Theorems · Theorem · group theory

AddSubmonoid.closure_induction

∀ {M : Type u_1} [inst : AddZeroClass M] {s : Set M} {motive : (x : M) → x ∈ AddSubmonoid.closure s → Prop},
  (∀ (x : M) (h : x ∈ s), motive x ⋯) →
    motive 0 ⋯ →
      (∀ (x y : M) (hx : x ∈ AddSubmonoid.closure s) (hy : y ∈ AddSubmonoid.closure s),
          motive x hx → motive y hy → motive (x + y) ⋯) →
        ∀ {x : M} (hx : x ∈ AddSubmonoid.closure s), motive x hx

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

Defined in

Mathlib.Algebra.Group.Submonoid.Basic

Cited by

30 results in Mathlib

Foundations

Depth 64 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddZeroClass