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
value |
|---|---|---|---|---|---|---|---|---|---|---|
frobenius_mul : frobenius R p (x * y) = frobenius R p x * frobenius R p y | (frobenius R p).map_mul x y | theorem | frobenius_mul | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"frobenius",
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_one : frobenius R p 1 = 1 | one_pow _ | theorem | frobenius_one | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"frobenius",
"one_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.map_frobenius : f (frobenius R p x) = frobenius S p (f x) | f.map_pow x p | theorem | monoid_hom.map_frobenius | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"frobenius"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.map_frobenius : g (frobenius R p x) = frobenius S p (g x) | g.map_pow x p | theorem | ring_hom.map_frobenius | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"frobenius"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.map_iterate_frobenius (n : ℕ) :
f (frobenius R p^[n] x) = (frobenius S p^[n] (f x)) | function.semiconj.iterate_right (f.map_frobenius p) n x | theorem | monoid_hom.map_iterate_frobenius | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"frobenius",
"function.semiconj.iterate_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.map_iterate_frobenius (n : ℕ) :
g (frobenius R p^[n] x) = (frobenius S p^[n] (g x)) | g.to_monoid_hom.map_iterate_frobenius p x n | theorem | ring_hom.map_iterate_frobenius | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"frobenius"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.iterate_map_frobenius (f : R →* R) (p : ℕ) [fact p.prime] [char_p R p] (n : ℕ) :
f^[n] (frobenius R p x) = frobenius R p (f^[n] x) | f.iterate_map_pow _ _ _ | theorem | monoid_hom.iterate_map_frobenius | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"fact",
"frobenius"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.iterate_map_frobenius (f : R →+* R) (p : ℕ) [fact p.prime] [char_p R p] (n : ℕ) :
f^[n] (frobenius R p x) = frobenius R p (f^[n] x) | f.iterate_map_pow _ _ _ | theorem | ring_hom.iterate_map_frobenius | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"fact",
"frobenius"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_zero : frobenius R p 0 = 0 | (frobenius R p).map_zero | theorem | frobenius_zero | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"frobenius"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_add : frobenius R p (x + y) = frobenius R p x + frobenius R p y | (frobenius R p).map_add x y | theorem | frobenius_add | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"frobenius"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_nat_cast (n : ℕ) : frobenius R p n = n | map_nat_cast (frobenius R p) n | theorem | frobenius_nat_cast | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"frobenius",
"map_nat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list_sum_pow_char (l : list R) : l.sum ^ p = (l.map (^ p)).sum | (frobenius R p).map_list_sum _ | lemma | list_sum_pow_char | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"frobenius"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiset_sum_pow_char (s : multiset R) : s.sum ^ p = (s.map (^ p)).sum | (frobenius R p).map_multiset_sum _ | lemma | multiset_sum_pow_char | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"frobenius",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_pow_char {ι : Type*} (s : finset ι) (f : ι → R) :
(∑ i in s, f i) ^ p = ∑ i in s, f i ^ p | (frobenius R p).map_sum _ _ | lemma | sum_pow_char | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"finset",
"frobenius"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_neg : frobenius R p (-x) = -frobenius R p x | (frobenius R p).map_neg x | theorem | frobenius_neg | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"frobenius"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_sub : frobenius R p (x - y) = frobenius R p x - frobenius R p y | (frobenius R p).map_sub x y | theorem | frobenius_sub | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"frobenius"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius_inj [comm_ring R] [is_reduced R]
(p : ℕ) [fact p.prime] [char_p R p] :
function.injective (frobenius R p) | λ x h H, by { rw ← sub_eq_zero at H ⊢, rw ← frobenius_sub at H, exact is_reduced.eq_zero _ ⟨_,H⟩ } | theorem | frobenius_inj | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"comm_ring",
"fact",
"frobenius",
"frobenius_sub",
"is_reduced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_square_of_char_two' {R : Type*} [finite R] [comm_ring R] [is_reduced R] [char_p R 2]
(a : R) : is_square a | by { casesI nonempty_fintype R, exact exists_imp_exists (λ b h, pow_two b ▸ eq.symm h)
(((fintype.bijective_iff_injective_and_card _).mpr ⟨frobenius_inj R 2, rfl⟩).surjective a) } | lemma | is_square_of_char_two' | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"comm_ring",
"finite",
"fintype.bijective_iff_injective_and_card",
"is_reduced",
"is_square",
"nonempty_fintype",
"pow_two"
] | If `ring_char R = 2`, where `R` is a finite reduced commutative ring,
then every `a : R` is a square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
char_p_to_char_zero (R : Type*) [add_group_with_one R] [char_p R 0] :
char_zero R | char_zero_of_inj_zero $
λ n h0, eq_zero_of_zero_dvd ((cast_eq_zero_iff R 0 n).mp h0) | lemma | char_p.char_p_to_char_zero | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"add_group_with_one",
"char_p",
"char_zero",
"char_zero_of_inj_zero",
"eq_zero_of_zero_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_eq_mod (p : ℕ) [char_p R p] (k : ℕ) : (k : R) = (k % p : ℕ) | calc (k : R) = ↑(k % p + p * (k / p)) : by rw [nat.mod_add_div]
... = ↑(k % p) : by simp [cast_eq_zero] | lemma | char_p.cast_eq_mod | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_ne_zero_of_finite (p : ℕ) [char_p R p] [finite R] : p ≠ 0 | begin
unfreezingI { rintro rfl },
haveI : char_zero R := char_p_to_char_zero R,
casesI nonempty_fintype R,
exact absurd nat.cast_injective (not_injective_infinite_finite (coe : ℕ → R))
end | theorem | char_p.char_ne_zero_of_finite | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"char_zero",
"finite",
"nat.cast_injective",
"nonempty_fintype",
"not_injective_infinite_finite"
] | The characteristic of a finite ring cannot be zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_char_ne_zero_of_finite [finite R] : ring_char R ≠ 0 | char_ne_zero_of_finite R (ring_char R) | lemma | char_p.ring_char_ne_zero_of_finite | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"finite",
"ring_char"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_prime_pow_mul_eq_one_iff (p k m : ℕ) [fact p.prime]
[char_p R p] (x : R) :
x ^ (p ^ k * m) = 1 ↔ x ^ m = 1 | begin
induction k with k hk,
{ rw [pow_zero, one_mul] },
{ refine ⟨λ h, _, λ h, _⟩,
{ rw [pow_succ, mul_assoc, pow_mul', ← frobenius_def, ← frobenius_one p] at h,
exact hk.1 (frobenius_inj R p h) },
{ rw [pow_mul', h, one_pow] } }
