statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
pow_of_prime (h : is_primitive_root ζ k) {p : ℕ} (hprime : nat.prime p) (hdiv : ¬ p ∣ k) :
is_primitive_root (ζ ^ p) k | h.pow_of_coprime p (hprime.coprime_iff_not_dvd.2 hdiv) | lemma | is_primitive_root.pow_of_prime | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"nat.prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_iff_coprime (h : is_primitive_root ζ k) (h0 : 0 < k) (i : ℕ) :
is_primitive_root (ζ ^ i) k ↔ i.coprime k | begin
refine ⟨_, h.pow_of_coprime i⟩,
intro hi,
obtain ⟨a, ha⟩ := i.gcd_dvd_left k,
obtain ⟨b, hb⟩ := i.gcd_dvd_right k,
suffices : b = k,
{ rwa [this, ← one_mul k, mul_left_inj' h0.ne', eq_comm] at hb { occs := occurrences.pos [1] } },
rw [ha] at hi,
rw [mul_comm] at hb,
apply nat.dvd_antisymm ⟨i.gcd... | lemma | is_primitive_root.pow_iff_coprime | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"mul_assoc",
"mul_comm",
"mul_left_inj'",
"one_mul",
"one_pow",
"pow_mul",
"pow_mul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_of (ζ : M) : is_primitive_root ζ (order_of ζ) | ⟨pow_order_of_eq_one ζ, λ l, order_of_dvd_of_pow_eq_one⟩ | lemma | is_primitive_root.order_of | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"order_of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique {ζ : M} (hk : is_primitive_root ζ k) (hl : is_primitive_root ζ l) : k = l | nat.dvd_antisymm (hk.2 _ hl.1) (hl.2 _ hk.1) | lemma | is_primitive_root.unique | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_order_of : k = order_of ζ | h.unique (is_primitive_root.order_of ζ) | lemma | is_primitive_root.eq_order_of | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root.order_of",
"order_of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff (hk : 0 < k) :
is_primitive_root ζ k ↔ ζ ^ k = 1 ∧ ∀ l : ℕ, 0 < l → l < k → ζ ^ l ≠ 1 | begin
refine ⟨λ h, ⟨h.pow_eq_one, λ l hl' hl, _⟩, λ ⟨hζ, hl⟩, is_primitive_root.mk_of_lt ζ hk hζ hl⟩,
rw h.eq_order_of at hl,
exact pow_ne_one_of_lt_order_of' hl'.ne' hl,
end | lemma | is_primitive_root.iff | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"is_primitive_root.mk_of_lt",
"pow_ne_one_of_lt_order_of'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_iff : ¬ is_primitive_root ζ k ↔ order_of ζ ≠ k | ⟨λ h hk, h $ hk ▸ is_primitive_root.order_of ζ,
λ h hk, h.symm $ hk.unique $ is_primitive_root.order_of ζ⟩ | lemma | is_primitive_root.not_iff | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"is_primitive_root.order_of",
"not_iff",
"order_of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_of_dvd (h : is_primitive_root ζ k) {p : ℕ} (hp : p ≠ 0) (hdiv : p ∣ k) :
is_primitive_root (ζ ^ p) (k / p) | begin
suffices : order_of (ζ ^ p) = k / p,
{ exact this ▸ is_primitive_root.order_of (ζ ^ p) },
rw [order_of_pow' _ hp, ← eq_order_of h, nat.gcd_eq_right hdiv]
end | lemma | is_primitive_root.pow_of_dvd | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"is_primitive_root.order_of",
"nat.gcd_eq_right",
"order_of",
"order_of_pow'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_roots_of_unity {ζ : Mˣ} {n : ℕ+} (h : is_primitive_root ζ n) : ζ ∈ roots_of_unity n M | h.pow_eq_one | lemma | is_primitive_root.mem_roots_of_unity | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"mem_roots_of_unity",
"roots_of_unity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow {n : ℕ} {a b : ℕ} (hn : 0 < n) (h : is_primitive_root ζ n) (hprod : n = a * b) :
is_primitive_root (ζ ^ a) b | begin
subst n,
simp only [iff_def, ← pow_mul, h.pow_eq_one, eq_self_iff_true, true_and],
intros l hl,
have ha0 : a ≠ 0, { rintro rfl, simpa only [nat.not_lt_zero, zero_mul] using hn },
rwa ← mul_dvd_mul_iff_left ha0,
exact h.dvd_of_pow_eq_one _ hl
end | lemma | is_primitive_root.pow | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"iff_def",
"is_primitive_root",
"mul_dvd_mul_iff_left",
"pow_mul",
"zero_mul"
] | If there is a `n`-th primitive root of unity in `R` and `b` divides `n`,
then there is a `b`-th primitive root of unity in `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_of_injective [monoid_hom_class F M N] (h : is_primitive_root ζ k) (hf : injective f) :
is_primitive_root (f ζ) k | { pow_eq_one := by rw [←map_pow, h.pow_eq_one, _root_.map_one],
dvd_of_pow_eq_one := begin
rw h.eq_order_of,
intros l hl,
rw [←map_pow, ←map_one f] at hl,
exact order_of_dvd_of_pow_eq_one (hf hl)
end } | lemma | is_primitive_root.map_of_injective | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"monoid_hom_class",
"order_of_dvd_of_pow_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_map_of_injective [monoid_hom_class F M N] (h : is_primitive_root (f ζ) k)
(hf : injective f) : is_primitive_root ζ k | { pow_eq_one := by { apply_fun f, rw [map_pow, _root_.map_one, h.pow_eq_one] },
dvd_of_pow_eq_one := begin
rw h.eq_order_of,
intros l hl,
apply_fun f at hl,
rw [map_pow, _root_.map_one] at hl,
exact order_of_dvd_of_pow_eq_one hl
end } | lemma | is_primitive_root.of_map_of_injective | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"map_pow",
"monoid_hom_class",
"order_of_dvd_of_pow_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_iff_of_injective [monoid_hom_class F M N] (hf : injective f) :
is_primitive_root (f ζ) k ↔ is_primitive_root ζ k | ⟨λ h, h.of_map_of_injective hf, λ h, h.map_of_injective hf⟩ | lemma | is_primitive_root.map_iff_of_injective | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"monoid_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero [nontrivial M₀] : is_primitive_root (0 : M₀) 0 | ⟨pow_zero 0, λ l hl, by simpa [zero_pow_eq, show ∀ p, ¬p → false ↔ p, from @not_not] using hl⟩ | lemma | is_primitive_root.zero | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"nontrivial",
"not_not",
"zero_pow_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero [nontrivial M₀] {ζ : M₀} (h : is_primitive_root ζ k) : k ≠ 0 → ζ ≠ 0 | mt $ λ hn, h.unique (hn.symm ▸ is_primitive_root.zero) | lemma | is_primitive_root.ne_zero | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"is_primitive_root.zero",
