Theorems · Theorem · group theory

Submonoid.mem_biSup_of_directedOn

∀ {M : Type u_1} [inst : MulOneClass M] {ι : Type u_4} {p : ι → Prop},
  (∃ i, p i) →
    ∀ {S : ι → Submonoid M},
      DirectedOn (Function.onFun (fun x1 x2 => x1 ≤ x2) S) {i | p i} →
        ∀ {x : M}, x ∈ ⨆ i, ⨆ (_ : p i), S i ↔ ∃ i, p i ∧ x ∈ S i

Defined in

Mathlib.Algebra.Group.Submonoid.Membership

Cited by

0 results in Mathlib

Foundations

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

Assumes

MulOneClass