fact stringlengths 6 14.3k | statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 12
values | symbolic_name stringlengths 0 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 8 10.2k ⌀ | line_start int64 6 4.24k | line_end int64 7 4.25k | has_proof bool 2
classes | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
sub_iff_left :
a₁ ≡ b₁ [PMOD p] → (a₁ - a₂ ≡ b₁ - b₂ [PMOD p] ↔ a₂ ≡ b₂ [PMOD p]) :=
λ ⟨m, hm⟩, (equiv.sub_left m).symm.exists_congr_left.trans $
by simpa [sub_sub_sub_comm, hm, sub_smul] | sub_iff_left :
a₁ ≡ b₁ [PMOD p] → (a₁ - a₂ ≡ b₁ - b₂ [PMOD p] ↔ a₂ ≡ b₂ [PMOD p]) | λ ⟨m, hm⟩, (equiv.sub_left m).symm.exists_congr_left.trans $
by simpa [sub_sub_sub_comm, hm, sub_smul] | lemma | add_comm_group.modeq.sub_iff_left | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [
"sub_smul"
] | null | 134 | 137 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_iff_right :
a₂ ≡ b₂ [PMOD p] → (a₁ - a₂ ≡ b₁ - b₂ [PMOD p] ↔ a₁ ≡ b₁ [PMOD p]) :=
λ ⟨m, hm⟩, (equiv.sub_right m).symm.exists_congr_left.trans $
by simpa [sub_sub_sub_comm, hm, sub_smul] | sub_iff_right :
a₂ ≡ b₂ [PMOD p] → (a₁ - a₂ ≡ b₁ - b₂ [PMOD p] ↔ a₁ ≡ b₁ [PMOD p]) | λ ⟨m, hm⟩, (equiv.sub_right m).symm.exists_congr_left.trans $
by simpa [sub_sub_sub_comm, hm, sub_smul] | lemma | add_comm_group.modeq.sub_iff_right | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [
"sub_smul"
] | null | 139 | 142 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_left (c : α) (h : a ≡ b [PMOD p]) : c + a ≡ c + b [PMOD p] := modeq_rfl.add h | add_left (c : α) (h : a ≡ b [PMOD p]) : c + a ≡ c + b [PMOD p] | modeq_rfl.add h | lemma | add_comm_group.modeq.add_left | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | null | 151 | 151 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_left (c : α) (h : a ≡ b [PMOD p]) : c - a ≡ c - b [PMOD p] := modeq_rfl.sub h | sub_left (c : α) (h : a ≡ b [PMOD p]) : c - a ≡ c - b [PMOD p] | modeq_rfl.sub h | lemma | add_comm_group.modeq.sub_left | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | null | 152 | 152 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_right (c : α) (h : a ≡ b [PMOD p]) : a + c ≡ b + c [PMOD p] := h.add modeq_rfl | add_right (c : α) (h : a ≡ b [PMOD p]) : a + c ≡ b + c [PMOD p] | h.add modeq_rfl | lemma | add_comm_group.modeq.add_right | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | null | 153 | 153 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_right (c : α) (h : a ≡ b [PMOD p]) : a - c ≡ b - c [PMOD p] := h.sub modeq_rfl | sub_right (c : α) (h : a ≡ b [PMOD p]) : a - c ≡ b - c [PMOD p] | h.sub modeq_rfl | lemma | add_comm_group.modeq.sub_right | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | null | 154 | 154 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_left_cancel' (c : α) : c + a ≡ c + b [PMOD p] → a ≡ b [PMOD p] :=
modeq_rfl.add_left_cancel | add_left_cancel' (c : α) : c + a ≡ c + b [PMOD p] → a ≡ b [PMOD p] | modeq_rfl.add_left_cancel | lemma | add_comm_group.modeq.add_left_cancel' | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | null | 156 | 157 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_right_cancel' (c : α) : a + c ≡ b + c [PMOD p] → a ≡ b [PMOD p] :=
modeq_rfl.add_right_cancel | add_right_cancel' (c : α) : a + c ≡ b + c [PMOD p] → a ≡ b [PMOD p] | modeq_rfl.add_right_cancel | lemma | add_comm_group.modeq.add_right_cancel' | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | null | 159 | 160 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_left_cancel' (c : α) : c - a ≡ c - b [PMOD p] → a ≡ b [PMOD p] :=
modeq_rfl.sub_left_cancel | sub_left_cancel' (c : α) : c - a ≡ c - b [PMOD p] → a ≡ b [PMOD p] | modeq_rfl.sub_left_cancel | lemma | add_comm_group.modeq.sub_left_cancel' | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | null | 162 | 163 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_right_cancel' (c : α) : a - c ≡ b - c [PMOD p] → a ≡ b [PMOD p] :=
modeq_rfl.sub_right_cancel | sub_right_cancel' (c : α) : a - c ≡ b - c [PMOD p] → a ≡ b [PMOD p] | modeq_rfl.sub_right_cancel | lemma | add_comm_group.modeq.sub_right_cancel' | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | null | 165 | 166 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