"ne_zero",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_eq_one (h : is_primitive_root ζ k) : ζ ^ (k : ℤ) = 1 | by { rw zpow_coe_nat, exact h.pow_eq_one } | lemma | is_primitive_root.zpow_eq_one | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"zpow_coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_eq_one_iff_dvd (h : is_primitive_root ζ k) (l : ℤ) :
ζ ^ l = 1 ↔ (k : ℤ) ∣ l | begin
by_cases h0 : 0 ≤ l,
{ lift l to ℕ using h0, rw [zpow_coe_nat], norm_cast, exact h.pow_eq_one_iff_dvd l },
{ have : 0 ≤ -l, { simp only [not_le, neg_nonneg] at h0 ⊢, exact le_of_lt h0 },
lift -l to ℕ using this with l' hl',
rw [← dvd_neg, ← hl'],
norm_cast,
rw [← h.pow_eq_one_iff_dvd, ← inv_... | lemma | is_primitive_root.zpow_eq_one_iff_dvd | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"dvd_neg",
"inv_inj",
"inv_one",
"is_primitive_root",
"lift",
"zpow_coe_nat",
"zpow_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv (h : is_primitive_root ζ k) : is_primitive_root ζ⁻¹ k | { pow_eq_one := by simp only [h.pow_eq_one, inv_one, eq_self_iff_true, inv_pow],
dvd_of_pow_eq_one :=
begin
intros l hl,
apply h.dvd_of_pow_eq_one l,
rw [← inv_inj, ← inv_pow, hl, inv_one]
end } | lemma | is_primitive_root.inv | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"inv_inj",
"inv_one",
"inv_pow",
"is_primitive_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_iff : is_primitive_root ζ⁻¹ k ↔ is_primitive_root ζ k | by { refine ⟨_, λ h, inv h⟩, intro h, rw [← inv_inv ζ], exact inv h } | lemma | is_primitive_root.inv_iff | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"inv_inv",
"is_primitive_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_of_gcd_eq_one (h : is_primitive_root ζ k) (i : ℤ) (hi : i.gcd k = 1) :
is_primitive_root (ζ ^ i) k | begin
by_cases h0 : 0 ≤ i,
{ lift i to ℕ using h0,
rw zpow_coe_nat,
exact h.pow_of_coprime i hi },
have : 0 ≤ -i, { simp only [not_le, neg_nonneg] at h0 ⊢, exact le_of_lt h0 },
lift -i to ℕ using this with i' hi',
rw [← inv_iff, ← zpow_neg, ← hi', zpow_coe_nat],
apply h.pow_of_coprime,
rw [int.gcd... | lemma | is_primitive_root.zpow_of_gcd_eq_one | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"lift",
"zpow_coe_nat",
"zpow_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
primitive_roots_one : primitive_roots 1 R = {(1 : R)} | begin
apply finset.eq_singleton_iff_unique_mem.2,
split,
{ simp only [is_primitive_root.one_right_iff, mem_primitive_roots zero_lt_one] },
{ intros x hx,
rw [mem_primitive_roots zero_lt_one, is_primitive_root.one_right_iff] at hx,
exact hx }
end | lemma | is_primitive_root.primitive_roots_one | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root.one_right_iff",
"mem_primitive_roots",
"primitive_roots",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero' {n : ℕ+} (hζ : is_primitive_root ζ n) : ne_zero ((n : ℕ) : R) | begin
let p := ring_char R,
have hfin := (multiplicity.finite_nat_iff.2 ⟨char_p.char_ne_one R p, n.pos⟩),
obtain ⟨m, hm⟩ := multiplicity.exists_eq_pow_mul_and_not_dvd hfin,
by_cases hp : p ∣ n,
{ obtain ⟨k, hk⟩ := nat.exists_eq_succ_of_ne_zero (multiplicity.pos_of_dvd hfin hp).ne',
haveI : ne_zero p := ne... | lemma | is_primitive_root.ne_zero' | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"char_p.char_is_prime_of_pos",
"fact",
"frobenius_def",
"frobenius_inj",
"frobenius_one",
"is_primitive_root",
"lt_mul_of_one_lt_right",
"mul_assoc",
"mul_comm",
"multiplicity.exists_eq_pow_mul_and_not_dvd",
"multiplicity.pos_of_dvd",
"ne_zero",
"ne_zero.of_not_dvd",
"ne_zero.of_pos",
"p... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nth_roots_finset (hζ : is_primitive_root ζ k) (hk : 0 < k) :
ζ ∈ nth_roots_finset k R | (mem_nth_roots_finset hk).2 hζ.pow_eq_one | lemma | is_primitive_root.mem_nth_roots_finset | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_neg_one_of_two_right [no_zero_divisors R] {ζ : R} (h : is_primitive_root ζ 2) : ζ = -1 | begin
apply (eq_or_eq_neg_of_sq_eq_sq ζ 1 _).resolve_left,
{ rw [← pow_one ζ], apply h.pow_ne_one_of_pos_of_lt; dec_trivial },
{ simp only [h.pow_eq_one, one_pow] }
end | lemma | is_primitive_root.eq_neg_one_of_two_right | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"eq_or_eq_neg_of_sq_eq_sq",
"is_primitive_root",
"no_zero_divisors",
"one_pow",
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_one (p : ℕ) [nontrivial R] [h : char_p R p] (hp : p ≠ 2) : is_primitive_root (-1 : R) 2 | begin
convert is_primitive_root.order_of (-1 : R),
rw [order_of_neg_one, if_neg],
rwa ring_char.eq_iff.mpr h
end | lemma | is_primitive_root.neg_one | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"char_p",
"is_primitive_root",
"is_primitive_root.order_of",
"nontrivial",
"order_of_neg_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_sum_eq_zero [is_domain R] {ζ : R} (hζ : is_primitive_root ζ k) (hk : 1 < k) :
(∑ i in range k, ζ ^ i) = 0 | begin
refine eq_zero_of_ne_zero_of_mul_left_eq_zero (sub_ne_zero_of_ne (hζ.ne_one hk).symm) _,
rw [mul_neg_geom_sum, hζ.pow_eq_one, sub_self]
end | lemma | is_primitive_root.geom_sum_eq_zero | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"eq_zero_of_ne_zero_of_mul_left_eq_zero",
"is_domain",
"is_primitive_root",
"mul_neg_geom_sum"
] | If `1 < k` then `(∑ i in range k, ζ ^ i) = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_sub_one_eq [is_domain R] {ζ : R} (hζ : is_primitive_root ζ k) (hk : 1 < k) :
ζ ^ k.pred = -(∑ i in range k.pred, ζ ^ i) | by rw [eq_neg_iff_add_eq_zero, add_comm, ←sum_range_succ, ←nat.succ_eq_add_one,
nat.succ_pred_eq_of_pos (pos_of_gt hk), hζ.geom_sum_eq_zero hk] | lemma | is_primitive_root.pow_sub_one_eq | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_domain",
"is_primitive_root",
"pos_of_gt"
] | If `1 < k`, then `ζ ^ k.pred = -(∑ i in range k.pred, ζ ^ i)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zmod_equiv_zpowers (h : is_primitive_root ζ k) : zmod k ≃+ additive (subgroup.zpowers ζ) | add_equiv.of_bijective
