continuous_conj_prod — Mathlib

English

Conjugation is jointly continuous on G × G when both multiplication and inversion are continuous.

Русский

Связь конъюгации по паре элементов в произведении G × G непрерывна при условии непрерывности умножения и инверсии.

LaTeX

$$$\text{Continuous}(g\mapsto g_{1} g_{2} g_{1}^{-1})$ for $g=(g_1,g_2) \in G\times G$.$$

Lean4

/-- Conjugation is jointly continuous on `G × G` when both `mul` and `inv` are continuous. -/
@[to_additive continuous_addConj_prod /--
    Conjugation is jointly continuous on `G × G` when both `add` and `neg` are continuous. -/
    ]
theorem continuous_conj_prod [ContinuousInv G] : Continuous fun g : G × G => g.fst * g.snd * g.fst⁻¹ :=
  continuous_mul.mul (continuous_inv.comp continuous_fst)

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