ext_iff₂ — Mathlib

English

Equality of MonoidHom M (MonoidHom N P) is equivalent to pointwise equality of the induced maps; this extends extensionality to function spaces.

Русский

Равенство гомоморфизмов MonoidHom M (MonoidHom N P) эквивалентно попиксному равенству соответствующих отображений.

LaTeX

$$$f=g\;\Longleftrightarrow\;\forall x,y, f x y = g x y$ with appropriate currying, i.e. extensionality for higher-arity homs.$$

Lean4

@[to_additive]
theorem ext_iff₂ {_ : MulOneClass M} {_ : MulOneClass N} {_ : CommMonoid P} {f g : M →* N →* P} :
    f = g ↔ ∀ x y, f x y = g x y :=
  DFunLike.ext_iff.trans <| forall_congr' fun _ => DFunLike.ext_iff

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