(add_monoid_hom.lift_of_right_inverse (int.cast_add_hom $ zmod k) _ zmod.int_cast_right_inverse
⟨{ to_fun := λ i, additive.of_mul (⟨_, i, rfl⟩ : subgroup.zpowers ζ),
map_zero' := by { simp only [zpow_zero], refl },
map_add' := by { intros i j, simp only [zpow_add], refl } },
... | def | is_primitive_root.zmod_equiv_zpowers | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"additive",
"additive.of_mul",
"char_p.int_cast_eq_zero_iff",
"int.cast_add_hom",
"int.coe_cast_add_hom",
"is_primitive_root",
"one_zpow",
"subgroup.zpowers",
"subtype.ext_iff",
"zmod",
"zmod.int_cast_right_inverse",
"zpow_add",
"zpow_coe_nat",
"zpow_mul",
"zpow_zero"
] | The (additive) monoid equivalence between `zmod k`
and the powers of a primitive root of unity `ζ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zmod_equiv_zpowers_apply_coe_int (i : ℤ) :
h.zmod_equiv_zpowers i = additive.of_mul (⟨ζ ^ i, i, rfl⟩ : subgroup.zpowers ζ) | add_monoid_hom.lift_of_right_inverse_comp_apply _ _ zmod.int_cast_right_inverse _ _ | lemma | is_primitive_root.zmod_equiv_zpowers_apply_coe_int | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"additive.of_mul",
"subgroup.zpowers",
"zmod.int_cast_right_inverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zmod_equiv_zpowers_apply_coe_nat (i : ℕ) :
h.zmod_equiv_zpowers i = additive.of_mul (⟨ζ ^ i, i, rfl⟩ : subgroup.zpowers ζ) | begin
have : (i : zmod k) = (i : ℤ), by norm_cast,
simp only [this, zmod_equiv_zpowers_apply_coe_int, zpow_coe_nat],
refl
end | lemma | is_primitive_root.zmod_equiv_zpowers_apply_coe_nat | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"additive.of_mul",
"subgroup.zpowers",
"zmod",
"zpow_coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zmod_equiv_zpowers_symm_apply_zpow (i : ℤ) :
h.zmod_equiv_zpowers.symm (additive.of_mul (⟨ζ ^ i, i, rfl⟩ : subgroup.zpowers ζ)) = i | by rw [← h.zmod_equiv_zpowers.symm_apply_apply i, zmod_equiv_zpowers_apply_coe_int] | lemma | is_primitive_root.zmod_equiv_zpowers_symm_apply_zpow | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"additive.of_mul",
"subgroup.zpowers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zmod_equiv_zpowers_symm_apply_zpow' (i : ℤ) :
h.zmod_equiv_zpowers.symm ⟨ζ ^ i, i, rfl⟩ = i | h.zmod_equiv_zpowers_symm_apply_zpow i | lemma | is_primitive_root.zmod_equiv_zpowers_symm_apply_zpow' | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zmod_equiv_zpowers_symm_apply_pow (i : ℕ) :
h.zmod_equiv_zpowers.symm (additive.of_mul (⟨ζ ^ i, i, rfl⟩ : subgroup.zpowers ζ)) = i | by rw [← h.zmod_equiv_zpowers.symm_apply_apply i, zmod_equiv_zpowers_apply_coe_nat] | lemma | is_primitive_root.zmod_equiv_zpowers_symm_apply_pow | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"additive.of_mul",
"subgroup.zpowers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zmod_equiv_zpowers_symm_apply_pow' (i : ℕ) :
h.zmod_equiv_zpowers.symm ⟨ζ ^ i, i, rfl⟩ = i | h.zmod_equiv_zpowers_symm_apply_pow i | lemma | is_primitive_root.zmod_equiv_zpowers_symm_apply_pow' | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpowers_eq {k : ℕ+} {ζ : Rˣ} (h : is_primitive_root ζ k) :
subgroup.zpowers ζ = roots_of_unity k R | begin
apply set_like.coe_injective,
haveI F : fintype (subgroup.zpowers ζ) := fintype.of_equiv _ (h.zmod_equiv_zpowers).to_equiv,
refine @set.eq_of_subset_of_card_le Rˣ (subgroup.zpowers ζ) (roots_of_unity k R)
F (roots_of_unity.fintype R k)
(subgroup.zpowers_le_of_mem $ show ζ ∈ roots_of_unity k R, from ... | lemma | is_primitive_root.zpowers_eq | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"card_roots_of_unity",
"fintype",
"fintype.card",
"fintype.card_congr",
"fintype.of_equiv",
"is_primitive_root",
"roots_of_unity",
"roots_of_unity.fintype",
"set.eq_of_subset_of_card_le",
"set_like.coe_injective",
"subgroup.zpowers",
"zmod",
"zmod.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_pow_of_mem_roots_of_unity {k : ℕ+} {ζ ξ : Rˣ}
(h : is_primitive_root ζ k) (hξ : ξ ∈ roots_of_unity k R) :
∃ (i : ℕ) (hi : i < k), ζ ^ i = ξ | begin
obtain ⟨n, rfl⟩ : ∃ n : ℤ, ζ ^ n = ξ, by rwa [← h.zpowers_eq] at hξ,
have hk0 : (0 : ℤ) < k := by exact_mod_cast k.pos,
let i := n % k,
have hi0 : 0 ≤ i := int.mod_nonneg _ (ne_of_gt hk0),
lift i to ℕ using hi0 with i₀ hi₀,
refine ⟨i₀, _, _⟩,
{ zify, rw [hi₀], exact int.mod_lt_of_pos _ hk0 },
{ ha... | lemma | is_primitive_root.eq_pow_of_mem_roots_of_unity | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"aux",
"coe_coe",
"int.mod_add_div",
"int.mod_lt_of_pos",
"int.mod_nonneg",
"is_primitive_root",
"lift",
"mul_one",
"one_zpow",
"roots_of_unity",
"zpow_add",
"zpow_coe_nat",
"zpow_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_pow_of_pow_eq_one {k : ℕ} {ζ ξ : R}
(h : is_primitive_root ζ k) (hξ : ξ ^ k = 1) (h0 : 0 < k) :
∃ i < k, ζ ^ i = ξ | begin
lift ζ to Rˣ using h.is_unit h0,
lift ξ to Rˣ using is_unit_of_pow_eq_one hξ h0.ne',
lift k to ℕ+ using h0,
simp only [← units.coe_pow, ← units.ext_iff],
rw coe_units_iff at h,
apply h.eq_pow_of_mem_roots_of_unity,
rw [mem_roots_of_unity, units.ext_iff, units.coe_pow, hξ, units.coe_one]
end | lemma | is_primitive_root.eq_pow_of_pow_eq_one | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"is_unit_of_pow_eq_one",
"lift",
"mem_roots_of_unity",
"units.coe_one",
"units.coe_pow",
"units.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive_root_iff' {k : ℕ+} {ζ ξ : Rˣ} (h : is_primitive_root ζ k) :
is_primitive_root ξ k ↔ ∃ (i < (k : ℕ)) (hi : i.coprime k), ζ ^ i = ξ | begin
split,
{ intro hξ,
obtain ⟨i, hik, rfl⟩ := h.eq_pow_of_mem_roots_of_unity hξ.pow_eq_one,
rw h.pow_iff_coprime k.pos at hξ,
exact ⟨i, hik, hξ, rfl⟩ },
{ rintro ⟨i, -, hi, rfl⟩, exact h.pow_of_coprime i hi }
end | lemma | is_primitive_root.is_primitive_root_iff' | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive_root_iff {k : ℕ} {ζ ξ : R} (h : is_primitive_root ζ k) (h0 : 0 < k) :
is_primitive_root ξ k ↔ ∃ (i < k) (hi : i.coprime k), ζ ^ i = ξ | begin
split,
{ intro hξ,
