Theorems · Theorem · linear algebra

Submodule.mem_map

∀ {R : Type u_1} {R₂ : Type u_3} {M : Type u_5} {M₂ : Type u_7} [inst : Semiring R] [inst_1 : Semiring R₂]
  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module R₂ M₂] {σ₁₂ : R →+* R₂}
  [inst_6 : RingHomSurjective σ₁₂] {f : M →ₛₗ[σ₁₂] M₂} {p : Submodule R M} {x : M₂},
  x ∈ Submodule.map f p ↔ ∃ y ∈ p, f y = x

Defined in

Mathlib.Algebra.Module.Submodule.Map

Cited by

24 results in Mathlib

Foundations

Depth 25 from the axioms · uses propext

Assumes

SemiringSemiringAddCommMonoidAddCommMonoidModuleModuleRingHomSurjective