end | lemma | char_p.pow_prime_pow_mul_eq_one_iff | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"fact",
"frobenius_def",
"frobenius_inj",
"frobenius_one",
"mul_assoc",
"one_mul",
"one_pow",
"pow_mul'",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_ne_one [nontrivial R] (p : ℕ) [hc : char_p R p] : p ≠ 1 | assume hp : p = 1,
have ( 1 : R) = 0, by simpa using (cast_eq_zero_iff R p 1).mpr (hp ▸ dvd_refl p),
absurd this one_ne_zero | theorem | char_p.char_ne_one | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"dvd_refl",
"nontrivial",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_is_prime_of_two_le (p : ℕ) [hc : char_p R p] (hp : 2 ≤ p) : nat.prime p | suffices ∀d ∣ p, d = 1 ∨ d = p, from nat.prime_def_lt''.mpr ⟨hp, this⟩,
assume (d : ℕ) (hdvd : ∃ e, p = d * e),
let ⟨e, hmul⟩ := hdvd in
have (p : R) = 0, from (cast_eq_zero_iff R p p).mpr (dvd_refl p),
have (d : R) * e = 0, from (@cast_mul R _ d e) ▸ (hmul ▸ this),
or.elim (eq_zero_or_eq_zero_of_mul_eq_zero this)
(a... | theorem | char_p.char_is_prime_of_two_le | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"dvd_antisymm",
"dvd_refl",
"mul_right_cancel₀",
"nat.prime",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_is_prime_or_zero (p : ℕ) [hc : char_p R p] : nat.prime p ∨ p = 0 | match p, hc with
| 0, _ := or.inr rfl
| 1, hc := absurd (eq.refl (1 : ℕ)) (@char_ne_one R _ _ (1 : ℕ) hc)
| (m+2), hc := or.inl (@char_is_prime_of_two_le R _ _ (m+2) hc (nat.le_add_left 2 m))
end | theorem | char_p.char_is_prime_or_zero | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"nat.prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_is_prime_of_pos (p : ℕ) [ne_zero p] [char_p R p] : fact p.prime | ⟨(char_p.char_is_prime_or_zero R _).resolve_right $ ne_zero.ne p⟩ | lemma | char_p.char_is_prime_of_pos | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"char_p.char_is_prime_or_zero",
"fact",
"ne_zero",
"ne_zero.ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_is_prime (p : ℕ) [char_p R p] :
p.prime | or.resolve_right (char_is_prime_or_zero R p) (char_ne_zero_of_finite R p) | theorem | char_p.char_is_prime | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
false_of_nontrivial_of_char_one [nontrivial R] [char_p R 1] : false | false_of_nontrivial_of_subsingleton R | lemma | char_p.false_of_nontrivial_of_char_one | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"false_of_nontrivial_of_subsingleton",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_char_ne_one [nontrivial R] : ring_char R ≠ 1 | by { intros h, apply zero_ne_one' R, symmetry, rw [←nat.cast_one, ring_char.spec, h], } | lemma | char_p.ring_char_ne_one | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"nontrivial",
"ring_char",
"ring_char.spec",
"zero_ne_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nontrivial_of_char_ne_one {v : ℕ} (hv : v ≠ 1) [hr : char_p R v] :
nontrivial R | ⟨⟨(1 : ℕ), 0, λ h, hv $ by rwa [char_p.cast_eq_zero_iff _ v, nat.dvd_one] at h; assumption ⟩⟩ | lemma | char_p.nontrivial_of_char_ne_one | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"nat.dvd_one",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_char_of_prime_eq_zero [nontrivial R] {p : ℕ}
(hprime : nat.prime p) (hp0 : (p : R) = 0) : ring_char R = p | or.resolve_left ((nat.dvd_prime hprime).1 (ring_char.dvd hp0)) ring_char_ne_one | lemma | char_p.ring_char_of_prime_eq_zero | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"nat.dvd_prime",
"nat.prime",
"nontrivial",
"ring_char",
"ring_char.dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring.two_ne_zero {R : Type*} [non_assoc_semiring R] [nontrivial R] (hR : ring_char R ≠ 2) :
(2 : R) ≠ 0 | begin
rw [ne.def, (by norm_cast : (2 : R) = (2 : ℕ)), ring_char.spec, nat.dvd_prime nat.prime_two],
exact mt (or_iff_left hR).mp char_p.ring_char_ne_one,
end | lemma | ring.two_ne_zero | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p.ring_char_ne_one",
"nat.dvd_prime",
"nat.prime_two",
"non_assoc_semiring",
"nontrivial",
"or_iff_left",
"ring_char",
"ring_char.spec"
] | We have `2 ≠ 0` in a nontrivial ring whose characteristic is not `2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring.neg_one_ne_one_of_char_ne_two {R : Type*} [non_assoc_ring R] [nontrivial R]
(hR : ring_char R ≠ 2) :
(-1 : R) ≠ 1 | λ h, ring.two_ne_zero hR (neg_eq_iff_add_eq_zero.mp h) | lemma | ring.neg_one_ne_one_of_char_ne_two | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"non_assoc_ring",
"nontrivial",
"ring.two_ne_zero",
"ring_char"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring.eq_self_iff_eq_zero_of_char_ne_two {R : Type*} [non_assoc_ring R] [nontrivial R]
[no_zero_divisors R] (hR : ring_char R ≠ 2) {a : R} :
-a = a ↔ a = 0 | ⟨λ h, (mul_eq_zero.mp $ (two_mul a).trans $ neg_eq_iff_add_eq_zero.mp h).resolve_left
(ring.two_ne_zero hR),
λ h, ((congr_arg (λ x, - x) h).trans neg_zero).trans h.symm⟩ | lemma | ring.eq_self_iff_eq_zero_of_char_ne_two | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"no_zero_divisors",
"non_assoc_ring",
"nontrivial",
"ring.two_ne_zero",
"ring_char",
"two_mul"
] | Characteristic `≠ 2` in a domain implies that `-a = a` iff `a = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
char_p_of_ne_zero (hn : fintype.card R = n) (hR : ∀ i < n, (i : R) = 0 → i = 0) :
char_p R n | { cast_eq_zero_iff :=
begin
have H : (n : R) = 0, by { rw [← hn, char_p.cast_card_eq_zero] },
intro k,
split,
{ intro h,
rw [← nat.mod_add_div k n, nat.cast_add, nat.cast_mul, H, zero_mul, add_zero] at h,
rw nat.dvd_iff_mod_eq_zero,
apply hR _ (nat.mod_lt _ _) h,
rw [← hn, fint... | lemma | char_p_of_ne_zero | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"char_p.cast_card_eq_zero",
"fintype.card",
"fintype.card_pos_iff",
"nat.cast_add",
"nat.cast_mul",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_p_of_prime_pow_injective (R) [ring R] [fintype R] (p : ℕ) [hp : fact p.prime] (n : ℕ)
(hn : fintype.card R = p ^ n) (hR : ∀ i ≤ n, (p ^ i : R) = 0 → i = n) :
char_p R (p ^ n) | begin
obtain ⟨c, hc⟩ := char_p.exists R, resetI,
have hcpn : c ∣ p ^ n,
{ rw [← char_p.cast_eq_zero_iff R c, ← hn, char_p.cast_card_eq_zero], },
obtain ⟨i, hi, hc⟩ : ∃ i ≤ n, c = p ^ i, by rwa nat.dvd_prime_pow hp.1 at hcpn,
obtain rfl : i = n,
{ apply hR i hi, rw [← nat.cast_pow, ← hc, char_p.cast_eq_zero]... | lemma | char_p_of_prime_pow_injective | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"char_p.cast_card_eq_zero",
"char_p.cast_eq_zero",
"char_p.exists",
"fact",
"fintype",
"fintype.card",
"nat.cast_pow",
"nat.dvd_prime_pow",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.char_p [char_p S p] : char_p (R × S) p | by convert nat.lcm.char_p R S p p; simp | instance | prod.char_p | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p"
] | The characteristic of the product of two rings of the same characteristic
is the same as the characteristic of the rings | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ulift.char_p [add_monoid_with_one R] (p : ℕ) [char_p R p] : char_p (ulift.{v} R) p | { cast_eq_zero_iff := λ n, iff.trans (ulift.ext_iff _ _) $ char_p.cast_eq_zero_iff R p n } | instance | ulift.char_p | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"add_monoid_with_one",
