Theorems · Theorem · group theory
AddSubgroup.addCommutator_pi_pi_of_finite
∀ {η : Type u_2} [Finite η] {Gs : η → Type u_3} [inst : (i : η) → AddGroup (Gs i)] (H K : (i : η) → AddSubgroup (Gs i)),
⁅AddSubgroup.pi Set.univ H, AddSubgroup.pi Set.univ K⁆ = AddSubgroup.pi Set.univ fun i => ⁅H i, K i⁆The commutator of a finite direct product is contained in the direct product of the commutators.
- Defined in
- Mathlib.GroupTheory.Commutator.Finite
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites23
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- SetLike.coeproof · cited by 8,199
- AddGroupstatement and proof · cited by 4,410
- Set.univstatement and proof · cited by 3,945
- AddSubgroupstatement and proof · cited by 3,232
- Finitestatement and proof · cited by 3,029
- le_antisymmproof · cited by 2,068
- Bracket.bracketstatement and proof · cited by 642
- Pi.singleproof · cited by 518
- SetLike.mem_coeproof · cited by 302
- AddSubgroup.mapproof · cited by 189
- ZeroMemClass.zero_memproof · cited by 162
Cited by1
Results whose statement or proof uses this declaration.
- AddSubgroup.lowerCentralSeries_pi_of_finiteproof · cited by 1