Theorems · Theorem · commutative algebra

Submodule.smul_inductionOn_pointwise

∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}
  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] (N : Submodule R M) [inst_5 : SMulCommClass S R M] {a : S}
  {p : (x : M) → x ∈ a • N → Prop},
  (∀ (s : M) (hs : s ∈ N), p (a • s) ⋯) →
    (∀ (r : R) (m : M) (mem : m ∈ a • N), p m mem → p (r • m) ⋯) →
      (∀ (x y : M) (hx : x ∈ a • N) (hy : y ∈ a • N), p x hx → p y hy → p (x + y) ⋯) →
        p 0 ⋯ → ∀ {x : M} (hx : x ∈ a • N), p x hx

Defined in

Mathlib.Algebra.Module.Submodule.Pointwise

Cited by

1 results in Mathlib

Foundations

Depth 70 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidModuleMonoidDistribMulActionSMulCommClass