Theorems · Theorem · group theory
Function.Injective.noZeroDivisors
∀ {M₀ : Type u_1} {M₀' : Type u_3} [inst : Mul M₀] [inst_1 : Zero M₀] [inst_2 : Mul M₀'] [inst_3 : Zero M₀']
(f : M₀ → M₀'),
Function.Injective f → f 0 = 0 → (∀ (x y : M₀), f (x * y) = f x * f y) → ∀ [NoZeroDivisors M₀'], NoZeroDivisors M₀Pull back a NoZeroDivisors instance along an injective function.
- Defined in
- Mathlib.Algebra.GroupWithZero.InjSurj
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 7 from the axioms · uses no axioms
- Assumes
- MulZeroMulZeroNoZeroDivisors
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites2
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- NoZeroDivisorsstatement and proof · cited by 545
- NoZeroDivisors.eq_zero_or_eq_zero_of_mul_eq_zeroproof · cited by 14
Cited by8
Results whose statement or proof uses this declaration.
- IsAlgebraic.restrictScalars_of_isIntegralproof · cited by 4
- AlgebraicIndependent.extendScalarsproof · cited by 4
- IsBaseChange.lift_rank_eqproof · cited by 3
- lift_trdeg_add_eqproof · cited by 2
- NoZeroDivisors.of_faithfulSMulproof · cited by 1
- MulEquiv.noZeroDivisorsproof · cited by 1
- IsFractionRing.of_fieldproof · cited by 0
- Polynomial.noZeroDivisors_iffproof · cited by 0