norm_prod_le' — Mathlib

English

The product of two nonunital seminormed rings forms a nonunital seminormed ring with sup norm.

Русский

Произведение двух ненулевых нормированных колец образует ненулевое нормированное кольцо с суп-нормой.

LaTeX

$$$\text{If } \alpha,\beta \text{ are nonunital seminormed rings, then } \alpha\times\beta \text{ is a nonunital seminormed ring with } \| (x,y) \| = \max(\|x\|,\|y\|).$$$

Lean4

theorem norm_prod_le' {α : Type*} [NormedCommRing α] (s : Finset ι) (hs : s.Nonempty) (f : ι → α) :
    ‖∏ i ∈ s, f i‖ ≤ ∏ i ∈ s, ‖f i‖ := by
  rcases s with ⟨⟨l⟩, hl⟩
  have : l.map f ≠ [] := by simpa using hs
  simpa using List.norm_prod_le' this

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