obtain ⟨i, hik, rfl⟩ := h.eq_pow_of_pow_eq_one hξ.pow_eq_one h0,
rw h.pow_iff_coprime h0 at hξ,
exact ⟨i, hik, hξ, rfl⟩ },
{ rintro ⟨i, -, hi, rfl⟩, exact h.pow_of_coprime i hi }
end | lemma | is_primitive_root.is_primitive_root_iff | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_roots_of_unity' {n : ℕ+} (h : is_primitive_root ζ n) :
fintype.card (roots_of_unity n R) = n | begin
let e := h.zmod_equiv_zpowers,
haveI F : fintype (subgroup.zpowers ζ) := fintype.of_equiv _ e.to_equiv,
calc fintype.card (roots_of_unity n R)
= fintype.card (subgroup.zpowers ζ) : fintype.card_congr $ by rw h.zpowers_eq
... = fintype.card (zmod n) : fintype.card_congr e.to_equiv.symm
... | lemma | is_primitive_root.card_roots_of_unity' | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"fintype",
"fintype.card",
"fintype.card_congr",
"fintype.of_equiv",
"is_primitive_root",
"roots_of_unity",
"subgroup.zpowers",
"zmod",
"zmod.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_roots_of_unity {ζ : R} {n : ℕ+} (h : is_primitive_root ζ n) :
fintype.card (roots_of_unity n R) = n | begin
obtain ⟨ζ, hζ⟩ := h.is_unit n.pos,
rw [← hζ, is_primitive_root.coe_units_iff] at h,
exact h.card_roots_of_unity'
end | lemma | is_primitive_root.card_roots_of_unity | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"card_roots_of_unity",
"fintype.card",
"is_primitive_root",
"is_primitive_root.coe_units_iff",
"roots_of_unity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_nth_roots {ζ : R} {n : ℕ} (h : is_primitive_root ζ n) :
(nth_roots n (1 : R)).card = n | begin
cases nat.eq_zero_or_pos n with hzero hpos,
{ simp only [hzero, multiset.card_zero, nth_roots_zero] },
rw eq_iff_le_not_lt,
use card_nth_roots n 1,
{ rw [not_lt],
have hcard : fintype.card {x // x ∈ nth_roots n (1 : R)}
≤ (nth_roots n (1 : R)).attach.card := multiset.card_le_of_le (multiset.de... | lemma | is_primitive_root.card_nth_roots | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"card_roots_of_unity",
"eq_iff_le_not_lt",
"fintype.card",
"fintype.card_congr",
"is_primitive_root",
"multiset.card_attach",
"multiset.card_le_of_le",
"multiset.card_zero",
"multiset.dedup_le",
"nat.to_pnat'",
"pnat.to_pnat'_coe",
"roots_of_unity_equiv_nth_roots"
] | The cardinality of the multiset `nth_roots ↑n (1 : R)` is `n`
if there is a primitive root of unity in `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nth_roots_nodup {ζ : R} {n : ℕ} (h : is_primitive_root ζ n) : (nth_roots n (1 : R)).nodup | begin
cases nat.eq_zero_or_pos n with hzero hpos,
{ simp only [hzero, multiset.nodup_zero, nth_roots_zero] },
apply (@multiset.dedup_eq_self R _ _).1,
rw eq_iff_le_not_lt,
split,
{ exact multiset.dedup_le (nth_roots n (1 : R)) },
{ by_contra ha,
replace ha := multiset.card_lt_of_lt ha,
rw card_nth... | lemma | is_primitive_root.nth_roots_nodup | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"by_contra",
"card_roots_of_unity",
"eq_iff_le_not_lt",
"finset",
"finset.card_mk",
"finset.mem_mk",
"fintype.card",
"fintype.card_congr",
"fintype.card_of_subtype",
"is_primitive_root",
"multiset.card_lt_of_lt",
"multiset.dedup_eq_self",
"multiset.dedup_le",
"multiset.mem_dedup",
"multi... | The multiset `nth_roots ↑n (1 : R)` has no repeated elements
if there is a primitive root of unity in `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
card_nth_roots_finset {ζ : R} {n : ℕ} (h : is_primitive_root ζ n) :
(nth_roots_finset n R).card = n | by rw [nth_roots_finset, ← multiset.to_finset_eq (nth_roots_nodup h), card_mk, h.card_nth_roots] | lemma | is_primitive_root.card_nth_roots_finset | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"multiset.to_finset_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_primitive_roots {ζ : R} {k : ℕ} (h : is_primitive_root ζ k) :
(primitive_roots k R).card = φ k | begin
by_cases h0 : k = 0,
{ simp [h0], },
symmetry,
refine finset.card_congr (λ i _, ζ ^ i) _ _ _,
{ simp only [true_and, and_imp, mem_filter, mem_range, mem_univ],
rintro i - hi,
rw mem_primitive_roots (nat.pos_of_ne_zero h0),
exact h.pow_of_coprime i hi.symm },
{ simp only [true_and, and_imp,... | lemma | is_primitive_root.card_primitive_roots | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"and_imp",
"exists_prop",
"finset.card_congr",
"is_primitive_root",
"mem_primitive_roots",
"primitive_roots"
] | If an integral domain has a primitive `k`-th root of unity, then it has `φ k` of them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
disjoint {k l : ℕ} (h : k ≠ l) :
disjoint (primitive_roots k R) (primitive_roots l R) | finset.disjoint_left.2 $ λ z hk hl, h $ (is_primitive_root_of_mem_primitive_roots hk).unique $
is_primitive_root_of_mem_primitive_roots hl | lemma | is_primitive_root.disjoint | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"disjoint",
"is_primitive_root_of_mem_primitive_roots",
"primitive_roots",
"unique"
] | The sets `primitive_roots k R` are pairwise disjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nth_roots_one_eq_bUnion_primitive_roots' {ζ : R} {n : ℕ+} (h : is_primitive_root ζ n) :
nth_roots_finset n R = (nat.divisors ↑n).bUnion (λ i, (primitive_roots i R)) | begin
symmetry,
apply finset.eq_of_subset_of_card_le,
{ intros x,
simp only [nth_roots_finset, ← multiset.to_finset_eq (nth_roots_nodup h),
exists_prop, finset.mem_bUnion, finset.mem_filter, finset.mem_range, mem_nth_roots,
finset.mem_mk, nat.mem_divisors, and_true, ne.def, pnat.ne_zero, pnat.pos,... | lemma | is_primitive_root.nth_roots_one_eq_bUnion_primitive_roots' | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"disjoint",
"exists_prop",
"finset.card_bUnion",
"finset.eq_of_subset_of_card_le",
"finset.mem_bUnion",
"finset.mem_filter",
"finset.mem_mk",
"finset.mem_range",
"is_primitive_root",
"mem_primitive_roots",
"mul_comm",
"multiset.to_finset_eq",
"nat.divisors",
"nat.mem_divisors",
"nat.sum_... | `nth_roots n` as a `finset` is equal to the union of `primitive_roots i R` for `i ∣ n`
if there is a primitive root of unity in `R`.
