Theorem ricqusker

Description: The image H of a ring homomorphism F is isomorphic with the quotient ring Q over F 's kernel K . This a part of what is sometimes called the first isomorphism theorem for rings. (Contributed by Thierry Arnoux, 10-Mar-2025)

	Ref	Expression
Hypotheses	rhmqusker.1	|- .0. = ( 0g ` H )
	rhmqusker.f	|- ( ph -> F e. ( G RingHom H ) )
	rhmqusker.k	|- K = ( `' F " { .0. } )
	rhmqusker.q	|- Q = ( G /s ( G ~QG K ) )
	rhmqusker.s	|- ( ph -> ran F = ( Base ` H ) )
	rhmqusker.2	|- ( ph -> G e. CRing )
Assertion	ricqusker	|- ( ph -> Q ~=r H )

Proof

Step	Hyp	Ref	Expression
1		rhmqusker.1	|- .0. = ( 0g ` H )
2		rhmqusker.f	|- ( ph -> F e. ( G RingHom H ) )
3		rhmqusker.k	|- K = ( `' F " { .0. } )
4		rhmqusker.q	|- Q = ( G /s ( G ~QG K ) )
5		rhmqusker.s	|- ( ph -> ran F = ( Base ` H ) )
6		rhmqusker.2	|- ( ph -> G e. CRing )
7		imaeq2	|- ( p = q -> ( F " p ) = ( F " q ) )
8	7	unieqd	|- ( p = q -> U. ( F " p ) = U. ( F " q ) )
9	8	cbvmptv	|- ( p e. ( Base ` Q ) |-> U. ( F " p ) ) = ( q e. ( Base ` Q ) |-> U. ( F " q ) )
10	1 2 3 4 5 6 9	rhmqusker	|- ( ph -> ( p e. ( Base ` Q ) |-> U. ( F " p ) ) e. ( Q RingIso H ) )
11		brrici	|- ( ( p e. ( Base ` Q ) |-> U. ( F " p ) ) e. ( Q RingIso H ) -> Q ~=r H )
12	10 11	syl	|- ( ph -> Q ~=r H )