feat(algebra/gcd_monoid/*): assorted lemmas (#10508)

From flt-regular.



Co-authored-by: Johan Commelin <johan@commelin.net>

src/algebra/gcd_monoid/finset.lean

@@ -147,6 +147,11 @@ dvd_gcd (λ b hb, (gcd_dvd hb).trans (h b hb))
 lemma gcd_mono (h : s₁ ⊆ s₂) : s₂.gcd f ∣ s₁.gcd f :=
 dvd_gcd $ assume b hb, gcd_dvd (h hb)
 
+theorem gcd_image {g : γ → β} (s: finset γ) [decidable_eq β] [is_idempotent α gcd_monoid.gcd] :
+  (s.image g).gcd f = s.gcd (f ∘ g) := by simp [gcd, fold_image_idem]
+
+theorem gcd_eq_gcd_image [decidable_eq α] [is_idempotent α gcd_monoid.gcd] :
+  s.gcd f = (s.image f).gcd id := (@gcd_image _ _ _ _ _ id _ _ _ _).symm
 
 theorem gcd_eq_zero_iff : s.gcd f = 0 ↔ ∀ (x : β), x ∈ s → f x = 0 :=
 begin

src/algebra/gcd_monoid/nat.lean

@@ -0,0 +1,45 @@
+/-
+Copyright (c) 2021 Eric Rodriguez. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eric Rodriguez
+-/
+import algebra.gcd_monoid.finset
+import number_theory.padics.padic_norm
+
+/-!
+# Basic results about setwise gcds on ℕ
+
+This file proves some basic results about `finset.gcd` on `ℕ`.
+
+## Main results
+* `finset.coprime_of_div_gcd`: The elements of a set divided through by their gcd are coprime.
+
+-/
+
+instance : is_idempotent ℕ gcd_monoid.gcd := ⟨nat.gcd_self⟩
+
+namespace finset
+
+theorem coprime_of_div_gcd (s : finset ℕ) {x : ℕ} (hx : x ∈ s) (hnz : x ≠ 0) :
+  s.gcd (/ (s.gcd id)) = 1 :=
+begin
+  rw nat.eq_one_iff_not_exists_prime_dvd,
+  intros p hp hdvd,
+  haveI : fact p.prime := ⟨hp⟩,
+  rw dvd_gcd_iff at hdvd,
+  replace hdvd : ∀ b ∈ s, s.gcd id * p ∣ b,
+  { intros b hb,
+    specialize hdvd b hb,
+    rwa nat.dvd_div_iff at hdvd,
+    apply gcd_dvd hb },
+  have : s.gcd id ≠ 0 := (not_iff_not.mpr gcd_eq_zero_iff).mpr (λ h, hnz $ h x hx),
+  apply @pow_succ_padic_val_nat_not_dvd p _ _ this.bot_lt,
+  apply dvd_gcd,
+  intros b hb,
+  obtain ⟨k, rfl⟩ := hdvd b hb,
+  rw [id, mul_right_comm, pow_succ', mul_dvd_mul_iff_right hp.ne_zero],
+  apply dvd_mul_of_dvd_left,
+  exact pow_padic_val_nat_dvd
+end
+
+end finset