This holds for any `nat`, not just `pnat`, see `nth_roots_one_eq_bUnion_primitive_roots`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nth_roots_one_eq_bUnion_primitive_roots {ζ : R} {n : ℕ}
(h : is_primitive_root ζ n) :
nth_roots_finset n R = (nat.divisors n).bUnion (λ i, (primitive_roots i R)) | begin
by_cases hn : n = 0,
{ simp [hn], },
exact @nth_roots_one_eq_bUnion_primitive_roots' _ _ _ _ ⟨n, nat.pos_of_ne_zero hn⟩ h
end | lemma | is_primitive_root.nth_roots_one_eq_bUnion_primitive_roots | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"nat.divisors",
"primitive_roots"
] | `nth_roots n` as a `finset` is equal to the union of `primitive_roots i R` for `i ∣ n`
if there is a primitive root of unity in `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
aut_to_pow : (S ≃ₐ[R] S) →* (zmod n)ˣ | let μ' := hμ.to_roots_of_unity in
have ho : order_of μ' = n :=
by rw [hμ.eq_order_of, ←hμ.coe_to_roots_of_unity_coe, order_of_units, order_of_subgroup],
monoid_hom.to_hom_units
{ to_fun := λ σ, (map_root_of_unity_eq_pow_self σ.to_alg_hom μ').some,
map_one' := begin
generalize_proofs h1,
have h := h1.some_sp... | def | is_primitive_root.aut_to_pow | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"alg_equiv.coe_ring_equiv",
"alg_equiv.mul_apply",
"alg_equiv.one_apply",
"alg_equiv.to_ring_equiv_eq_coe",
"map_root_of_unity_eq_pow_self",
"monoid_hom.to_hom_units",
"nat.cast_one",
"order_of",
"order_of_subgroup",
"order_of_units",
"ring_equiv.coe_to_ring_hom",
"ring_equiv.to_ring_hom_eq_co... | The `monoid_hom` that takes an automorphism to the power of μ that μ gets mapped to under it. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_aut_to_pow_apply (f : S ≃ₐ[R] S) : (aut_to_pow R hμ f : zmod n) =
((map_root_of_unity_eq_pow_self f hμ.to_roots_of_unity).some : zmod n) | rfl | lemma | is_primitive_root.coe_aut_to_pow_apply | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"map_root_of_unity_eq_pow_self",
"zmod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aut_to_pow_spec (f : S ≃ₐ[R] S) :
μ ^ (hμ.aut_to_pow R f : zmod n).val = f μ | begin
rw is_primitive_root.coe_aut_to_pow_apply,
generalize_proofs h,
have := h.some_spec,
dsimp only [alg_equiv.to_alg_hom_eq_coe, alg_equiv.coe_alg_hom] at this,
refine (_ : ↑hμ.to_roots_of_unity ^ _ = _).trans this.symm,
rw [←roots_of_unity.coe_pow, ←roots_of_unity.coe_pow],
congr' 1,
rw [pow_eq_pow_... | lemma | is_primitive_root.aut_to_pow_spec | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"alg_equiv.coe_alg_hom",
"alg_equiv.to_alg_hom_eq_coe",
"is_primitive_root.coe_aut_to_pow_apply",
"nat.mod_modeq",
"pow_eq_pow_iff_modeq",
"zmod",
"zmod.val_nat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive_root_exp_of_coprime (i n : ℕ) (h0 : n ≠ 0) (hi : i.coprime n) :
is_primitive_root (exp (2 * π * I * (i / n))) n | begin
rw is_primitive_root.iff_def,
simp only [← exp_nat_mul, exp_eq_one_iff],
have hn0 : (n : ℂ) ≠ 0, by exact_mod_cast h0,
split,
{ use i,
field_simp [hn0, mul_comm (i : ℂ), mul_comm (n : ℂ)] },
{ simp only [hn0, mul_right_comm _ _ ↑n, mul_left_inj' two_pi_I_ne_zero, ne.def, not_false_iff,
mul_c... | lemma | complex.is_primitive_root_exp_of_coprime | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/complex.lean | [
"analysis.special_functions.complex.log",
"ring_theory.roots_of_unity.basic"
] | [
"dvd_mul_left",
"exists_imp_distrib",
"exp",
"int.coe_nat_dvd",
"is_primitive_root",
"is_primitive_root.iff_def",
"mul_assoc",
"mul_comm",
"mul_left_inj'",
"mul_right_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive_root_exp (n : ℕ) (h0 : n ≠ 0) : is_primitive_root (exp (2 * π * I / n)) n | by simpa only [nat.cast_one, one_div]
using is_primitive_root_exp_of_coprime 1 n h0 n.coprime_one_left | lemma | complex.is_primitive_root_exp | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/complex.lean | [
"analysis.special_functions.complex.log",
"ring_theory.roots_of_unity.basic"
] | [
"exp",
"is_primitive_root",
"nat.cast_one",
"one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive_root_iff (ζ : ℂ) (n : ℕ) (hn : n ≠ 0) :
is_primitive_root ζ n ↔ (∃ (i < (n : ℕ)) (hi : i.coprime n), exp (2 * π * I * (i / n)) = ζ) | begin
have hn0 : (n : ℂ) ≠ 0 := by exact_mod_cast hn,
split, swap,
{ rintro ⟨i, -, hi, rfl⟩, exact is_primitive_root_exp_of_coprime i n hn hi },
intro h,
obtain ⟨i, hi, rfl⟩ :=
(is_primitive_root_exp n hn).eq_pow_of_pow_eq_one h.pow_eq_one (nat.pos_of_ne_zero hn),
refine ⟨i, hi, ((is_primitive_root_exp ... | lemma | complex.is_primitive_root_iff | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/complex.lean | [
"analysis.special_functions.complex.log",
"ring_theory.roots_of_unity.basic"
] | [
"exp",
"is_primitive_root",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_roots_of_unity (n : ℕ+) (x : units ℂ) :
x ∈ roots_of_unity n ℂ ↔ (∃ i < (n : ℕ), exp (2 * π * I * (i / n)) = x) | begin
rw [mem_roots_of_unity, units.ext_iff, units.coe_pow, units.coe_one],
have hn0 : (n : ℂ) ≠ 0 := by exact_mod_cast (n.ne_zero),
split,
{ intro h,
obtain ⟨i, hi, H⟩ : ∃ i < (n : ℕ), exp (2 * π * I / n) ^ i = x,
{ simpa only using (is_primitive_root_exp n n.ne_zero).eq_pow_of_pow_eq_one h n.pos },
... | lemma | complex.mem_roots_of_unity | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/complex.lean | [
"analysis.special_functions.complex.log",
"ring_theory.roots_of_unity.basic"
] | [
"exp",
"mem_roots_of_unity",
"mul_comm",
"roots_of_unity",
"units",
"units.coe_one",
"units.coe_pow",
"units.ext_iff"
] | The complex `n`-th roots of unity are exactly the
complex numbers of the form `e ^ (2 * real.pi * complex.I * (i / n))` for some `i < n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
card_roots_of_unity (n : ℕ+) : fintype.card (roots_of_unity n ℂ) = n | (is_primitive_root_exp n n.ne_zero).card_roots_of_unity | lemma | complex.card_roots_of_unity | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/complex.lean | [
"analysis.special_functions.complex.log",
"ring_theory.roots_of_unity.basic"
] | [
"card_roots_of_unity",
"fintype.card",
"roots_of_unity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_primitive_roots (k : ℕ) : (primitive_roots k ℂ).card = φ k | begin
by_cases h : k = 0,
{ simp [h] },
exact (is_primitive_root_exp k h).card_primitive_roots,
end | lemma | complex.card_primitive_roots | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/complex.lean | [
"analysis.special_functions.complex.log",
"ring_theory.roots_of_unity.basic"
] | [
"primitive_roots"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive_root.norm'_eq_one {ζ : ℂ} {n : ℕ} (h : is_primitive_root ζ n) (hn : n ≠ 0) :
‖ζ‖ = 1 | complex.norm_eq_one_of_pow_eq_one h.pow_eq_one hn | lemma | is_primitive_root.norm'_eq_one | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/complex.lean | [
"analysis.special_functions.complex.log",
"ring_theory.roots_of_unity.basic"
] | [
"complex.norm_eq_one_of_pow_eq_one",
"is_primitive_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive_root.nnnorm_eq_one {ζ : ℂ} {n : ℕ} (h : is_primitive_root ζ n) (hn : n ≠ 0) :