"char_p"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_opposite.char_p [add_monoid_with_one R] (p : ℕ) [char_p R p] : char_p (Rᵐᵒᵖ) p | { cast_eq_zero_iff := λ n, mul_opposite.unop_inj.symm.trans $ char_p.cast_eq_zero_iff R p n } | instance | mul_opposite.char_p | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"add_monoid_with_one",
"char_p"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.cast_inj_on_of_ring_char_ne_two {R : Type*} [non_assoc_ring R] [nontrivial R]
(hR : ring_char R ≠ 2) :
({0, 1, -1} : set ℤ).inj_on (coe : ℤ → R) | begin
intros a ha b hb h,
apply eq_of_sub_eq_zero,
by_contra hf,
change a = 0 ∨ a = 1 ∨ a = -1 at ha,
change b = 0 ∨ b = 1 ∨ b = -1 at hb,
have hh : a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2 := by
{ rcases ha with ha | ha | ha; rcases hb with hb | hb | hb,
swap 5, swap 9, -- move goals with `a = b... | lemma | int.cast_inj_on_of_ring_char_ne_two | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"by_contra",
"non_assoc_ring",
"nontrivial",
"one_ne_zero",
"ring.two_ne_zero",
"ring_char"
] | If two integers from `{0, 1, -1}` result in equal elements in a ring `R`
that is nontrivial and of characteristic not `2`, then they are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_not_dvd [char_p R p] (h : ¬ p ∣ n) : ne_zero (n : R) | ⟨(char_p.cast_eq_zero_iff R p n).not.mpr h⟩ | lemma | ne_zero.of_not_dvd | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_char_dvd (p : ℕ) [char_p R p] (k : ℕ) [h : ne_zero (k : R)] : ¬ p ∣ k | by rwa [←char_p.cast_eq_zero_iff R p k, ←ne.def, ←ne_zero_iff] | lemma | ne_zero.not_char_dvd | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_iff_not_dvd_char_of_ring_char_ne_zero (R : Type*) [comm_ring R] (p : ℕ) [fact p.prime]
(hR : ring_char R ≠ 0) :
is_unit (p : R) ↔ ¬ p ∣ ring_char R | begin
have hch := char_p.cast_eq_zero R (ring_char R),
have hp : p.prime := fact.out p.prime,
split,
{ rintros h₁ ⟨q, hq⟩,
rcases is_unit.exists_left_inv h₁ with ⟨a, ha⟩,
have h₃ : ¬ ring_char R ∣ q :=
begin
rintro ⟨r, hr⟩,
rw [hr, ← mul_assoc, mul_comm p, mul_assoc] at hq,
nth_rew... | lemma | is_unit_iff_not_dvd_char_of_ring_char_ne_zero | algebra.char_p | src/algebra/char_p/char_and_card.lean | [
"algebra.char_p.basic",
"group_theory.perm.cycle.type"
] | [
"char_p.cast_eq_zero",
"char_p.int_cast_eq_zero_iff",
"comm_ring",
"fact",
"is_coprime",
"is_unit",
"is_unit.exists_left_inv",
"is_unit_of_mul_eq_one",
"mul_assoc",
"mul_comm",
"mul_left_cancel₀",
"mul_one",
"mul_zero",
"nat.cast_mul",
"nat.prime.not_dvd_one",
"one_mul",
"ring_char"
... | A prime `p` is a unit in a commutative ring `R` of nonzero characterstic iff it does not divide
the characteristic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit_iff_not_dvd_char (R : Type*) [comm_ring R] (p : ℕ) [fact p.prime] [finite R] :
is_unit (p : R) ↔ ¬ p ∣ ring_char R | is_unit_iff_not_dvd_char_of_ring_char_ne_zero R p $ char_p.char_ne_zero_of_finite R (ring_char R) | lemma | is_unit_iff_not_dvd_char | algebra.char_p | src/algebra/char_p/char_and_card.lean | [
"algebra.char_p.basic",
"group_theory.perm.cycle.type"
] | [
"char_p.char_ne_zero_of_finite",
"comm_ring",
"fact",
"finite",
"is_unit",
"is_unit_iff_not_dvd_char_of_ring_char_ne_zero",
"ring_char"
] | A prime `p` is a unit in a finite commutative ring `R`
iff it does not divide the characteristic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prime_dvd_char_iff_dvd_card {R : Type*} [comm_ring R] [fintype R] (p : ℕ) [fact p.prime] :
p ∣ ring_char R ↔ p ∣ fintype.card R | begin
refine ⟨λ h, h.trans $ int.coe_nat_dvd.mp $ (char_p.int_cast_eq_zero_iff R (ring_char R)
(fintype.card R)).mp $ by exact_mod_cast char_p.cast_card_eq_zero R, λ h, _⟩,
by_contra h₀,
rcases exists_prime_add_order_of_dvd_card p h with ⟨r, hr⟩,
have hr₁ := add_order_of_nsmul_eq_zero r,
rw [hr, nsmul_eq_... | lemma | prime_dvd_char_iff_dvd_card | algebra.char_p | src/algebra/char_p/char_and_card.lean | [
"algebra.char_p.basic",
"group_theory.perm.cycle.type"
] | [
"by_contra",
"char_p.cast_card_eq_zero",
"char_p.int_cast_eq_zero_iff",
"comm_ring",
"exists_prime_add_order_of_dvd_card",
"fact",
"fintype",
"fintype.card",
"is_unit.exists_left_inv",
"is_unit_iff_not_dvd_char",
"mul_assoc",
"mul_zero",
"nat.prime.ne_one",
"nsmul_eq_mul",
"one_mul",
"... | The prime divisors of the characteristic of a finite commutative ring are exactly
the prime divisors of its cardinality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_is_unit_prime_of_dvd_card {R : Type*} [comm_ring R] [fintype R] (p : ℕ) [fact p.prime]
(hp : p ∣ fintype.card R) : ¬ is_unit (p : R) | mt (is_unit_iff_not_dvd_char R p).mp (not_not.mpr ((prime_dvd_char_iff_dvd_card p).mpr hp)) | lemma | not_is_unit_prime_of_dvd_card | algebra.char_p | src/algebra/char_p/char_and_card.lean | [
"algebra.char_p.basic",
"group_theory.perm.cycle.type"
] | [
"comm_ring",
"fact",
"fintype",
"fintype.card",
"is_unit",
"is_unit_iff_not_dvd_char",
"prime_dvd_char_iff_dvd_card"
] | A prime that does not divide the cardinality of a finite commutative ring `R`
is a unit in `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_char (R : Type u) [semiring R] : ℕ → Prop
| zero [char_zero R] : exp_char 1
| prime {q : ℕ} (hprime : q.prime) [hchar : char_p R q] : exp_char q | class inductive | exp_char | algebra.char_p | src/algebra/char_p/exp_char.lean | [
"algebra.char_p.basic",
"data.nat.prime"
] | [
"char_p",
"char_zero",
"prime",
"semiring"
] | The definition of the exponential characteristic of a semiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_char_one_of_char_zero (q : ℕ) [hp : char_p R 0] [hq : exp_char R q] :
q = 1 | begin
casesI hq with q hq_one hq_prime,
{ refl },
{ exact false.elim (lt_irrefl _ ((hp.eq R hq_hchar).symm ▸ hq_prime : (0 : ℕ).prime).pos) }
end | lemma | exp_char_one_of_char_zero | algebra.char_p | src/algebra/char_p/exp_char.lean | [
"algebra.char_p.basic",
"data.nat.prime"
] | [
"char_p",
"exp_char",
"prime"
] | The exponential characteristic is one if the characteristic is zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
char_eq_exp_char_iff (p q : ℕ) [hp : char_p R p] [hq : exp_char R q] :
p = q ↔ p.prime | begin
casesI hq with q hq_one hq_prime,
{ apply iff_of_false,
{ unfreezingI {rintro rfl},
exact one_ne_zero (hp.eq R (char_p.of_char_zero R)) },
{ intro pprime,
rw (char_p.eq R hp infer_instance : p = 0) at pprime,
exact nat.not_prime_zero pprime } },
{ exact ⟨λ hpq, hpq.symm ▸ hq_prime,... | theorem | char_eq_exp_char_iff | algebra.char_p | src/algebra/char_p/exp_char.lean | [
"algebra.char_p.basic",
"data.nat.prime"
] | [
"char_p",
"char_p.eq",
"char_p.of_char_zero",
"exp_char",
"iff_of_false",
"nat.not_prime_zero",
"one_ne_zero"
] | The characteristic equals the exponential characteristic iff the former is prime. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
char_zero_of_exp_char_one (p : ℕ) [hp : char_p R p] [hq : exp_char R 1] :
p = 0 | begin
casesI hq,
{ exact char_p.eq R hp infer_instance, },
