * h * g⁻¹ | rfl | lemma | mul_aut.conj_apply | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_symm_apply [group G] (g h : G) : (conj g).symm h = g⁻¹ * h * g | rfl | lemma | mul_aut.conj_symm_apply | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_inv_apply [group G] (g h : G) : (conj g)⁻¹ h = g⁻¹ * h * g | rfl | lemma | mul_aut.conj_inv_apply | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group : group (add_aut A) | by refine_struct
{ mul := λ g h, add_equiv.trans h g,
one := add_equiv.refl A,
inv := add_equiv.symm,
div := _,
npow := @npow_rec _ ⟨add_equiv.refl A⟩ ⟨λ g h, add_equiv.trans h g⟩,
zpow := @zpow_rec _ ⟨add_equiv.refl A⟩ ⟨λ g h, add_equiv.trans h g⟩ ⟨add_equiv.symm⟩ };
intros; ext; try { refl }; apply equiv.le... | instance | add_aut.group | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"group",
"npow_rec",
"zpow_rec"
] | The group operation on additive automorphisms is defined by
`λ g h, add_equiv.trans h g`.
This means that multiplication agrees with composition, `(g*h)(x) = g (h x)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mul (e₁ e₂ : add_aut A) : ⇑(e₁ * e₂) = e₁ ∘ e₂ | rfl | lemma | add_aut.coe_mul | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ⇑(1 : add_aut A) = id | rfl | lemma | add_aut.coe_one | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_def (e₁ e₂ : add_aut A) : e₁ * e₂ = e₂.trans e₁ | rfl | lemma | add_aut.mul_def | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_def : (1 : add_aut A) = add_equiv.refl _ | rfl | lemma | add_aut.one_def | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_def (e₁ : add_aut A) : e₁⁻¹ = e₁.symm | rfl | lemma | add_aut.inv_def | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_apply (e₁ e₂ : add_aut A) (a : A) : (e₁ * e₂) a = e₁ (e₂ a) | rfl | lemma | add_aut.mul_apply | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_apply (a : A) : (1 : add_aut A) a = a | rfl | lemma | add_aut.one_apply | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_inv_self (e : add_aut A) (a : A) : e⁻¹ (e a) = a | add_equiv.apply_symm_apply _ _ | lemma | add_aut.apply_inv_self | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_apply_self (e : add_aut A) (a : A) : e (e⁻¹ a) = a | add_equiv.apply_symm_apply _ _ | lemma | add_aut.inv_apply_self | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_perm : add_aut A →* equiv.perm A | by refine_struct { to_fun := add_equiv.to_equiv }; intros; refl | def | add_aut.to_perm | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"equiv.perm"
] | Monoid hom from the group of multiplicative automorphisms to the group of permutations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_distrib_mul_action {A} [add_monoid A] : distrib_mul_action (add_aut A) A | { smul := ($),
smul_zero := add_equiv.map_zero,
smul_add := add_equiv.map_add,
one_smul := λ _, rfl,
mul_smul := λ _ _ _, rfl } | instance | add_aut.apply_distrib_mul_action | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"add_monoid",
"distrib_mul_action",
"one_smul",
"smul_add",
"smul_zero"
] | The tautological action by `add_aut A` on `A`.
