noncommCoprod_comp_inr — Mathlib

English

If f and g are monoid homomorphisms with commuting images and the product map is injective, then f and g are injective and their ranges are disjoint.

Русский

Если gомоморфизмы f и g имеют коммутирующие образы и копродукт инъективен, то f и g инъективны и их образы несовместимы (дисjoint).

LaTeX

$$$\\text{Injective}(f\\noncommCoprod g comm) \\iff (\\text{Injective}(f) \\wedge \\text{Injective}(g) \\wedge \\mathrm{Disjoint}(f\\text{range}, g\\text{range})).$$$

Lean4

@[to_additive (attr := simp)]
theorem noncommCoprod_comp_inr : (f.noncommCoprod g comm).comp (inr M N) = g :=
  ext fun x => by simp

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