{ exact false.elim (char_p.char_ne_one R 1 rfl), }
end | lemma | char_zero_of_exp_char_one | algebra.char_p | src/algebra/char_p/exp_char.lean | [
"algebra.char_p.basic",
"data.nat.prime"
] | [
"char_p",
"char_p.char_ne_one",
"char_p.eq",
"exp_char"
] | The exponential characteristic is one if the characteristic is zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
char_zero_of_exp_char_one' [hq : exp_char R 1] : char_zero R | begin
casesI hq,
{ assumption, },
{ exact false.elim (char_p.char_ne_one R 1 rfl), }
end | instance | char_zero_of_exp_char_one' | algebra.char_p | src/algebra/char_p/exp_char.lean | [
"algebra.char_p.basic",
"data.nat.prime"
] | [
"char_p.char_ne_one",
"char_zero",
"exp_char"
] | The characteristic is zero if the exponential characteristic is one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_char_one_iff_char_zero (p q : ℕ) [char_p R p] [exp_char R q] :
q = 1 ↔ p = 0 | begin
split,
{ unfreezingI {rintro rfl},
exact char_zero_of_exp_char_one R p, },
{ unfreezingI {rintro rfl},
exact exp_char_one_of_char_zero R q, }
end | theorem | exp_char_one_iff_char_zero | algebra.char_p | src/algebra/char_p/exp_char.lean | [
"algebra.char_p.basic",
"data.nat.prime"
] | [
"char_p",
"char_zero_of_exp_char_one",
"exp_char",
"exp_char_one_of_char_zero"
] | The exponential characteristic is one iff the characteristic is zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
char_prime_of_ne_zero {p : ℕ} [hp : char_p R p] (p_ne_zero : p ≠ 0) : nat.prime p | begin
cases char_p.char_is_prime_or_zero R p with h h,
{ exact h, },
{ contradiction, }
end | lemma | char_prime_of_ne_zero | algebra.char_p | src/algebra/char_p/exp_char.lean | [
"algebra.char_p.basic",
"data.nat.prime"
] | [
"char_p",
"char_p.char_is_prime_or_zero",
"nat.prime"
] | A helper lemma: the characteristic is prime if it is non-zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_char_is_prime_or_one (q : ℕ) [hq : exp_char R q] : nat.prime q ∨ q = 1 | or_iff_not_imp_right.mpr $ λ h,
begin
casesI char_p.exists R with p hp,
have p_ne_zero : p ≠ 0,
{ intro p_zero,
haveI : char_p R 0, { rwa ←p_zero },
have : q = 1 := exp_char_one_of_char_zero R q,
contradiction, },
have p_eq_q : p = q := (char_eq_exp_char_iff R p q).mpr (char_prime_of_ne_zero R p_ne_... | theorem | exp_char_is_prime_or_one | algebra.char_p | src/algebra/char_p/exp_char.lean | [
"algebra.char_p.basic",
"data.nat.prime"
] | [
"char_eq_exp_char_iff",
"char_p",
"char_p.char_is_prime_or_zero",
"char_p.exists",
"char_prime_of_ne_zero",
"exp_char",
"exp_char_one_of_char_zero",
"nat.prime"
] | The exponential characteristic is a prime number or one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
invertible_of_ring_char_not_dvd
{t : ℕ} (not_dvd : ¬(ring_char K ∣ t)) : invertible (t : K) | invertible_of_nonzero (λ h, not_dvd ((ring_char.spec K t).mp h)) | def | invertible_of_ring_char_not_dvd | algebra.char_p | src/algebra/char_p/invertible.lean | [
"algebra.invertible",
"algebra.char_p.basic"
] | [
"invertible",
"invertible_of_nonzero",
"ring_char",
"ring_char.spec"
] | A natural number `t` is invertible in a field `K` if the charactistic of `K` does not divide
`t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_ring_char_dvd_of_invertible {t : ℕ} [invertible (t : K)] :
¬(ring_char K ∣ t) | begin
rw [← ring_char.spec, ← ne.def],
exact nonzero_of_invertible (t : K)
end | lemma | not_ring_char_dvd_of_invertible | algebra.char_p | src/algebra/char_p/invertible.lean | [
"algebra.invertible",
"algebra.char_p.basic"
] | [
"invertible",
"nonzero_of_invertible",
"ring_char",
"ring_char.spec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
invertible_of_char_p_not_dvd {p : ℕ} [char_p K p]
{t : ℕ} (not_dvd : ¬(p ∣ t)) : invertible (t : K) | invertible_of_nonzero (λ h, not_dvd ((char_p.cast_eq_zero_iff K p t).mp h)) | def | invertible_of_char_p_not_dvd | algebra.char_p | src/algebra/char_p/invertible.lean | [
"algebra.invertible",
"algebra.char_p.basic"
] | [
"char_p",
"invertible",
"invertible_of_nonzero"
] | A natural number `t` is invertible in a field `K` of charactistic `p` if `p` does not divide
`t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
invertible_of_pos [char_zero K] (n : ℕ) [ne_zero n] : invertible (n : K) | invertible_of_nonzero $ ne_zero.out | instance | invertible_of_pos | algebra.char_p | src/algebra/char_p/invertible.lean | [
"algebra.invertible",
"algebra.char_p.basic"
] | [
"char_zero",
"invertible",
"invertible_of_nonzero",
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
invertible_succ (n : ℕ) : invertible (n.succ : K) | invertible_of_nonzero (nat.cast_ne_zero.mpr (nat.succ_ne_zero _)) | instance | invertible_succ | algebra.char_p | src/algebra/char_p/invertible.lean | [
"algebra.invertible",
"algebra.char_p.basic"
] | [
"invertible",
"invertible_of_nonzero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
invertible_two : invertible (2 : K) | invertible_of_nonzero (by exact_mod_cast (dec_trivial : 2 ≠ 0)) | instance | invertible_two | algebra.char_p | src/algebra/char_p/invertible.lean | [
"algebra.invertible",
"algebra.char_p.basic"
] | [
"invertible",
"invertible_of_nonzero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
invertible_three : invertible (3 : K) | invertible_of_nonzero (by exact_mod_cast (dec_trivial : 3 ≠ 0)) | instance | invertible_three | algebra.char_p | src/algebra/char_p/invertible.lean | [
"algebra.invertible",
"algebra.char_p.basic"
] | [
"invertible",
"invertible_of_nonzero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_p_zero_or_prime_power (R : Type*) [comm_ring R] [local_ring R] (q : ℕ)
[char_R_q : char_p R q] : q = 0 ∨ is_prime_pow q | begin
/- Assume `q := char(R)` is not zero. -/
apply or_iff_not_imp_left.2,
intro q_pos,
let K := local_ring.residue_field R,
haveI RM_char := ring_char.char_p K,
let r := ring_char K,
let n := (q.factorization) r,
/- `r := char(R/m)` is either prime or zero: -/
cases char_p.char_is_prime_or_zero K r... | theorem | char_p_zero_or_prime_power | algebra.char_p | src/algebra/char_p/local_ring.lean | [
"algebra.char_p.basic",
"ring_theory.ideal.local_ring",
"algebra.is_prime_pow",
"data.nat.factorization.basic"
] | [
"by_contradiction",
"char_p",
"char_p.cast_eq_zero",
"char_p.char_is_prime_or_zero",
"char_p.char_ne_one",
"char_p.char_p_to_char_zero",
"char_p.eq",
"char_p.of_char_zero",
"char_zero",
"comm_ring",
"ideal.quotient.eq_zero_iff_mem",
"is_prime_pow",
"is_unit",
"local_ring",
"local_ring.re... | In a local ring the characteristics is either zero or a prime power. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mixed_char_zero (p : ℕ) : Prop | [to_char_zero : char_zero R]
(char_p_quotient : ∃ (I : ideal R), (I ≠ ⊤) ∧ char_p (R ⧸ I) p) | class | mixed_char_zero | algebra.char_p | src/algebra/char_p/mixed_char_zero.lean | [
"algebra.char_p.algebra",
"algebra.char_p.local_ring",
"ring_theory.ideal.quotient",
"tactic.field_simp"
] | [
"char_p",
"char_zero",
"ideal"
] | A ring of characteristic zero is of "mixed characteristic `(0, p)`" if there exists an ideal
such that the quotient `R ⧸ I` has caracteristic `p`.
