English
Two monoid homomorphisms from the quotient c.Quotient to P are equal whenever their compositions with c.mk' agree; equivalently, the lift is an extensional auxiliary for equality of maps on the quotient.
Русский
Два моноид-гомоморфизма из квотыquотинга совпадают, когда их композиции с c.mk' совпадают; иначе это экстенсивный критерий равенства гомоморфизмов на квоти.
LaTeX
$$$\forall f,g:\; c.Quotient \to P,\; (f \circ c.mk') = (g \circ c.mk') \Rightarrow f = g$$$
Lean4
/-- Homomorphisms on the quotient of a monoid by a congruence relation `c` are equal if their
compositions with `c.mk'` are equal. -/
@[to_additive (attr := ext) /-- Homomorphisms on the quotient of an `AddMonoid` by an additive
congruence relation `c` are equal if their compositions with `c.mk'` are equal. -/
]
theorem hom_ext {f g : c.Quotient →* P} (h : f.comp c.mk' = g.comp c.mk') : f = g :=
by
rw [← lift_apply_mk' f, ← lift_apply_mk' g]
congr 1
