Theorems · Theorem · group theory
AddSubsemigroup.mem_iSup_of_directed
∀ {M : Type u_2} [inst : Add M] {ι : Sort u_3} {S : ι → AddSubsemigroup M},
Directed (fun x1 x2 => x1 ≤ x2) S → ∀ {x : M}, x ∈ ⨆ i, S i ↔ ∃ i, x ∈ S i- Cited by
- 3 results in Mathlib
- Foundations
- Depth 66 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Add
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- SetLike.coeproof · cited by 8,199
- Set.iUnionproof · cited by 2,483
- iSupstatement and proof · cited by 2,415
- SetLike.mem_coeproof · cited by 302
- AddSubsemigroupstatement and proof · cited by 262
- AddMemClass.add_memproof · cited by 229
- Directedstatement and proof · cited by 213
- Set.mem_iUnionproof · cited by 212
- le_iSupproof · cited by 207
- AddSubsemigroup.closureproof · cited by 34
- AddSubsemigroup.closure_inductionproof · cited by 5
- AddSubsemigroup.closure_eqproof · cited by 3
Cited by3
Results whose statement or proof uses this declaration.
- AddSubsemigroup.coe_iSup_of_directedproof · cited by 1
- AddSubsemigroup.mem_sSup_of_directed_onproof · cited by 1
- AddSubsemigroup.mem_biSup_of_directedOnproof · cited by 0