Theorems · Theorem · linear algebra

Submodule.span_induction

∀ {R : Type u_1} {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {s : Set M}
  {p : (x : M) → x ∈ Submodule.span R s → Prop},
  (∀ (x : M) (h : x ∈ s), p x ⋯) →
    p 0 ⋯ →
      (∀ (x y : M) (hx : x ∈ Submodule.span R s) (hy : y ∈ Submodule.span R s), p x hx → p y hy → p (x + y) ⋯) →
        (∀ (a : R) (x : M) (hx : x ∈ Submodule.span R s), p x hx → p (a • x) ⋯) →
          ∀ {x : M} (hx : x ∈ Submodule.span R s), p x hx

An induction principle for span membership. If p holds for 0 and all elements of s, and is preserved under addition and scalar multiplication, then p holds for all elements of the span of s.

Defined in

Mathlib.LinearAlgebra.Span.Defs

Cited by

77 results in Mathlib

Foundations

Depth 25 from the axioms · uses propext, Quot.sound

Assumes

SemiringAddCommMonoidModule