**Remark:** For `p = 0`, `mixed_char R 0` is a meaningless definition as `R ⧸ ⊥ ≅ R` has by
definition always characteristic zero.
One could require `(I ≠ ⊥)` in the defini... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reduce_to_p_prime {P : Prop} :
(∀ p > 0, mixed_char_zero R p → P) ↔
(∀ (p : ℕ), p.prime → mixed_char_zero R p → P) | begin
split,
{ intros h q q_prime q_mixed_char,
exact h q (nat.prime.pos q_prime) q_mixed_char },
{ intros h q q_pos q_mixed_char,
rcases q_mixed_char.char_p_quotient with ⟨I, hI_ne_top, hI_char⟩,
-- Krull's Thm: There exists a prime ideal `P` such that `I ≤ P`
rcases ideal.exists_le_maximal I hI... | lemma | mixed_char_zero.reduce_to_p_prime | algebra.char_p | src/algebra/char_p/mixed_char_zero.lean | [
"algebra.char_p.algebra",
"algebra.char_p.local_ring",
"ring_theory.ideal.quotient",
"tactic.field_simp"
] | [
"char_p.cast_eq_zero",
"char_p.char_is_prime_or_zero",
"char_zero",
"ideal.exists_le_maximal",
"ideal.quotient.factor",
"map_nat_cast",
"mixed_char_zero",
"nat.prime",
"nat.prime.pos",
"ne_zero_of_dvd_ne_zero",
"ring_char",
"ring_char.of_eq"
] | Reduction to `p` prime: When proving any statement `P` about mixed characteristic rings we
can always assume that `p` is prime. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reduce_to_maximal_ideal {p : ℕ} (hp : nat.prime p) :
(∃ (I : ideal R), (I ≠ ⊤) ∧ char_p (R ⧸ I) p) ↔
(∃ (I : ideal R), (I.is_maximal) ∧ char_p (R ⧸ I) p) | begin
split,
{ intro g,
rcases g with ⟨I, ⟨hI_not_top, hI⟩⟩,
-- Krull's Thm: There exists a prime ideal `M` such that `I ≤ M`.
rcases ideal.exists_le_maximal I hI_not_top with ⟨M, ⟨hM_max, hM⟩⟩,
use M,
split,
exact hM_max,
{ cases char_p.exists (R ⧸ M) with r hr,
convert hr,
... | lemma | mixed_char_zero.reduce_to_maximal_ideal | algebra.char_p | src/algebra/char_p/mixed_char_zero.lean | [
"algebra.char_p.algebra",
"algebra.char_p.local_ring",
"ring_theory.ideal.quotient",
"tactic.field_simp"
] | [
"char_p",
"char_p.cast_eq_zero",
"char_p.char_ne_one",
"char_p.exists",
"ideal",
"ideal.exists_le_maximal",
"ideal.quotient.factor",
"nat.prime",
"nat.prime.eq_one_or_self_of_dvd"
] | Reduction to `I` prime ideal: When proving statements about mixed characteristic rings,
after we reduced to `p` prime, we can assume that the ideal `I` in the definition is maximal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Q_algebra_to_equal_char_zero [nontrivial R] [algebra ℚ R] :
∀ (I : ideal R), I ≠ ⊤ → char_zero (R ⧸ I) | begin
haveI : char_zero R := algebra_rat.char_zero R,
intros I hI,
constructor,
intros a b h_ab,
contrapose! hI,
-- `↑a - ↑b` is a unit contained in `I`, which contradicts `I ≠ ⊤`.
refine I.eq_top_of_is_unit_mem _ (is_unit.map (algebra_map ℚ R) (is_unit.mk0 (a - b : ℚ) _)),
{ simpa only [← ideal.quotien... | lemma | Q_algebra_to_equal_char_zero | algebra.char_p | src/algebra/char_p/mixed_char_zero.lean | [
"algebra.char_p.algebra",
"algebra.char_p.local_ring",
"ring_theory.ideal.quotient",
"tactic.field_simp"
] | [
"algebra",
"algebra_map",
"algebra_rat.char_zero",
"char_zero",
"ideal",
"ideal.quotient.eq_zero_iff_mem",
"is_unit.map",
"is_unit.mk0",
"map_nat_cast",
"nat.cast_injective",
"nontrivial"
] | `ℚ`-algebra implies equal characteristic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equal_char_zero.pnat_coe_is_unit [h : fact (∀ (I : ideal R), I ≠ ⊤ → char_zero (R ⧸ I))]
(n : ℕ+) : is_unit (n : R) | begin
-- `n : R` is a unit iff `(n)` is not a proper ideal in `R`.
rw ← ideal.span_singleton_eq_top,
-- So by contrapositive, we should show the quotient does not have characteristic zero.