modeq_sub_iff_add_modeq' : a ≡ b - c [PMOD p] ↔ c + a ≡ b [PMOD p] := by simp [modeq, sub_sub] | modeq_sub_iff_add_modeq' : a ≡ b - c [PMOD p] ↔ c + a ≡ b [PMOD p] | by simp [modeq, sub_sub] | lemma | add_comm_group.modeq_sub_iff_add_modeq' | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | null | 170 | 170 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
modeq_sub_iff_add_modeq : a ≡ b - c [PMOD p] ↔ a + c ≡ b [PMOD p] :=
modeq_sub_iff_add_modeq'.trans $ by rw add_comm | modeq_sub_iff_add_modeq : a ≡ b - c [PMOD p] ↔ a + c ≡ b [PMOD p] | modeq_sub_iff_add_modeq'.trans $ by rw add_comm | lemma | add_comm_group.modeq_sub_iff_add_modeq | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | null | 171 | 172 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_modeq_iff_modeq_add' : a - b ≡ c [PMOD p] ↔ a ≡ b + c [PMOD p] :=
modeq_comm.trans $ modeq_sub_iff_add_modeq'.trans modeq_comm | sub_modeq_iff_modeq_add' : a - b ≡ c [PMOD p] ↔ a ≡ b + c [PMOD p] | modeq_comm.trans $ modeq_sub_iff_add_modeq'.trans modeq_comm | lemma | add_comm_group.sub_modeq_iff_modeq_add' | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | null | 173 | 174 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_modeq_iff_modeq_add : a - b ≡ c [PMOD p] ↔ a ≡ c + b [PMOD p] :=
modeq_comm.trans $ modeq_sub_iff_add_modeq.trans modeq_comm | sub_modeq_iff_modeq_add : a - b ≡ c [PMOD p] ↔ a ≡ c + b [PMOD p] | modeq_comm.trans $ modeq_sub_iff_add_modeq.trans modeq_comm | lemma | add_comm_group.sub_modeq_iff_modeq_add | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | null | 175 | 176 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_modeq_zero : a - b ≡ 0 [PMOD p] ↔ a ≡ b [PMOD p] :=
by simp [sub_modeq_iff_modeq_add] | sub_modeq_zero : a - b ≡ 0 [PMOD p] ↔ a ≡ b [PMOD p] | by simp [sub_modeq_iff_modeq_add] | lemma | add_comm_group.sub_modeq_zero | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | null | 178 | 179 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_modeq_left : a + b ≡ a [PMOD p] ↔ b ≡ 0 [PMOD p] :=
by simp [←modeq_sub_iff_add_modeq'] | add_modeq_left : a + b ≡ a [PMOD p] ↔ b ≡ 0 [PMOD p] | by simp [←modeq_sub_iff_add_modeq'] | lemma | add_comm_group.add_modeq_left | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | null | 181 | 182 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_modeq_right : a + b ≡ b [PMOD p] ↔ a ≡ 0 [PMOD p] :=
by simp [←modeq_sub_iff_add_modeq] | add_modeq_right : a + b ≡ b [PMOD p] ↔ a ≡ 0 [PMOD p] | by simp [←modeq_sub_iff_add_modeq] | lemma | add_comm_group.add_modeq_right | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | null | 184 | 185 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
modeq_iff_eq_add_zsmul : a ≡ b [PMOD p] ↔ ∃ z : ℤ, b = a + z • p :=
by simp_rw [modeq, sub_eq_iff_eq_add'] | modeq_iff_eq_add_zsmul : a ≡ b [PMOD p] ↔ ∃ z : ℤ, b = a + z • p | by simp_rw [modeq, sub_eq_iff_eq_add'] | lemma | add_comm_group.modeq_iff_eq_add_zsmul | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | null | 187 | 188 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_modeq_iff_ne_add_zsmul : ¬a ≡ b [PMOD p] ↔ ∀ z : ℤ, b ≠ a + z • p :=
by rw [modeq_iff_eq_add_zsmul, not_exists] | not_modeq_iff_ne_add_zsmul : ¬a ≡ b [PMOD p] ↔ ∀ z : ℤ, b ≠ a + z • p | by rw [modeq_iff_eq_add_zsmul, not_exists] | lemma | add_comm_group.not_modeq_iff_ne_add_zsmul | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [
"not_exists"
] | null | 190 | 191 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
modeq_iff_eq_mod_zmultiples : a ≡ b [PMOD p] ↔ (b : α ⧸ add_subgroup.zmultiples p) = a :=
by simp_rw [modeq_iff_eq_add_zsmul, quotient_add_group.eq_iff_sub_mem,
add_subgroup.mem_zmultiples_iff, eq_sub_iff_add_eq', eq_comm] | modeq_iff_eq_mod_zmultiples : a ≡ b [PMOD p] ↔ (b : α ⧸ add_subgroup.zmultiples p) = a | by simp_rw [modeq_iff_eq_add_zsmul, quotient_add_group.eq_iff_sub_mem,
add_subgroup.mem_zmultiples_iff, eq_sub_iff_add_eq', eq_comm] | lemma | add_comm_group.modeq_iff_eq_mod_zmultiples | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [
"add_subgroup.zmultiples"
] | null | 193 | 195 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_modeq_iff_ne_mod_zmultiples :
¬a ≡ b [PMOD p] ↔ (b : α ⧸ add_subgroup.zmultiples p) ≠ a :=
modeq_iff_eq_mod_zmultiples.not | not_modeq_iff_ne_mod_zmultiples :
¬a ≡ b [PMOD p] ↔ (b : α ⧸ add_subgroup.zmultiples p) ≠ a | modeq_iff_eq_mod_zmultiples.not | lemma | add_comm_group.not_modeq_iff_ne_mod_zmultiples | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [
"add_subgroup.zmultiples"
] | null | 197 | 199 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
modeq_iff_int_modeq {a b z : ℤ} : a ≡ b [PMOD z] ↔ a ≡ b [ZMOD z] :=
by simp [modeq, dvd_iff_exists_eq_mul_left, int.modeq_iff_dvd] | modeq_iff_int_modeq {a b z : ℤ} : a ≡ b [PMOD z] ↔ a ≡ b [ZMOD z] | by simp [modeq, dvd_iff_exists_eq_mul_left, int.modeq_iff_dvd] | lemma | add_comm_group.modeq_iff_int_modeq | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [
"dvd_iff_exists_eq_mul_left",
"int.modeq_iff_dvd"
] | null | 203 | 204 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
int_cast_modeq_int_cast {a b z : ℤ} : a ≡ b [PMOD (z : α)] ↔ a ≡ b [PMOD z] :=
by simp_rw [modeq, ←int.cast_mul_eq_zsmul_cast]; norm_cast | int_cast_modeq_int_cast {a b z : ℤ} : a ≡ b [PMOD (z : α)] ↔ a ≡ b [PMOD z] | by simp_rw [modeq, ←int.cast_mul_eq_zsmul_cast]; norm_cast | lemma | add_comm_group.int_cast_modeq_int_cast | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | null | 209 | 211 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_cast_modeq_nat_cast {a b n : ℕ} : a ≡ b [PMOD (n : α)] ↔ a ≡ b [MOD n] :=
by simp_rw [←int.coe_nat_modeq_iff, ←modeq_iff_int_modeq, ←@int_cast_modeq_int_cast α,
int.cast_coe_nat] | nat_cast_modeq_nat_cast {a b n : ℕ} : a ≡ b [PMOD (n : α)] ↔ a ≡ b [MOD n] | by simp_rw [←int.coe_nat_modeq_iff, ←modeq_iff_int_modeq, ←@int_cast_modeq_int_cast α,
int.cast_coe_nat] | lemma | add_comm_group.nat_cast_modeq_nat_cast | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [
"int.cast_coe_nat"
] | null | 213 | 216 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ne_zero {R} [has_zero R] (n : R) : Prop := (out : n ≠ 0) | ne_zero {R} [has_zero R] (n : R) : Prop | (out : n ≠ 0) | class | ne_zero | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [] | A type-class version of `n ≠ 0`. | 23 | 23 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ne_zero.ne {R} [has_zero R] (n : R) [h : ne_zero n] : n ≠ 0 := h.out | ne_zero.ne {R} [has_zero R] (n : R) [h : ne_zero n] : n ≠ 0 | h.out | lemma | ne_zero.ne | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero"
] | null | 25 | 25 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ne_zero.ne' {R} [has_zero R] (n : R) [h : ne_zero n] : 0 ≠ n := h.out.symm | ne_zero.ne' {R} [has_zero R] (n : R) [h : ne_zero n] : 0 ≠ n | h.out.symm | lemma | ne_zero.ne' | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero"
] | null | 27 | 27 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ne_zero_iff {R : Type*} [has_zero R] {n : R} : ne_zero n ↔ n ≠ 0 :=
⟨λ h, h.out, ne_zero.mk⟩ | ne_zero_iff {R : Type*} [has_zero R] {n : R} : ne_zero n ↔ n ≠ 0 | ⟨λ h, h.out, ne_zero.mk⟩ | lemma | ne_zero_iff | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero"
] | null | 29 | 30 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_ne_zero {R : Type*} [has_zero R] {n : R} : ¬ ne_zero n ↔ n = 0 :=
by simp [ne_zero_iff] | not_ne_zero {R : Type*} [has_zero R] {n : R} : ¬ ne_zero n ↔ n = 0 | by simp [ne_zero_iff] | lemma | not_ne_zero | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero",
"ne_zero_iff"
] | null | 32 | 33 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_zero_or_ne_zero {α} [has_zero α] (a : α) : a = 0 ∨ ne_zero a :=
(eq_or_ne a 0).imp_right ne_zero.mk | eq_zero_or_ne_zero {α} [has_zero α] (a : α) : a = 0 ∨ ne_zero a | (eq_or_ne a 0).imp_right ne_zero.mk | lemma | eq_zero_or_ne_zero | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"eq_or_ne",
"ne_zero"