‖ζ‖₊ = 1 | subtype.ext $ h.norm'_eq_one hn | lemma | is_primitive_root.nnnorm_eq_one | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/complex.lean | [
"analysis.special_functions.complex.log",
"ring_theory.roots_of_unity.basic"
] | [
"is_primitive_root",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive_root.arg_ext {n m : ℕ} {ζ μ : ℂ} (hζ : is_primitive_root ζ n)
(hμ : is_primitive_root μ m) (hn : n ≠ 0) (hm : m ≠ 0) (h : ζ.arg = μ.arg) : ζ = μ | complex.ext_abs_arg ((hζ.norm'_eq_one hn).trans (hμ.norm'_eq_one hm).symm) h | lemma | is_primitive_root.arg_ext | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/complex.lean | [
"analysis.special_functions.complex.log",
"ring_theory.roots_of_unity.basic"
] | [
"complex.ext_abs_arg",
"is_primitive_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive_root.arg_eq_zero_iff {n : ℕ} {ζ : ℂ} (hζ : is_primitive_root ζ n)
(hn : n ≠ 0) : ζ.arg = 0 ↔ ζ = 1 | ⟨λ h, hζ.arg_ext is_primitive_root.one hn one_ne_zero (h.trans complex.arg_one.symm),
λ h, h.symm ▸ complex.arg_one⟩ | lemma | is_primitive_root.arg_eq_zero_iff | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/complex.lean | [
"analysis.special_functions.complex.log",
"ring_theory.roots_of_unity.basic"
] | [
"is_primitive_root",
"is_primitive_root.one",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive_root.arg_eq_pi_iff {n : ℕ} {ζ : ℂ} (hζ : is_primitive_root ζ n)
(hn : n ≠ 0) : ζ.arg = real.pi ↔ ζ = -1 | ⟨λ h, hζ.arg_ext (is_primitive_root.neg_one 0 two_ne_zero.symm) hn two_ne_zero
(h.trans complex.arg_neg_one.symm), λ h, h.symm ▸ complex.arg_neg_one⟩ | lemma | is_primitive_root.arg_eq_pi_iff | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/complex.lean | [
"analysis.special_functions.complex.log",
"ring_theory.roots_of_unity.basic"
] | [
"is_primitive_root",
"is_primitive_root.neg_one",
"real.pi",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive_root.arg {n : ℕ} {ζ : ℂ} (h : is_primitive_root ζ n) (hn : n ≠ 0) :
∃ i : ℤ, ζ.arg = i / n * (2 * real.pi) ∧ is_coprime i n ∧ i.nat_abs < n | begin
rw complex.is_primitive_root_iff _ _ hn at h,
obtain ⟨i, h, hin, rfl⟩ := h,
rw [mul_comm, ←mul_assoc, complex.exp_mul_I],
refine ⟨if i * 2 ≤ n then i else i - n, _, _, _⟩,
work_on_goal 2
{ replace hin := nat.is_coprime_iff_coprime.mpr hin,
split_ifs with _,
{ exact hin },
{ convert hin.add... | lemma | is_primitive_root.arg | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/complex.lean | [
"analysis.special_functions.complex.log",
"ring_theory.roots_of_unity.basic"
] | [
"complex.arg_cos_add_sin_mul_I",
"complex.exp_mul_I",
"complex.is_primitive_root_iff",
"div_le_iff'",
"div_lt_iff",
"div_nonneg",
"div_self",
"int.nat_abs_eq_iff",
"is_coprime",
"is_primitive_root",
"mul_comm",
"mul_div_assoc",
"mul_le_of_le_one_right",
"mul_lt_iff_lt_one_right",
"mul_ne... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral : is_integral ℤ μ | begin
use (X ^ n - 1),
split,
{ exact (monic_X_pow_sub_C 1 (ne_of_lt hpos).symm) },
{ simp only [((is_primitive_root.iff_def μ n).mp h).left, eval₂_one, eval₂_X_pow, eval₂_sub,
sub_self] }
end | lemma | is_primitive_root.is_integral | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/minpoly.lean | [
"ring_theory.roots_of_unity.basic",
"field_theory.minpoly.is_integrally_closed"
] | [
"is_integral",
"is_primitive_root.iff_def"
] | `μ` is integral over `ℤ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
minpoly_dvd_X_pow_sub_one : minpoly ℤ μ ∣ X ^ n - 1 | begin
rcases n.eq_zero_or_pos with rfl | hpos,
{ simp },
letI : is_integrally_closed ℤ := gcd_monoid.to_is_integrally_closed,
apply minpoly.is_integrally_closed_dvd (is_integral h hpos),
simp only [((is_primitive_root.iff_def μ n).mp h).left, aeval_X_pow, eq_int_cast,
int.cast_one, aeval_one, alg_hom.map_su... | lemma | is_primitive_root.minpoly_dvd_X_pow_sub_one | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/minpoly.lean | [
"ring_theory.roots_of_unity.basic",
"field_theory.minpoly.is_integrally_closed"
] | [
"alg_hom.map_sub",
"eq_int_cast",
"gcd_monoid.to_is_integrally_closed",
"int.cast_one",
"is_integral",
"is_integrally_closed",
"is_primitive_root.iff_def",
"minpoly",
"minpoly.is_integrally_closed_dvd"
] | The minimal polynomial of a root of unity `μ` divides `X ^ n - 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
separable_minpoly_mod {p : ℕ} [fact p.prime] (hdiv : ¬p ∣ n) :
separable (map (int.cast_ring_hom (zmod p)) (minpoly ℤ μ)) | begin
have hdvd : (map (int.cast_ring_hom (zmod p))
(minpoly ℤ μ)) ∣ X ^ n - 1,
{ simpa [polynomial.map_pow, map_X, polynomial.map_one, polynomial.map_sub] using
ring_hom.map_dvd (map_ring_hom (int.cast_ring_hom (zmod p)))
(minpoly_dvd_X_pow_sub_one h) },
refine separable.of_dvd (separable_X_pow... | lemma | is_primitive_root.separable_minpoly_mod | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/minpoly.lean | [
"ring_theory.roots_of_unity.basic",
"field_theory.minpoly.is_integrally_closed"
] | [
"by_contra",
"fact",
"int.cast_ring_hom",
"minpoly",
"one_ne_zero",
"polynomial.map_one",
"polynomial.map_pow",
"polynomial.map_sub",
"ring_hom.map_dvd",
"zmod",
"zmod.nat_coe_zmod_eq_zero_iff_dvd"
] | The reduction modulo `p` of the minimal polynomial of a root of unity `μ` is separable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
squarefree_minpoly_mod {p : ℕ} [fact p.prime] (hdiv : ¬ p ∣ n) :
squarefree (map (int.cast_ring_hom (zmod p)) (minpoly ℤ μ)) | (separable_minpoly_mod h hdiv).squarefree | lemma | is_primitive_root.squarefree_minpoly_mod | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/minpoly.lean | [
"ring_theory.roots_of_unity.basic",
"field_theory.minpoly.is_integrally_closed"
] | [
"fact",
"int.cast_ring_hom",
"minpoly",
"squarefree",
"zmod"
] | The reduction modulo `p` of the minimal polynomial of a root of unity `μ` is squarefree. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
minpoly_dvd_expand {p : ℕ} (hdiv : ¬ p ∣ n) : minpoly ℤ μ ∣ expand ℤ p (minpoly ℤ (μ ^ p)) | begin
rcases n.eq_zero_or_pos with rfl | hpos,
{ simp * at *, },
letI : is_integrally_closed ℤ := gcd_monoid.to_is_integrally_closed,
refine minpoly.is_integrally_closed_dvd (h.is_integral hpos) _,
{ rw [aeval_def, coe_expand, ← comp, eval₂_eq_eval_map, map_comp, polynomial.map_pow, map_X,
eval_comp, ... | lemma | is_primitive_root.minpoly_dvd_expand | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/minpoly.lean | [
"ring_theory.roots_of_unity.basic",
"field_theory.minpoly.is_integrally_closed"
] | [
"gcd_monoid.to_is_integrally_closed",
"is_integrally_closed",
"map_comp",
"minpoly",
"minpoly.aeval",
"minpoly.is_integrally_closed_dvd",
"polynomial.map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
minpoly_dvd_pow_mod {p : ℕ} [hprime : fact p.prime] (hdiv : ¬ p ∣ n) :
map (int.cast_ring_hom (zmod p)) (minpoly ℤ μ) ∣
map (int.cast_ring_hom (zmod p)) (minpoly ℤ (μ ^ p)) ^ p | begin
set Q := minpoly ℤ (μ ^ p),
have hfrob : map (int.cast_ring_hom (zmod p)) Q ^ p =
map (int.cast_ring_hom (zmod p)) (expand ℤ p Q),
