Theorems · Theorem · group theory

Submonoid.iSup_induction

∀ {M : Type u_1} [inst : MulOneClass M] {ι : Sort u_4} (S : ι → Submonoid M) {motive : M → Prop} {x : M},
  x ∈ ⨆ i, S i →
    (∀ (i : ι), ∀ x ∈ S i, motive x) → motive 1 → (∀ (x y : M), motive x → motive y → motive (x * y)) → motive x

An induction principle for elements of ⨆ i, S i. If C holds for 1 and all elements of S i for all i, and is preserved under multiplication, then it holds for all elements of the supremum of S.

Defined in

Mathlib.Algebra.Group.Submonoid.Membership

Cited by

1 results in Mathlib

Foundations

Depth 67 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MulOneClass