] | null | 35 | 36 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_ne_one [ne_zero (1 : α)] : (0 : α) ≠ 1 := ne_zero.ne' (1 : α) | zero_ne_one [ne_zero (1 : α)] : (0 : α) ≠ 1 | ne_zero.ne' (1 : α) | lemma | zero_ne_one | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero",
"ne_zero.ne'"
] | null | 41 | 41 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_ne_zero [ne_zero (1 : α)] : (1 : α) ≠ 0 := ne_zero.ne (1 : α) | one_ne_zero [ne_zero (1 : α)] : (1 : α) ≠ 0 | ne_zero.ne (1 : α) | lemma | one_ne_zero | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero",
"ne_zero.ne"
] | null | 42 | 42 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
two_ne_zero [has_add α] [ne_zero (2 : α)] : (2 : α) ≠ 0 := ne_zero.ne (2 : α) | two_ne_zero [has_add α] [ne_zero (2 : α)] : (2 : α) ≠ 0 | ne_zero.ne (2 : α) | lemma | two_ne_zero | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero",
"ne_zero.ne"
] | null | 43 | 43 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
three_ne_zero [has_add α] [ne_zero (3 : α)] : (3 : α) ≠ 0 := ne_zero.ne (3 : α) | three_ne_zero [has_add α] [ne_zero (3 : α)] : (3 : α) ≠ 0 | ne_zero.ne (3 : α) | lemma | three_ne_zero | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero",
"ne_zero.ne"
] | null | 44 | 44 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
four_ne_zero [has_add α] [ne_zero (4 : α)] : (4 : α) ≠ 0 := ne_zero.ne (4 : α) | four_ne_zero [has_add α] [ne_zero (4 : α)] : (4 : α) ≠ 0 | ne_zero.ne (4 : α) | lemma | four_ne_zero | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero",
"ne_zero.ne"
] | null | 45 | 45 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ne_zero_of_eq_one [ne_zero (1 : α)] {a : α} (h : a = 1) : a ≠ 0 :=
calc a = 1 : h
... ≠ 0 : one_ne_zero | ne_zero_of_eq_one [ne_zero (1 : α)] {a : α} (h : a = 1) : a ≠ 0 | calc a = 1 : h
... ≠ 0 : one_ne_zero | lemma | ne_zero_of_eq_one | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero",
"one_ne_zero"
] | null | 47 | 49 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_ne_one' [ne_zero (1 : α)] : (0 : α) ≠ 1 := ne_zero.ne' (1 : α) | zero_ne_one' [ne_zero (1 : α)] : (0 : α) ≠ 1 | ne_zero.ne' (1 : α) | lemma | zero_ne_one' | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero",
"ne_zero.ne'"
] | null | 53 | 53 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_ne_zero' [ne_zero (1 : α)] : (1 : α) ≠ 0 := ne_zero.ne (1 : α) | one_ne_zero' [ne_zero (1 : α)] : (1 : α) ≠ 0 | ne_zero.ne (1 : α) | lemma | one_ne_zero' | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero",
"ne_zero.ne"
] | null | 54 | 54 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
two_ne_zero' [has_add α] [ne_zero (2 : α)] : (2 : α) ≠ 0 := ne_zero.ne (2 : α) | two_ne_zero' [has_add α] [ne_zero (2 : α)] : (2 : α) ≠ 0 | ne_zero.ne (2 : α) | lemma | two_ne_zero' | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero",
"ne_zero.ne"
] | null | 55 | 55 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
three_ne_zero' [has_add α] [ne_zero (3 : α)] : (3 : α) ≠ 0 := ne_zero.ne (3 : α) | three_ne_zero' [has_add α] [ne_zero (3 : α)] : (3 : α) ≠ 0 | ne_zero.ne (3 : α) | lemma | three_ne_zero' | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero",
"ne_zero.ne"
] | null | 56 | 56 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
four_ne_zero' [has_add α] [ne_zero (4 : α)] : (4 : α) ≠ 0 := ne_zero.ne (4 : α) | four_ne_zero' [has_add α] [ne_zero (4 : α)] : (4 : α) ≠ 0 | ne_zero.ne (4 : α) | lemma | four_ne_zero' | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero",
"ne_zero.ne"
] | null | 57 | 57 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
succ : ne_zero (n + 1) := ⟨n.succ_ne_zero⟩ | succ : ne_zero (n + 1) | ⟨n.succ_ne_zero⟩ | instance | ne_zero.succ | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero"
] | null | 65 | 65 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_pos [preorder M] [has_zero M] (h : 0 < x) : ne_zero x := ⟨ne_of_gt h⟩ | of_pos [preorder M] [has_zero M] (h : 0 < x) : ne_zero x | ⟨ne_of_gt h⟩ | lemma | ne_zero.of_pos | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero"
] | null | 67 | 67 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_trans [has_zero M] [has_coe R S] [has_coe_t S M] [h : ne_zero (r : M)] :