apply not_imp_comm.mp (h.elim (ideal.span {n})),
unfreezingI { intro h_char_zero },
-- In particular, the image of `n` ... | lemma | equal_char_zero.pnat_coe_is_unit | algebra.char_p | src/algebra/char_p/mixed_char_zero.lean | [
"algebra.char_p.algebra",
"algebra.char_p.local_ring",
"ring_theory.ideal.quotient",
"tactic.field_simp"
] | [
"char_zero",
"fact",
"ideal",
"ideal.quotient.eq_zero_iff_mem",
"ideal.quotient.mk",
"ideal.span",
"ideal.span_singleton_eq_top",
"ideal.subset_span",
"is_unit",
"nat.cast_zero",
"set.mem_singleton"
] | Internal: Not intended to be used outside this local construction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equal_char_zero.pnat_has_coe_units
[fact (∀ (I : ideal R), I ≠ ⊤ → char_zero (R ⧸ I))] : has_coe_t ℕ+ Rˣ | ⟨λn, (equal_char_zero.pnat_coe_is_unit R n).unit⟩ | instance | equal_char_zero.pnat_has_coe_units | algebra.char_p | src/algebra/char_p/mixed_char_zero.lean | [
"algebra.char_p.algebra",
"algebra.char_p.local_ring",
"ring_theory.ideal.quotient",
"tactic.field_simp"
] | [
"char_zero",
"equal_char_zero.pnat_coe_is_unit",
"fact",
"ideal"
] | Internal: Not intended to be used outside this local construction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equal_char_zero.pnat_coe_units_eq_one [fact (∀ (I : ideal R), I ≠ ⊤ → char_zero (R ⧸ I))] :
((1 : ℕ+) : Rˣ) = 1 | begin
apply units.ext,
rw units.coe_one,
change ((equal_char_zero.pnat_coe_is_unit R 1).unit : R) = 1,
rw is_unit.unit_spec (equal_char_zero.pnat_coe_is_unit R 1),
rw [coe_coe, pnat.one_coe, nat.cast_one],
end | lemma | equal_char_zero.pnat_coe_units_eq_one | algebra.char_p | src/algebra/char_p/mixed_char_zero.lean | [
"algebra.char_p.algebra",
"algebra.char_p.local_ring",
"ring_theory.ideal.quotient",
"tactic.field_simp"
] | [
"char_zero",
"coe_coe",
"equal_char_zero.pnat_coe_is_unit",
"fact",
"ideal",
"is_unit.unit_spec",
"nat.cast_one",
"pnat.one_coe",
"units.coe_one",
"units.ext"
] | Internal: Not intended to be used outside this local construction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equal_char_zero.pnat_coe_units_coe_eq_coe
[fact (∀ (I : ideal R), I ≠ ⊤ → char_zero (R ⧸ I))] (n : ℕ+) :
((n : Rˣ) : R) = ↑n | begin
change ((equal_char_zero.pnat_coe_is_unit R n).unit : R) = ↑n,
simp only [is_unit.unit_spec],
end | lemma | equal_char_zero.pnat_coe_units_coe_eq_coe | algebra.char_p | src/algebra/char_p/mixed_char_zero.lean | [
"algebra.char_p.algebra",
"algebra.char_p.local_ring",
"ring_theory.ideal.quotient",
"tactic.field_simp"
] | [
"char_zero",
"equal_char_zero.pnat_coe_is_unit",
"fact",
"ideal",
"is_unit.unit_spec"
] | Internal: Not intended to be used outside this local construction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equal_char_zero_to_Q_algebra (h : ∀ (I : ideal R), I ≠ ⊤ → char_zero (R ⧸ I)) :
algebra ℚ R | by haveI : fact (∀ (I : ideal R), I ≠ ⊤ → char_zero (R ⧸ I)) := ⟨h⟩; exact
ring_hom.to_algebra
{ to_fun := λ x, x.num /ₚ ↑(x.pnat_denom),
map_zero' := by simp [divp],
map_one' := by simp [equal_char_zero.pnat_coe_units_eq_one],
map_mul' :=
begin
intros a b,
field_simp,
repeat { rw eq... | def | equal_char_zero_to_Q_algebra | algebra.char_p | src/algebra/char_p/mixed_char_zero.lean | [
"algebra.char_p.algebra",
"algebra.char_p.local_ring",
"ring_theory.ideal.quotient",
"tactic.field_simp"
] | [
"algebra",
"char_zero",
"coe_coe",
"divp",
"equal_char_zero.pnat_coe_units_coe_eq_coe",
"equal_char_zero.pnat_coe_units_eq_one",
"fact",
"ideal",
"int.cast_coe_nat",
"int.cast_mul",
"num",
"rat.add_num_denom'",
"rat.coe_pnat_denom",
"rat.mul_num_denom'",
"ring",
"ring_hom.to_algebra"
] | Equal characteristic implies `ℚ`-algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_mixed_char_to_equal_char_zero [char_zero R] (h : ∀ p > 0, ¬mixed_char_zero R p) :
∀ (I : ideal R), I ≠ ⊤ → char_zero (R ⧸ I) | begin
intros I hI_ne_top,
apply char_p.char_p_to_char_zero _,
cases char_p.exists (R ⧸ I) with p hp,
cases p,
{ exact hp },
{ have h_mixed : mixed_char_zero R p.succ := ⟨⟨I, ⟨hI_ne_top, hp⟩⟩⟩,
exact absurd h_mixed (h p.succ p.succ_pos) }
end | lemma | not_mixed_char_to_equal_char_zero | algebra.char_p | src/algebra/char_p/mixed_char_zero.lean | [
"algebra.char_p.algebra",
"algebra.char_p.local_ring",
"ring_theory.ideal.quotient",
"tactic.field_simp"
] | [
"char_p.char_p_to_char_zero",
"char_p.exists",
"char_zero",
"ideal",
"mixed_char_zero"
] | Not mixed characteristic implies equal characteristic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equal_char_zero_to_not_mixed_char (h : ∀ (I : ideal R), I ≠ ⊤ → char_zero (R ⧸ I)) :
∀ p > 0, ¬mixed_char_zero R p | begin
intros p p_pos,
by_contradiction hp_mixed_char,
rcases hp_mixed_char.char_p_quotient with ⟨I, hI_ne_top, hI_p⟩,
replace hI_zero : char_p (R ⧸ I) 0 := @char_p.of_char_zero _ _ (h I hI_ne_top),
exact absurd (char_p.eq (R ⧸ I) hI_p hI_zero) (ne_of_gt p_pos),
end | lemma | equal_char_zero_to_not_mixed_char | algebra.char_p | src/algebra/char_p/mixed_char_zero.lean | [
"algebra.char_p.algebra",
"algebra.char_p.local_ring",
"ring_theory.ideal.quotient",
"tactic.field_simp"
] | [
"by_contradiction",
"char_p",
"char_p.eq",
"char_p.of_char_zero",
"char_zero",
"ideal",
"mixed_char_zero"
] | Equal characteristic implies not mixed characteristic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equal_char_zero_iff_not_mixed_char [char_zero R] :
(∀ (I : ideal R), I ≠ ⊤ → char_zero (R ⧸ I)) ↔ (∀ p > 0, ¬mixed_char_zero R p) | ⟨equal_char_zero_to_not_mixed_char R, not_mixed_char_to_equal_char_zero R⟩ | lemma | equal_char_zero_iff_not_mixed_char | algebra.char_p | src/algebra/char_p/mixed_char_zero.lean | [
