Theorems · Theorem · commutative algebra
GCDMonoid.dvd_gcd
∀ {α : Type u_2} {inst : CommMonoidWithZero α} [self : GCDMonoid α] {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c bAny common divisor of both elements is a divisor of the GCD.
- Defined in
- Mathlib.Algebra.GCDMonoid.Basic
- Cited by
- 23 results in Mathlib
- Foundations
- Depth 6 from the axioms · uses no axioms
- Assumes
- GCDMonoid
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites3
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommMonoidWithZerostatement and proof · cited by 913
- GCDMonoid.gcdstatement · cited by 143
- GCDMonoidstatement and proof · cited by 96
Cited by23
Results whose statement or proof uses this declaration.
- dvd_gcd_iffproof · cited by 8
- gcd_sameproof · cited by 8
- gcd_dvd_gcdproof · cited by 8
- gcd_commproof · cited by 6
- gcd_mul_left'proof · cited by 4
- dvd_gcd_mul_of_dvd_mulproof · cited by 3
- gcd_zero_rightproof · cited by 3
- gcd_assocproof · cited by 3
- gcd_eq_zero_iffproof · cited by 3
- gcd_mul_dvd_mul_gcdproof · cited by 2
- gcd_mul_leftproof · cited by 2
- gcd_zero_leftproof · cited by 2