ne_zero ((r : S) : M) := ⟨h.out⟩ | coe_trans [has_zero M] [has_coe R S] [has_coe_t S M] [h : ne_zero (r : M)] :
ne_zero ((r : S) : M) | ⟨h.out⟩ | instance | ne_zero.coe_trans | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero"
] | null | 69 | 70 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans [has_zero M] [has_coe R S] [has_coe_t S M] (h : ne_zero ((r : S) : M)) :
ne_zero (r : M) := ⟨h.out⟩ | trans [has_zero M] [has_coe R S] [has_coe_t S M] (h : ne_zero ((r : S) : M)) :
ne_zero (r : M) | ⟨h.out⟩ | lemma | ne_zero.trans | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero"
] | null | 72 | 73 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_opposite (α : Type u) : Type u := α | mul_opposite (α : Type u) : Type u | α | def | mul_opposite | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | Multiplicative opposite of a type. This type inherits all additive structures on `α` and
reverses left and right in multiplication. | 41 | 43 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op : α → αᵐᵒᵖ := id | op : α → αᵐᵒᵖ | id | def | mul_opposite.op | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | The element of `mul_opposite α` that represents `x : α`. | 53 | 54 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop : αᵐᵒᵖ → α := id | unop : αᵐᵒᵖ → α | id | def | mul_opposite.unop | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | The element of `α` represented by `x : αᵐᵒᵖ`. | 57 | 58 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_op (x : α) : unop (op x) = x := rfl | unop_op (x : α) : unop (op x) = x | rfl | lemma | mul_opposite.unop_op | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 62 | 62 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_unop (x : αᵐᵒᵖ) : op (unop x) = x := rfl | op_unop (x : αᵐᵒᵖ) : op (unop x) = x | rfl | lemma | mul_opposite.op_unop | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 63 | 63 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_comp_unop : (op : α → αᵐᵒᵖ) ∘ unop = id := rfl | op_comp_unop : (op : α → αᵐᵒᵖ) ∘ unop = id | rfl | lemma | mul_opposite.op_comp_unop | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 64 | 64 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_comp_op : (unop : αᵐᵒᵖ → α) ∘ op = id := rfl | unop_comp_op : (unop : αᵐᵒᵖ → α) ∘ op = id | rfl | lemma | mul_opposite.unop_comp_op | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 65 | 65 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rec {F : Π (X : αᵐᵒᵖ), Sort v} (h : Π X, F (op X)) : Π X, F X :=
λ X, h (unop X) | rec {F : Π (X : αᵐᵒᵖ), Sort v} (h : Π X, F (op X)) : Π X, F X | λ X, h (unop X) | def | mul_opposite.rec | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | A recursor for `mul_opposite`. Use as `induction x using mul_opposite.rec`. | 70 | 72 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_equiv : α ≃ αᵐᵒᵖ := ⟨op, unop, unop_op, op_unop⟩ | op_equiv : α ≃ αᵐᵒᵖ | ⟨op, unop, unop_op, op_unop⟩ | def | mul_opposite.op_equiv | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | The canonical bijection between `α` and `αᵐᵒᵖ`. | 75 | 77 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_bijective : bijective (op : α → αᵐᵒᵖ) := op_equiv.bijective | op_bijective : bijective (op : α → αᵐᵒᵖ) | op_equiv.bijective | lemma | mul_opposite.op_bijective | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 79 | 79 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_bijective : bijective (unop : αᵐᵒᵖ → α) := op_equiv.symm.bijective | unop_bijective : bijective (unop : αᵐᵒᵖ → α) | op_equiv.symm.bijective | lemma | mul_opposite.unop_bijective | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 80 | 80 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_injective : injective (op : α → αᵐᵒᵖ) := op_bijective.injective | op_injective : injective (op : α → αᵐᵒᵖ) | op_bijective.injective | lemma | mul_opposite.op_injective | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 81 | 81 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_surjective : surjective (op : α → αᵐᵒᵖ) := op_bijective.surjective | op_surjective : surjective (op : α → αᵐᵒᵖ) | op_bijective.surjective | lemma | mul_opposite.op_surjective | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 82 | 82 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_injective : injective (unop : αᵐᵒᵖ → α) := unop_bijective.injective | unop_injective : injective (unop : αᵐᵒᵖ → α) | unop_bijective.injective | lemma | mul_opposite.unop_injective | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 83 | 83 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_surjective : surjective (unop : αᵐᵒᵖ → α) := unop_bijective.surjective | unop_surjective : surjective (unop : αᵐᵒᵖ → α) | unop_bijective.surjective | lemma | mul_opposite.unop_surjective | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 84 | 84 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_inj {x y : α} : op x = op y ↔ x = y := op_injective.eq_iff | op_inj {x y : α} : op x = op y ↔ x = y | op_injective.eq_iff | lemma | mul_opposite.op_inj | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 86 | 86 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_inj {x y : αᵐᵒᵖ} : unop x = unop y ↔ x = y := unop_injective.eq_iff | unop_inj {x y : αᵐᵒᵖ} : unop x = unop y ↔ x = y | unop_injective.eq_iff | lemma | mul_opposite.unop_inj | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 87 | 87 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
[nontrivial α] : nontrivial αᵐᵒᵖ := op_injective.nontrivial | [nontrivial α] : nontrivial αᵐᵒᵖ | op_injective.nontrivial | instance | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [
"nontrivial"
] | null | 91 | 91 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
[inhabited α] : inhabited αᵐᵒᵖ := ⟨op default⟩ | [inhabited α] : inhabited αᵐᵒᵖ | ⟨op default⟩ | instance | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 92 | 92 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
[subsingleton α] : subsingleton αᵐᵒᵖ := unop_injective.subsingleton | [subsingleton α] : subsingleton αᵐᵒᵖ | unop_injective.subsingleton | instance | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 93 | 93 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
[unique α] : unique αᵐᵒᵖ := unique.mk' _ | [unique α] : unique αᵐᵒᵖ | unique.mk' _ | instance | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [
"unique",
"unique.mk'"
] | null | 94 | 94 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
[is_empty α] : is_empty αᵐᵒᵖ := function.is_empty unop | [is_empty α] : is_empty αᵐᵒᵖ | function.is_empty unop | instance | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [
"function.is_empty",
"is_empty"
] | null | 95 | 95 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
[has_zero α] : has_zero αᵐᵒᵖ := { zero := op 0 } | [has_zero α] : has_zero αᵐᵒᵖ | { zero := op 0 } | instance | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 97 | 97 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
[has_one α] : has_one αᵐᵒᵖ := { one := op 1 } | [has_one α] : has_one αᵐᵒᵖ | { one := op 1 } | instance | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 99 | 99 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
[has_add α] : has_add αᵐᵒᵖ :=
{ add := λ x y, op (unop x + unop y) } | [has_add α] : has_add αᵐᵒᵖ | { add := λ x y, op (unop x + unop y) } | instance | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 101 | 102 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
[has_sub α] : has_sub αᵐᵒᵖ :=
{ sub := λ x y, op (unop x - unop y) } | [has_sub α] : has_sub αᵐᵒᵖ | { sub := λ x y, op (unop x - unop y) } | instance | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 104 | 105 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
[has_neg α] : has_neg αᵐᵒᵖ :=
{ neg := λ x, op $ -(unop x) } | [has_neg α] : has_neg αᵐᵒᵖ | { neg := λ x, op $ -(unop x) } | instance | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 107 | 108 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
[has_involutive_neg α] : has_involutive_neg αᵐᵒᵖ :=
{ neg_neg := λ a, unop_injective $ neg_neg _,
..mul_opposite.has_neg α } | [has_involutive_neg α] : has_involutive_neg αᵐᵒᵖ | { neg_neg := λ a, unop_injective $ neg_neg _,
