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Theorems · Theorem · group theory

Subgroup.mem_closure

∀ {G : Type u_1} [inst : Group G] {k : Set G} {x : G}, x ∈ Subgroup.closure k ↔ ∀ (K : Subgroup G), k ⊆ ↑K → x ∈ K
Defined in
Mathlib.Algebra.Group.Subgroup.Lattice
Cited by
2 results in Mathlib
Foundations
Depth 67 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
Group
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Subgroup.subset_closure · cited by 53Subgroup.subset_closureMonoidWithZeroHom.mem_valueGroup · cited by 4MonoidWithZeroHom.mem_val…Set · cited by 53352SetSetLike.coe · cited by 8199SetLike.coeGroup · cited by 6238GroupSubgroup · cited by 3593SubgroupSubgroup.closure · cited by 196Subgroup.closureSubgroup.mem_sInf · cited by 2Subgroup.mem_sInfSubgroup.mem_closureCITED BYCITES

Cites6

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by2

Results whose statement or proof uses this declaration.