by rw [← zmod.expand_card, map_expand],
rw [hfrob],
apply ring_hom.map_dvd (map_ring_hom (int.cast_ring_hom (zmod p))),
exact minpoly_dvd_expand h hdiv
end | lemma | is_primitive_root.minpoly_dvd_pow_mod | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/minpoly.lean | [
"ring_theory.roots_of_unity.basic",
"field_theory.minpoly.is_integrally_closed"
] | [
"fact",
"int.cast_ring_hom",
"minpoly",
"ring_hom.map_dvd",
"zmod",
"zmod.expand_card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
minpoly_dvd_mod_p {p : ℕ} [hprime : fact p.prime] (hdiv : ¬ p ∣ n) :
map (int.cast_ring_hom (zmod p)) (minpoly ℤ μ) ∣
map (int.cast_ring_hom (zmod p)) (minpoly ℤ (μ ^ p)) | (unique_factorization_monoid.dvd_pow_iff_dvd_of_squarefree (squarefree_minpoly_mod h
hdiv) hprime.1.ne_zero).1 (minpoly_dvd_pow_mod h hdiv) | lemma | is_primitive_root.minpoly_dvd_mod_p | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/minpoly.lean | [
"ring_theory.roots_of_unity.basic",
"field_theory.minpoly.is_integrally_closed"
] | [
"fact",
"int.cast_ring_hom",
"minpoly",
"unique_factorization_monoid.dvd_pow_iff_dvd_of_squarefree",
"zmod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
minpoly_eq_pow {p : ℕ} [hprime : fact p.prime] (hdiv : ¬ p ∣ n) :
minpoly ℤ μ = minpoly ℤ (μ ^ p) | begin
classical,
by_cases hn : n = 0, { simp * at *, },
have hpos := nat.pos_of_ne_zero hn,
by_contra hdiff,
set P := minpoly ℤ μ,
set Q := minpoly ℤ (μ ^ p),
have Pmonic : P.monic := minpoly.monic (h.is_integral hpos),
have Qmonic : Q.monic := minpoly.monic ((h.pow_of_prime hprime.1 hdiv).is_integral h... | lemma | is_primitive_root.minpoly_eq_pow | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/minpoly.lean | [
"ring_theory.roots_of_unity.basic",
"field_theory.minpoly.is_integrally_closed"
] | [
"associated_of_dvd_dvd",
"aux",
"by_contra",
"dvd_or_coprime",
"dvd_trans",
"eq_int_cast",
"fact",
"int.cast_injective",
"int.cast_one",
"int.cast_ring_hom",
"irreducible",
"is_coprime.mul_dvd",
"is_integral",
"minpoly",
"minpoly.degree_pos",
"minpoly.irreducible",
"minpoly.monic",
... | If `p` is a prime that does not divide `n`,
then the minimal polynomials of a primitive `n`-th root of unity `μ`
and of `μ ^ p` are the same. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
minpoly_eq_pow_coprime {m : ℕ} (hcop : nat.coprime m n) :
minpoly ℤ μ = minpoly ℤ (μ ^ m) | begin
revert n hcop,
refine unique_factorization_monoid.induction_on_prime m _ _ _,
{ intros n hn h,
congr,
simpa [(nat.coprime_zero_left n).mp hn] using h },
{ intros u hunit n hcop h,
congr,
simp [nat.is_unit_iff.mp hunit] },
{ intros a p ha hprime hind n hcop h,
rw hind (nat.coprime.cop... | lemma | is_primitive_root.minpoly_eq_pow_coprime | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/minpoly.lean | [
"ring_theory.roots_of_unity.basic",
"field_theory.minpoly.is_integrally_closed"
] | [
"minpoly",
"nat.coprime.coprime_mul_left",
"nat.coprime.coprime_mul_right",
"nat.coprime_zero_left",
"nat.prime.coprime_iff_not_dvd",
"unique_factorization_monoid.induction_on_prime"
] | If `m : ℕ` is coprime with `n`,
then the minimal polynomials of a primitive `n`-th root of unity `μ`
and of `μ ^ m` are the same. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_is_root_minpoly {m : ℕ} (hcop : nat.coprime m n) :
is_root (map (int.cast_ring_hom K) (minpoly ℤ μ)) (μ ^ m) | by simpa [minpoly_eq_pow_coprime h hcop, eval_map, aeval_def (μ ^ m) _]
using minpoly.aeval ℤ (μ ^ m) | lemma | is_primitive_root.pow_is_root_minpoly | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/minpoly.lean | [
"ring_theory.roots_of_unity.basic",
"field_theory.minpoly.is_integrally_closed"
] | [
"int.cast_ring_hom",
"minpoly",
"minpoly.aeval"
] | If `m : ℕ` is coprime with `n`,
then the minimal polynomial of a primitive `n`-th root of unity `μ`
has `μ ^ m` as root. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_roots_of_minpoly [decidable_eq K] : primitive_roots n K ⊆ (map (int.cast_ring_hom K)
(minpoly ℤ μ)).roots.to_finset | begin
by_cases hn : n = 0, { simp * at *, },
have hpos := nat.pos_of_ne_zero hn,
intros x hx,
obtain ⟨m, hle, hcop, rfl⟩ := (is_primitive_root_iff h hpos).1 ((mem_primitive_roots hpos).1 hx),
simpa [multiset.mem_to_finset,
mem_roots (map_monic_ne_zero $ minpoly.monic $ is_integral h hpos)]
using pow_i... | lemma | is_primitive_root.is_roots_of_minpoly | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/minpoly.lean | [
"ring_theory.roots_of_unity.basic",
"field_theory.minpoly.is_integrally_closed"
] | [
"int.cast_ring_hom",
"is_integral",
"mem_primitive_roots",
"minpoly",
"minpoly.monic",
"multiset.mem_to_finset",
"primitive_roots"
] | `primitive_roots n K` is a subset of the roots of the minimal polynomial of a primitive
`n`-th root of unity `μ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
totient_le_degree_minpoly : nat.totient n ≤ (minpoly ℤ μ).nat_degree | begin
classical,
let P : ℤ[X] := minpoly ℤ μ,-- minimal polynomial of `μ`
let P_K : K[X] := map (int.cast_ring_hom K) P, -- minimal polynomial of `μ` sent to `K[X]`
calc
n.totient = (primitive_roots n K).card : h.card_primitive_roots.symm
... ≤ P_K.roots.to_finset.card : finset.card_le_of_subset (is_r... | lemma | is_primitive_root.totient_le_degree_minpoly | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/minpoly.lean | [
"ring_theory.roots_of_unity.basic",
"field_theory.minpoly.is_integrally_closed"
] | [
"finset.card_le_of_subset",
"int.cast_ring_hom",
"minpoly",
"multiset.to_finset_card_le",
"nat.totient",
"primitive_roots"
] | The degree of the minimal polynomial of `μ` is at least `totient n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subring_class (S : Type*) (R : Type u) [ring R] [set_like S R]
extends subsemiring_class S R, neg_mem_class S R : Prop | class | subring_class | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"neg_mem_class",
"ring",
"set_like",
"subsemiring_class"
] | `subring_class S R` states that `S` is a type of subsets `s ⊆ R` that
are both a multiplicative submonoid and an additive subgroup. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subring_class.add_subgroup_class (S : Type*) (R : Type u) [set_like S R]
[ring R] [h : subring_class S R] : add_subgroup_class S R | { .. h } | instance | subring_class.add_subgroup_class | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"add_subgroup_class",
"ring",
"set_like",
"subring_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_int_mem (n : ℤ) : (n : R) ∈ s | by simp only [← zsmul_one, zsmul_mem, one_mem] | lemma | coe_int_mem | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"zsmul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_has_int_cast : has_int_cast s | ⟨λ n, ⟨n, coe_int_mem s n⟩⟩ | instance | subring_class.to_has_int_cast | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"coe_int_mem",
"has_int_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ring : ring s | subtype.coe_injective.ring coe rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) (λ _ _, rfl)
(λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) (λ _, rfl) | instance | subring_class.to_ring | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"ring"
] | A subring of a ring inherits a ring structure | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_comm_ring {R} [comm_ring R] [set_like S R] [subring_class S R] : comm_ring s | subtype.coe_injective.comm_ring coe rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) (λ _ _, rfl)
(λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) (λ _, rfl) | instance | subring_class.to_comm_ring | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"comm_ring",
"set_like",
"subring_class"
] | A subring of a `comm_ring` is a `comm_ring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_ordered_ring {R} [ordered_ring R] [set_like S R] [subring_class S R] :
ordered_ring s | subtype.coe_injective.ordered_ring coe rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) (λ _ _, rfl)
(λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) (λ _, rfl) | instance | subring_class.to_ordered_ring | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"ordered_ring",
"set_like",
"subring_class"
] | A subring of an `ordered_ring` is an `ordered_ring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_ordered_comm_ring {R} [ordered_comm_ring R] [set_like S R] [subring_class S R] :
ordered_comm_ring s | subtype.coe_injective.ordered_comm_ring coe rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl)
(λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) (λ _, rfl) | instance | subring_class.to_ordered_comm_ring | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"ordered_comm_ring",
"set_like",
"subring_class"
] | A subring of an `ordered_comm_ring` is an `ordered_comm_ring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_linear_ordered_ring {R} [linear_ordered_ring R] [set_like S R] [subring_class S R] :
linear_ordered_ring s | subtype.coe_injective.linear_ordered_ring coe rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl)
(λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) (λ _, rfl) (λ _ _, rfl)
(λ _ _, rfl) | instance | subring_class.to_linear_ordered_ring | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"linear_ordered_ring",
"set_like",
"subring_class"
] | A subring of a `linear_ordered_ring` is a `linear_ordered_ring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_linear_ordered_comm_ring {R} [linear_ordered_comm_ring R] [set_like S R]
[subring_class S R] : linear_ordered_comm_ring s | subtype.coe_injective.linear_ordered_comm_ring coe rfl rfl (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl)
(λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) (λ _, rfl) (λ _, rfl) (λ _ _, rfl)
(λ _ _, rfl) | instance | subring_class.to_linear_ordered_comm_ring | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"linear_ordered_comm_ring",
"set_like",
"subring_class"
] | A subring of a `linear_ordered_comm_ring` is a `linear_ordered_comm_ring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subtype (s : S) : s →+* R | { to_fun := coe,
.. submonoid_class.subtype s,
.. add_subgroup_class.subtype s } | def | subring_class.subtype | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"submonoid_class.subtype"
] | The natural ring hom from a subring of ring `R` to `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_nat_cast (n : ℕ) : ((n : s) : R) = n | map_nat_cast (subtype s) n | lemma | subring_class.coe_nat_cast | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"map_nat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_int_cast (n : ℤ) : ((n : s) : R) = n | map_int_cast (subtype s) n | lemma | subring_class.coe_int_cast | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"map_int_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subring (R : Type u) [ring R] extends subsemiring R, add_subgroup R | structure | subring | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"add_subgroup",
"ring",
"subsemiring"
] | `subring R` is the type of subrings of `R`. A subring of `R` is a subset `s` that is a
multiplicative submonoid and an additive subgroup. Note in particular that it shares the
same 0 and 1 as R. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_submonoid (s : subring R) : submonoid R | { carrier := s.carrier,
..s.to_subsemiring.to_submonoid } | def | subring.to_submonoid | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"submonoid",
"subring"
] | The underlying submonoid of a subring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_carrier {s : subring R} {x : R} : x ∈ s.carrier ↔ x ∈ s | iff.rfl | lemma | subring.mem_carrier | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_mk {S : set R} {x : R} (h₁ h₂ h₃ h₄ h₅) :
x ∈ (⟨S, h₁, h₂, h₃, h₄, h₅⟩ : subring R) ↔ x ∈ S | iff.rfl | lemma | subring.mem_mk | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_set_mk (S : set R) (h₁ h₂ h₃ h₄ h₅) :
((⟨S, h₁, h₂, h₃, h₄, h₅⟩ : subring R) : set R) = S | rfl | lemma | subring.coe_set_mk | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_le_mk {S S' : set R} (h₁ h₂ h₃ h₄ h₅ h₁' h₂' h₃' h₄' h₅') :
(⟨S, h₁, h₂, h₃, h₄, h₅⟩ : subring R) ≤ (⟨S', h₁', h₂', h₃', h₄', h₅'⟩ : subring R) ↔ S ⊆ S' | iff.rfl | lemma | subring.mk_le_mk | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {S T : subring R} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T | set_like.ext h | theorem | subring.ext | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"set_like.ext",
"subring"
] | Two subrings are equal if they have the same elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
copy (S : subring R) (s : set R) (hs : s = ↑S) : subring R | { carrier := s,
neg_mem' := λ _, hs.symm ▸ S.neg_mem',
..S.to_subsemiring.copy s hs } | def | subring.copy | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | Copy of a subring with a new `carrier` equal to the old one. Useful to fix definitional
equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (S : subring R) (s : set R) (hs : s = ↑S) :
(S.copy s hs : set R) = s | rfl | lemma | subring.coe_copy | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (S : subring R) (s : set R) (hs : s = ↑S) : S.copy s hs = S | set_like.coe_injective hs | lemma | subring.copy_eq | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"set_like.coe_injective",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_subsemiring_injective : function.injective (to_subsemiring : subring R → subsemiring R) | | r s h := ext (set_like.ext_iff.mp h : _) | lemma | subring.to_subsemiring_injective | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"subring",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_subsemiring_strict_mono : strict_mono (to_subsemiring : subring R → subsemiring R) | λ _ _, id | lemma | subring.to_subsemiring_strict_mono | ring_theory.subring | src/ring_theory/subring/basic.lean | [
"group_theory.subgroup.basic",
"ring_theory.subsemiring.basic"
] | [
"strict_mono",
"subring",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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