map_update_add — Mathlib

English

For f : ContinuousMultilinearMap R M₁ M₂×M₃, and g,h : ContinuousMultilinearMap R M₁ M₂, we have f = g.prod h if and only if fst ∘ f = g and snd ∘ f = h.

Русский

Для f: ContinuousMultilinearMap R M₁ (M₂×M₃) и g,h: ContinuousMultilinearMap R M₁ M₂ имеем f = g.prod h тогда и только тогда, когда fst ∘ f = g и snd ∘ f = h.

LaTeX

$$$f = g.\\prod h \\iff (\\mathrm{fst} \\circ f) = g \\land (\\mathrm{snd} \\circ f) = h$$$

Lean4

@[simp]
theorem map_update_add [DecidableEq ι] (m : ∀ i, M₁ i) (i : ι) (x y : M₁ i) :
    f (update m i (x + y)) = f (update m i x) + f (update m i y) :=
  f.map_update_add' m i x y

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