Theorems · Theorem · group theory
Subgroup.commutator_pi_pi_le
∀ {η : Type u_4} {Gs : η → Type u_5} [inst : (i : η) → Group (Gs i)] (H K : (i : η) → Subgroup (Gs i)),
⁅Subgroup.pi Set.univ H, Subgroup.pi Set.univ K⁆ ≤ Subgroup.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
- Group
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.
- Groupstatement and proof · cited by 6,238
- Set.univstatement and proof · cited by 3,945
- Subgroupstatement and proof · cited by 3,593
- Bracket.bracketstatement · cited by 642
- Subgroup.pistatement and proof · cited by 23
- Subgroup.commutator_mem_commutatorproof · cited by 13
- Subgroup.commutator_leproof · cited by 10
Cited by2
Results whose statement or proof uses this declaration.
- Subgroup.lowerCentralSeries_pi_leproof · cited by 2
- Subgroup.commutator_pi_pi_of_finiteproof · cited by 1