Theorems · Theorem · linear algebra
Submodule.apply_mem_span_image_of_mem_span
∀ {R : Type u_1} {R₂ : Type u_2} {M : Type u_4} {M₂ : Type u_5} [inst : Semiring R] [inst_1 : AddCommMonoid M]
[inst_2 : Module R M] [inst_3 : Semiring R₂] {σ₁₂ : R →+* R₂} [inst_4 : AddCommMonoid M₂] [inst_5 : Module R₂ M₂]
[RingHomSurjective σ₁₂] (f : M →ₛₗ[σ₁₂] M₂) {x : M} {s : Set M},
x ∈ Submodule.span R s → f x ∈ Submodule.span R₂ (⇑f '' s)- Defined in
- Mathlib.LinearAlgebra.Span.Basic
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 29 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Setstatement and proof · cited by 53,352
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- LinearMapstatement and proof · cited by 10,215
- RingHomstatement and proof · cited by 10,189
- Submodulestatement and proof · cited by 7,192
- Set.imagestatement · cited by 5,609
- Submodule.spanstatement and proof · cited by 1,504
- RingHomSurjectivestatement and proof · cited by 220
- Submodule.span_imageproof · cited by 25
Cited by3
Results whose statement or proof uses this declaration.
- Polynomial.Chebyshev.T_iterate_derivative_mem_span_Tproof · cited by 1
- Submodule.notMem_span_of_apply_notMem_span_imageproof · cited by 1
- Profinite.NobelingProof.GoodProducts.spanproof · cited by 1