This generalizes `function.End.apply_mul_action`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_def {A} [add_monoid A] (f : add_aut A) (a : A) : f • a = f a | rfl | lemma | add_aut.smul_def | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"add_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_has_faithful_smul {A} [add_monoid A] : has_faithful_smul (add_aut A) A | ⟨λ _ _, add_equiv.ext⟩ | instance | add_aut.apply_has_faithful_smul | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"add_monoid",
"has_faithful_smul"
] | `add_aut.apply_distrib_mul_action` is faithful. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conj [add_group G] : G →+ additive (add_aut G) | { to_fun := λ g, @additive.of_mul (add_aut G)
{ to_fun := λ h, g + h + -g, -- this definition is chosen to match `mul_aut.conj`
inv_fun := λ h, -g + h + g,
left_inv := λ _, by simp [add_assoc],
right_inv := λ _, by simp [add_assoc],
map_add' := by simp [add_assoc] },
map_add' := λ _ _, by apply addi... | def | add_aut.conj | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"add_group",
"additive",
"additive.of_mul",
"inv_fun"
] | Additive group conjugation, `add_aut.conj g h = g + h - g`, as an additive monoid
homomorphism mapping addition in `G` into multiplication in the automorphism group `add_aut G`
(written additively in order to define the map). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conj_apply [add_group G] (g h : G) : conj g h = g + h + -g | rfl | lemma | add_aut.conj_apply | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_symm_apply [add_group G] (g h : G) : (conj g).symm h = -g + h + g | rfl | lemma | add_aut.conj_symm_apply | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_inv_apply [add_group G] (g h : G) : (-(conj g)) h = -g + h + g | rfl | lemma | add_aut.conj_inv_apply | algebra.hom | src/algebra/hom/aut.lean | [
"group_theory.perm.basic"
] | [
"add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
centroid_hom (α : Type*) [non_unital_non_assoc_semiring α] extends α →+ α | (map_mul_left' (a b : α) : to_fun (a * b) = a * to_fun b)
(map_mul_right' (a b : α) : to_fun (a * b) = to_fun a * b) | structure | centroid_hom | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"non_unital_non_assoc_semiring"
] | The type of centroid homomorphisms from `α` to `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
centroid_hom_class (F : Type*) (α : out_param $ Type*) [non_unital_non_assoc_semiring α]
extends add_monoid_hom_class F α α | (map_mul_left (f : F) (a b : α) : f (a * b) = a * f b)
(map_mul_right (f : F) (a b : α) : f (a * b) = f a * b) | class | centroid_hom_class | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"add_monoid_hom_class",
"non_unital_non_assoc_semiring"
] | `centroid_hom_class F α` states that `F` is a type of centroid homomorphisms.
You should extend this class when you extend `centroid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_eq_coe {f : centroid_hom α} : f.to_fun = (f : α → α) | rfl | lemma | centroid_hom.to_fun_eq_coe | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : centroid_hom α} (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | centroid_hom.ext | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom",
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_add_monoid_hom (f : centroid_hom α) : ⇑(f : α →+ α) = f | rfl | lemma | centroid_hom.coe_to_add_monoid_hom | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_monoid_hom_eq_coe (f : centroid_hom α) : f.to_add_monoid_hom = f | rfl | lemma | centroid_hom.to_add_monoid_hom_eq_coe | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_add_monoid_hom_injective : injective (coe : centroid_hom α → α →+ α) | λ f g h, ext $ λ a, by { have := fun_like.congr_fun h a, exact this } | lemma | centroid_hom.coe_to_add_monoid_hom_injective | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom",
"fun_like.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_End (f : centroid_hom α) : add_monoid.End α | (f : α →+ α) | def | centroid_hom.to_End | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"add_monoid.End",
"centroid_hom"
] | Turn a centroid homomorphism into an additive monoid endomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_End_injective : injective (centroid_hom.to_End : centroid_hom α → add_monoid.End α) | coe_to_add_monoid_hom_injective | lemma | centroid_hom.to_End_injective | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"add_monoid.End",
"centroid_hom",
"centroid_hom.to_End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : centroid_hom α) (f' : α → α) (h : f' = f) :
centroid_hom α | { to_fun := f',
map_mul_left' := λ a b, by simp_rw [h, map_mul_left],
map_mul_right' := λ a b, by simp_rw [h, map_mul_right],
..f.to_add_monoid_hom.copy f' $ by exact h } | def | centroid_hom.copy | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | Copy of a `centroid_hom` with a new `to_fun` equal to the old one. Useful to fix
definitional equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (f : centroid_hom α) (f' : α → α) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | centroid_hom.coe_copy | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : centroid_hom α) (f' : α → α) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | centroid_hom.copy_eq | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom",
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : centroid_hom α | { map_mul_left' := λ _ _, rfl,
map_mul_right' := λ _ _, rfl,
.. add_monoid_hom.id α } | def | centroid_hom.id | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | `id` as a `centroid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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