prodEquiv — Mathlib

English

The product of subgroups H ≤ G and K ≤ N is canonically isomorphic to H × K as groups.

Русский

Произведение подгрупп H в G и K в N канонически изоморфно H × K как группам.

LaTeX

$$$ H \mathrm{prod} K \ \cong^*\ K \;\text{(isomorphism)}$$$

Lean4

/-- Product of subgroups is isomorphic to their product as groups. -/
@[to_additive prodEquiv /-- Product of additive subgroups is isomorphic to their product
          as additive groups -/
    ]
def prodEquiv (H : Subgroup G) (K : Subgroup N) : H.prod K ≃* H × K :=
  { Equiv.Set.prod (H : Set G) (K : Set N) with map_mul' := fun _ _ => rfl }

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