English
The dual opposite-product setting provides an isometric smul on the opposite product, i.e., (M×N)ᵐᵒᵖ acts isometrically on (M×N)ᵐᵒᵖ by componentwise right multiplications in the opposite sense.
Русский
Двойственное противоположное произведение даёт изометрическое действие на противоположное произведение пространств, компонентно справа.
LaTeX
$$$\forall c\in (M\times N)^{\mathrm{op}},\; d((c\cdot x)^{\mathrm{op}}, (c\cdot y)^{\mathrm{op}}) = d(x^{\mathrm{op}}, y^{\mathrm{op}})$$$
Lean4
@[to_additive]
instance isIsometricSMul' {N} [Mul M] [PseudoEMetricSpace M] [IsIsometricSMul M M] [Mul N] [PseudoEMetricSpace N]
[IsIsometricSMul N N] : IsIsometricSMul (M × N) (M × N) :=
⟨fun c => (isometry_smul M c.1).prodMap (isometry_smul N c.2)⟩