"algebra.char_p.algebra",
"algebra.char_p.local_ring",
"ring_theory.ideal.quotient",
"tactic.field_simp"
] | [
"char_zero",
"ideal",
"mixed_char_zero",
"not_mixed_char_to_equal_char_zero"
] | A ring of characteristic zero has equal characteristic iff it does not
have mixed characteristic for any `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Q_algebra_iff_equal_char_zero [nontrivial R] :
nonempty (algebra ℚ R) ↔ ∀ (I : ideal R), I ≠ ⊤ → char_zero (R ⧸ I) | begin
split,
{ intro h_alg,
haveI h_alg' : algebra ℚ R := h_alg.some,
apply Q_algebra_to_equal_char_zero },
{ intro h,
apply nonempty.intro,
exact equal_char_zero_to_Q_algebra R h }
end | theorem | Q_algebra_iff_equal_char_zero | algebra.char_p | src/algebra/char_p/mixed_char_zero.lean | [
"algebra.char_p.algebra",
"algebra.char_p.local_ring",
"ring_theory.ideal.quotient",
"tactic.field_simp"
] | [
"Q_algebra_to_equal_char_zero",
"algebra",
"char_zero",
"equal_char_zero_to_Q_algebra",
"ideal",
"nontrivial"
] | A ring is a `ℚ`-algebra iff it has equal characteristic zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_Q_algebra_iff_not_equal_char_zero [char_zero R] :
is_empty (algebra ℚ R) ↔ (∃ p > 0, mixed_char_zero R p) | begin
rw ←not_iff_not,
push_neg,
rw [not_is_empty_iff, ←equal_char_zero_iff_not_mixed_char],
apply Q_algebra_iff_equal_char_zero,
end | theorem | not_Q_algebra_iff_not_equal_char_zero | algebra.char_p | src/algebra/char_p/mixed_char_zero.lean | [
"algebra.char_p.algebra",
"algebra.char_p.local_ring",
"ring_theory.ideal.quotient",
"tactic.field_simp"
] | [
"Q_algebra_iff_equal_char_zero",
"algebra",
"char_zero",
"is_empty",
"mixed_char_zero",
"not_is_empty_iff"
] | A ring of characteristic zero is not a `ℚ`-algebra iff it has mixed characteristic for some `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
split_equal_mixed_char [char_zero R]
(h_equal : algebra ℚ R → P)
(h_mixed : ∀ (p : ℕ), (nat.prime p → mixed_char_zero R p → P)) : P | begin
by_cases h : ∃ p > 0, mixed_char_zero R p,
{ rcases h with ⟨p, ⟨H, hp⟩⟩,
rw ←mixed_char_zero.reduce_to_p_prime at h_mixed,
exact h_mixed p H hp },
{ apply h_equal,
rw [←not_Q_algebra_iff_not_equal_char_zero, not_is_empty_iff] at h,
exact h.some },
end | theorem | split_equal_mixed_char | algebra.char_p | src/algebra/char_p/mixed_char_zero.lean | [
"algebra.char_p.algebra",
"algebra.char_p.local_ring",
"ring_theory.ideal.quotient",
"tactic.field_simp"
] | [
"algebra",
"char_zero",
"mixed_char_zero",
"nat.prime",
"not_is_empty_iff"
] | Split a `Prop` in characteristic zero into equal and mixed characteristic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
split_by_characteristic
(h_pos : ∀ (p : ℕ), (p ≠ 0 → char_p R p → P))
(h_equal : algebra ℚ R → P)
(h_mixed : ∀ (p : ℕ), (nat.prime p → mixed_char_zero R p → P)) : P | begin
cases char_p.exists R with p p_char,
by_cases p = 0,
{ rw h at p_char,
resetI, -- make `p_char : char_p R 0` an instance.
haveI h0 : char_zero R := char_p.char_p_to_char_zero R,
exact split_equal_mixed_char R h_equal h_mixed },
exact h_pos p h p_char,
end | theorem | split_by_characteristic | algebra.char_p | src/algebra/char_p/mixed_char_zero.lean | [
"algebra.char_p.algebra",
"algebra.char_p.local_ring",
"ring_theory.ideal.quotient",
"tactic.field_simp"
] | [
"algebra",
"char_p",
"char_p.char_p_to_char_zero",
"char_p.exists",
"char_zero",
"mixed_char_zero",
"nat.prime",
"split_equal_mixed_char"
] | Split any `Prop` over `R` into the three cases:
- positive characteristic.
- equal characteristic zero.
- mixed characteristic `(0, p)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
split_by_characteristic_domain [is_domain R]
(h_pos : ∀ (p : ℕ), (nat.prime p → char_p R p → P))
(h_equal : algebra ℚ R → P)
(h_mixed : ∀ (p : ℕ), (nat.prime p → mixed_char_zero R p → P)) : P | begin
refine split_by_characteristic R _ h_equal h_mixed,
introsI p p_pos p_char,
have p_prime : nat.prime p :=
or_iff_not_imp_right.mp (char_p.char_is_prime_or_zero R p) p_pos,
exact h_pos p p_prime p_char,
end | theorem | split_by_characteristic_domain | algebra.char_p | src/algebra/char_p/mixed_char_zero.lean | [
"algebra.char_p.algebra",
"algebra.char_p.local_ring",
"ring_theory.ideal.quotient",
"tactic.field_simp"
] | [
"algebra",
"char_p",
"char_p.char_is_prime_or_zero",
"is_domain",
"mixed_char_zero",
"nat.prime",
"split_by_characteristic"
] | In a `is_domain R`, split any `Prop` over `R` into the three cases:
- *prime* characteristic.
- equal characteristic zero.
- mixed characteristic `(0, p)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
split_by_characteristic_local_ring [local_ring R]
(h_pos : ∀ (p : ℕ), (is_prime_pow p → char_p R p → P))
(h_equal : algebra ℚ R → P)
(h_mixed : ∀ (p : ℕ), (nat.prime p → mixed_char_zero R p → P)) : P | begin
refine split_by_characteristic R _ h_equal h_mixed,
introsI p p_pos p_char,
have p_ppow : is_prime_pow (p : ℕ) :=
or_iff_not_imp_left.mp (char_p_zero_or_prime_power R p) p_pos,
exact h_pos p p_ppow p_char,
end | theorem | split_by_characteristic_local_ring | algebra.char_p | src/algebra/char_p/mixed_char_zero.lean | [
"algebra.char_p.algebra",
"algebra.char_p.local_ring",
"ring_theory.ideal.quotient",
"tactic.field_simp"
] | [
"algebra",
"char_p",
"char_p_zero_or_prime_power",
"is_prime_pow",
"local_ring",
"mixed_char_zero",
"nat.prime",
"split_by_characteristic"
] | In a `local_ring R`, split any `Prop` over `R` into the three cases:
- *prime power* characteristic.
- equal characteristic zero.
