Theorems · Theorem · group theory
AddSubgroup.addCommutator_pi_pi_le
∀ {η : Type u_4} {Gs : η → Type u_5} [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 direct product is contained in the direct product of the
commutators. See commutator_pi_pi_of_finite for equality given Fintype η.
- Defined in
- Mathlib.GroupTheory.Commutator.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- AddGroup
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- AddGroupstatement and proof · cited by 4,410
- Set.univstatement and proof · cited by 3,945
- AddSubgroupstatement and proof · cited by 3,232
- Bracket.bracketstatement · cited by 642
- AddSubgroup.pistatement and proof · cited by 26
- AddSubgroup.addCommutator_leproof · cited by 11
- AddSubgroup.addCommutator_mem_addCommutatorproof · cited by 9
Cited by2
Results whose statement or proof uses this declaration.
- AddSubgroup.lowerCentralSeries_pi_leproof · cited by 1
- AddSubgroup.addCommutator_pi_pi_of_finiteproof · cited by 1