..mul_opposite.has_neg α } | instance | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [
"has_involutive_neg"
] | null | 110 | 112 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
[has_mul α] : has_mul αᵐᵒᵖ :=
{ mul := λ x y, op (unop y * unop x) } | [has_mul α] : has_mul αᵐᵒᵖ | { mul := λ x y, op (unop y * unop x) } | instance | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 114 | 115 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
[has_inv α] : has_inv αᵐᵒᵖ :=
{ inv := λ x, op $ (unop x)⁻¹ } | [has_inv α] : has_inv αᵐᵒᵖ | { inv := λ x, op $ (unop x)⁻¹ } | instance | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 117 | 118 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
[has_involutive_inv α] : has_involutive_inv αᵐᵒᵖ :=
{ inv_inv := λ a, unop_injective $ inv_inv _,
..mul_opposite.has_inv α } | [has_involutive_inv α] : has_involutive_inv αᵐᵒᵖ | { inv_inv := λ a, unop_injective $ inv_inv _,
..mul_opposite.has_inv α } | instance | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [
"has_involutive_inv",
"inv_inv"
] | null | 120 | 122 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
(R : Type*) [has_smul R α] : has_smul R αᵐᵒᵖ :=
{ smul := λ c x, op (c • unop x) } | (R : Type*) [has_smul R α] : has_smul R αᵐᵒᵖ | { smul := λ c x, op (c • unop x) } | instance | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [
"has_smul"
] | null | 124 | 125 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_zero [has_zero α] : op (0 : α) = 0 := rfl | op_zero [has_zero α] : op (0 : α) = 0 | rfl | lemma | mul_opposite.op_zero | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 130 | 130 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_zero [has_zero α] : unop (0 : αᵐᵒᵖ) = 0 := rfl | unop_zero [has_zero α] : unop (0 : αᵐᵒᵖ) = 0 | rfl | lemma | mul_opposite.unop_zero | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 131 | 131 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_one [has_one α] : op (1 : α) = 1 := rfl | op_one [has_one α] : op (1 : α) = 1 | rfl | lemma | mul_opposite.op_one | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 133 | 133 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_one [has_one α] : unop (1 : αᵐᵒᵖ) = 1 := rfl | unop_one [has_one α] : unop (1 : αᵐᵒᵖ) = 1 | rfl | lemma | mul_opposite.unop_one | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 134 | 134 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_add [has_add α] (x y : α) : op (x + y) = op x + op y := rfl | op_add [has_add α] (x y : α) : op (x + y) = op x + op y | rfl | lemma | mul_opposite.op_add | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 138 | 138 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_add [has_add α] (x y : αᵐᵒᵖ) : unop (x + y) = unop x + unop y := rfl | unop_add [has_add α] (x y : αᵐᵒᵖ) : unop (x + y) = unop x + unop y | rfl | lemma | mul_opposite.unop_add | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 139 | 139 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_neg [has_neg α] (x : α) : op (-x) = -op x := rfl | op_neg [has_neg α] (x : α) : op (-x) = -op x | rfl | lemma | mul_opposite.op_neg | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 141 | 141 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_neg [has_neg α] (x : αᵐᵒᵖ) : unop (-x) = -unop x := rfl | unop_neg [has_neg α] (x : αᵐᵒᵖ) : unop (-x) = -unop x | rfl | lemma | mul_opposite.unop_neg | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 142 | 142 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_mul [has_mul α] (x y : α) : op (x * y) = op y * op x := rfl | op_mul [has_mul α] (x y : α) : op (x * y) = op y * op x | rfl | lemma | mul_opposite.op_mul | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 144 | 144 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_mul [has_mul α] (x y : αᵐᵒᵖ) : unop (x * y) = unop y * unop x := rfl | unop_mul [has_mul α] (x y : αᵐᵒᵖ) : unop (x * y) = unop y * unop x | rfl | lemma | mul_opposite.unop_mul | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 145 | 145 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_inv [has_inv α] (x : α) : op (x⁻¹) = (op x)⁻¹ := rfl | op_inv [has_inv α] (x : α) : op (x⁻¹) = (op x)⁻¹ | rfl | lemma | mul_opposite.op_inv | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 147 | 147 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_inv [has_inv α] (x : αᵐᵒᵖ) : unop (x⁻¹) = (unop x)⁻¹ := rfl | unop_inv [has_inv α] (x : αᵐᵒᵖ) : unop (x⁻¹) = (unop x)⁻¹ | rfl | lemma | mul_opposite.unop_inv | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 148 | 148 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_sub [has_sub α] (x y : α) : op (x - y) = op x - op y := rfl | op_sub [has_sub α] (x y : α) : op (x - y) = op x - op y | rfl | lemma | mul_opposite.op_sub | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 150 | 150 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_sub [has_sub α] (x y : αᵐᵒᵖ) : unop (x - y) = unop x - unop y := rfl | unop_sub [has_sub α] (x y : αᵐᵒᵖ) : unop (x - y) = unop x - unop y | rfl | lemma | mul_opposite.unop_sub | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 151 | 151 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_smul {R : Type*} [has_smul R α] (c : R) (a : α) :
op (c • a) = c • op a := rfl | op_smul {R : Type*} [has_smul R α] (c : R) (a : α) :
op (c • a) = c • op a | rfl | lemma | mul_opposite.op_smul | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [
"has_smul"
] | null | 153 | 154 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_smul {R : Type*} [has_smul R α] (c : R) (a : αᵐᵒᵖ) :
unop (c • a) = c • unop a := rfl | unop_smul {R : Type*} [has_smul R α] (c : R) (a : αᵐᵒᵖ) :
unop (c • a) = c • unop a | rfl | lemma | mul_opposite.unop_smul | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [
"has_smul"
] | null | 156 | 157 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_eq_zero_iff [has_zero α] (a : αᵐᵒᵖ) : a.unop = (0 : α) ↔ a = (0 : αᵐᵒᵖ) :=
unop_injective.eq_iff' rfl | unop_eq_zero_iff [has_zero α] (a : αᵐᵒᵖ) : a.unop = (0 : α) ↔ a = (0 : αᵐᵒᵖ) | unop_injective.eq_iff' rfl | lemma | mul_opposite.unop_eq_zero_iff | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 163 | 164 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_eq_zero_iff [has_zero α] (a : α) : op a = (0 : αᵐᵒᵖ) ↔ a = (0 : α) :=
op_injective.eq_iff' rfl | op_eq_zero_iff [has_zero α] (a : α) : op a = (0 : αᵐᵒᵖ) ↔ a = (0 : α) | op_injective.eq_iff' rfl | lemma | mul_opposite.op_eq_zero_iff | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 166 | 167 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_ne_zero_iff [has_zero α] (a : αᵐᵒᵖ) : a.unop ≠ (0 : α) ↔ a ≠ (0 : αᵐᵒᵖ) :=
not_congr $ unop_eq_zero_iff a | unop_ne_zero_iff [has_zero α] (a : αᵐᵒᵖ) : a.unop ≠ (0 : α) ↔ a ≠ (0 : αᵐᵒᵖ) | not_congr $ unop_eq_zero_iff a | lemma | mul_opposite.unop_ne_zero_iff | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 169 | 170 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_ne_zero_iff [has_zero α] (a : α) : op a ≠ (0 : αᵐᵒᵖ) ↔ a ≠ (0 : α) :=
not_congr $ op_eq_zero_iff a | op_ne_zero_iff [has_zero α] (a : α) : op a ≠ (0 : αᵐᵒᵖ) ↔ a ≠ (0 : α) | not_congr $ op_eq_zero_iff a | lemma | mul_opposite.op_ne_zero_iff | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 172 | 173 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_eq_one_iff [has_one α] (a : αᵐᵒᵖ) : a.unop = 1 ↔ a = 1 :=
unop_injective.eq_iff' rfl | unop_eq_one_iff [has_one α] (a : αᵐᵒᵖ) : a.unop = 1 ↔ a = 1 | unop_injective.eq_iff' rfl | lemma | mul_opposite.unop_eq_one_iff | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 175 | 176 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_eq_one_iff [has_one α] (a : α) : op a = 1 ↔ a = 1 :=
op_injective.eq_iff' rfl | op_eq_one_iff [has_one α] (a : α) : op a = 1 ↔ a = 1 | op_injective.eq_iff' rfl | lemma | mul_opposite.op_eq_one_iff | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 178 | 179 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
[has_one α] : has_one αᵃᵒᵖ := { one := op 1 } | [has_one α] : has_one αᵃᵒᵖ | { one := op 1 } | instance | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | null | 185 | 185 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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