- mixed characteristic `(0, p)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi (ι : Type u) [hi : nonempty ι] (R : Type v) [semiring R] (p : ℕ) [char_p R p] :
char_p (ι → R) p | ⟨λ x, let ⟨i⟩ := hi in iff.symm $ (char_p.cast_eq_zero_iff R p x).symm.trans
⟨λ h, funext $ λ j, show pi.eval_ring_hom (λ _, R) j (↑x : ι → R) = 0,
by rw [map_nat_cast, h],
λ h, map_nat_cast (pi.eval_ring_hom (λ _: ι, R) i) x ▸ by rw [h, ring_hom.map_zero]⟩⟩ | instance | char_p.pi | algebra.char_p | src/algebra/char_p/pi.lean | [
"algebra.char_p.basic",
"algebra.ring.pi"
] | [
"char_p",
"map_nat_cast",
"pi.eval_ring_hom",
"ring_hom.map_zero",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi' (ι : Type u) [hi : nonempty ι] (R : Type v) [comm_ring R] (p : ℕ) [char_p R p] :
char_p (ι → R) p | char_p.pi ι R p | instance | char_p.pi' | algebra.char_p | src/algebra/char_p/pi.lean | [
"algebra.char_p.basic",
"algebra.ring.pi"
] | [
"char_p",
"char_p.pi",
"comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient (R : Type u) [comm_ring R] (p : ℕ) [hp1 : fact p.prime] (hp2 : ↑p ∈ nonunits R) :
char_p (R ⧸ (ideal.span {p} : ideal R)) p | have hp0 : (p : R ⧸ (ideal.span {p} : ideal R)) = 0,
from map_nat_cast (ideal.quotient.mk (ideal.span {p} : ideal R)) p ▸
ideal.quotient.eq_zero_iff_mem.2 (ideal.subset_span $ set.mem_singleton _),
ring_char.of_eq $ or.resolve_left ((nat.dvd_prime hp1.1).1 $ ring_char.dvd hp0) $ λ h1,
hp2 $ is_unit_iff_dvd_one.2 ... | theorem | char_p.quotient | algebra.char_p | src/algebra/char_p/quotient.lean | [
"algebra.char_p.basic",
"ring_theory.ideal.quotient"
] | [
"char_p",
"comm_ring",
"fact",
"ideal",
"ideal.quotient.mk",
"ideal.span",
"ideal.subset_span",
"map_nat_cast",
"nat.dvd_prime",
"nonunits",
"ring_char.dvd",
"ring_char.of_eq",
"set.mem_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient' {R : Type*} [comm_ring R] (p : ℕ) [char_p R p] (I : ideal R)
(h : ∀ x : ℕ, (x : R) ∈ I → (x : R) = 0) :
char_p (R ⧸ I) p | ⟨λ x, begin
rw [←cast_eq_zero_iff R p x, ←map_nat_cast (ideal.quotient.mk I)],
refine ideal.quotient.eq.trans (_ : ↑x - 0 ∈ I ↔ _),
rw sub_zero,
exact ⟨h x, λ h', h'.symm ▸ I.zero_mem⟩,
end⟩ | lemma | char_p.quotient' | algebra.char_p | src/algebra/char_p/quotient.lean | [
"algebra.char_p.basic",
"ring_theory.ideal.quotient"
] | [
"char_p",
"comm_ring",
"ideal",
"ideal.quotient.mk"
] | If an ideal does not contain any coercions of natural numbers other than zero, then its quotient
inherits the characteristic of the underlying ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.quotient.index_eq_zero {R : Type*} [comm_ring R] (I : ideal R) :
(I.to_add_subgroup.index : R ⧸ I) = 0 | begin
rw [add_subgroup.index, nat.card_eq],
split_ifs with hq, swap, simp,
by_contra h,
-- TODO: can we avoid rewriting the `I.to_add_subgroup` here?
letI : fintype (R ⧸ I) := @fintype.of_finite _ hq,
have h : (fintype.card (R ⧸ I) : R ⧸ I) ≠ 0 := h,
simpa using h
end | lemma | ideal.quotient.index_eq_zero | algebra.char_p | src/algebra/char_p/quotient.lean | [
"algebra.char_p.basic",
"ring_theory.ideal.quotient"
] | [
"by_contra",
"comm_ring",
"fintype",
"fintype.card",
"fintype.of_finite",
"ideal",
"nat.card_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsemiring (R : Type u) [semiring R] (p : ℕ) [char_p R p] (S : subsemiring R) :
char_p S p | ⟨λ x, iff.symm $ (char_p.cast_eq_zero_iff R p x).symm.trans
⟨λ h, subtype.eq $ show S.subtype x = 0, by rw [map_nat_cast, h],
λ h, map_nat_cast S.subtype x ▸ by rw [h, ring_hom.map_zero]⟩⟩ | instance | char_p.subsemiring | algebra.char_p | src/algebra/char_p/subring.lean | [
"algebra.char_p.basic",
"ring_theory.subring.basic"
] | [
"char_p",
"map_nat_cast",
"ring_hom.map_zero",
"semiring",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subring (R : Type u) [ring R] (p : ℕ) [char_p R p] (S : subring R) :
char_p S p | ⟨λ x, iff.symm $ (char_p.cast_eq_zero_iff R p x).symm.trans
⟨λ h, subtype.eq $ show S.subtype x = 0, by rw [map_nat_cast, h],
λ h, map_nat_cast S.subtype x ▸ by rw [h, ring_hom.map_zero]⟩⟩ | instance | char_p.subring | algebra.char_p | src/algebra/char_p/subring.lean | [
"algebra.char_p.basic",
"ring_theory.subring.basic"
] | [
"char_p",
"map_nat_cast",
"ring",
"ring_hom.map_zero",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subring' (R : Type u) [comm_ring R] (p : ℕ) [char_p R p] (S : subring R) :
char_p S p | char_p.subring R p S | instance | char_p.subring' | algebra.char_p | src/algebra/char_p/subring.lean | [
"algebra.char_p.basic",
"ring_theory.subring.basic"
] | [
"char_p",
"char_p.subring",
"comm_ring",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_eq_zero : (2 : R) = 0 | by rw [← nat.cast_two, char_p.cast_eq_zero] | lemma | char_two.two_eq_zero | algebra.char_p | src/algebra/char_p/two.lean | [
"algebra.char_p.basic"
] | [
"char_p.cast_eq_zero",
"nat.cast_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_self_eq_zero (x : R) : x + x = 0 | by rw [←two_smul R x, two_eq_zero, zero_smul] | lemma | char_two.add_self_eq_zero | algebra.char_p | src/algebra/char_p/two.lean | [
"algebra.char_p.basic"
] | [
"add_self_eq_zero",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_eq_zero : (bit0 : R → R) = 0 | by { funext, exact add_self_eq_zero _ } | lemma | char_two.bit0_eq_zero | algebra.char_p | src/algebra/char_p/two.lean | [
"algebra.char_p.basic"
] | [
"add_self_eq_zero",
"bit0_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_apply_eq_zero (x : R) : (bit0 x : R) = 0 | by simp | lemma | char_two.bit0_apply_eq_zero | algebra.char_p | src/algebra/char_p/two.lean | [
"algebra.char_p.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_eq_one : (bit1 : R → R) = 1 | by { funext, simp [bit1] } | lemma | char_two.bit1_eq_one | algebra.char_p | src/algebra/char_p/two.lean | [
"algebra.char_p.basic"
] | [
"bit1_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_apply_eq_one (x : R) : (bit1 x : R) = 1 | by simp | lemma | char_two.bit1_apply_eq_one | algebra.char_p | src/algebra/char_p/two.lean | [
"algebra.char_p.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_eq (x : R) : -x = x | by rw [neg_eq_iff_add_eq_zero, ←two_smul R x, two_eq_zero, zero_smul] | lemma | char_two.neg_eq | algebra.char_p | src/algebra/char_p/two.lean | [
"algebra.char_p.basic"
] | [
"neg_eq",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_eq' : has_neg.neg = (id : R → R) | funext neg_eq | lemma | char_two.neg_eq' | algebra.char_p | src/algebra/char_p/two.lean | [
"algebra.char_p.basic"
] | [
"neg_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_eq_add (x y : R) : x - y = x + y | by rw [sub_eq_add_neg, neg_eq] | lemma | char_two.sub_eq_add | algebra.char_p | src/algebra/char_p/two.lean | [
"algebra.char_p.basic"
] | [
"neg_eq",
"sub_eq_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_eq_add' : has_sub.sub = ((+) : R → R → R) | funext $ λ x, funext $ λ y, sub_eq_add x y | lemma | char_two.sub_eq_add' | algebra.char_p | src/algebra/char_p/two.lean | [
"algebra.char_p.basic"
] | [
"sub_eq_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_sq (x y : R) : (x + y) ^ 2 = x ^ 2 + y ^ 2 | add_pow_char _ _ _ | lemma | char_two.add_sq | algebra.char_p | src/algebra/char_p/two.lean | [
"algebra.char_p.basic"
] | [
"add_pow_char",
"add_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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