Theorem rngqiprngho

Description: F is a homomorphism of non-unital rings. (Contributed by AV, 21-Feb-2025)

	Ref	Expression
Hypotheses	rng2idlring.r	|- ( ph -> R e. Rng )
	rng2idlring.i	|- ( ph -> I e. ( 2Ideal ` R ) )
	rng2idlring.j	|- J = ( R |`s I )
	rng2idlring.u	|- ( ph -> J e. Ring )
	rng2idlring.b	|- B = ( Base ` R )
	rng2idlring.t	|- .x. = ( .r ` R )
	rng2idlring.1	|- .1. = ( 1r ` J )
	rngqiprngim.g	|- .~ = ( R ~QG I )
	rngqiprngim.q	|- Q = ( R /s .~ )
	rngqiprngim.c	|- C = ( Base ` Q )
	rngqiprngim.p	|- P = ( Q Xs. J )
	rngqiprngim.f	|- F = ( x e. B |-> <. [ x ] .~ , ( .1. .x. x ) >. )
Assertion	rngqiprngho	|- ( ph -> F e. ( R RngHom P ) )

Proof

Step	Hyp	Ref	Expression
1		rng2idlring.r	|- ( ph -> R e. Rng )
2		rng2idlring.i	|- ( ph -> I e. ( 2Ideal ` R ) )
3		rng2idlring.j	|- J = ( R |`s I )
4		rng2idlring.u	|- ( ph -> J e. Ring )
5		rng2idlring.b	|- B = ( Base ` R )
6		rng2idlring.t	|- .x. = ( .r ` R )
7		rng2idlring.1	|- .1. = ( 1r ` J )
8		rngqiprngim.g	|- .~ = ( R ~QG I )
9		rngqiprngim.q	|- Q = ( R /s .~ )
10		rngqiprngim.c	|- C = ( Base ` Q )
11		rngqiprngim.p	|- P = ( Q Xs. J )
12		rngqiprngim.f	|- F = ( x e. B |-> <. [ x ] .~ , ( .1. .x. x ) >. )
13	1 2 3 4 5 6 7 8 9 10 11	rngqiprng	|- ( ph -> P e. Rng )
14	1 2 3 4 5 6 7 8 9 10 11 12	rngqiprngghm	|- ( ph -> F e. ( R GrpHom P ) )
15	1 2 3 4 5 6 7 8 9 10 11 12	rngqiprnglin	|- ( ph -> A. a e. B A. b e. B ( F ` ( a .x. b ) ) = ( ( F ` a ) ( .r ` P ) ( F ` b ) ) )
16	14 15	jca	|- ( ph -> ( F e. ( R GrpHom P ) /\ A. a e. B A. b e. B ( F ` ( a .x. b ) ) = ( ( F ` a ) ( .r ` P ) ( F ` b ) ) ) )
17		eqid	|- ( .r ` P ) = ( .r ` P )
18	5 6 17	isrnghm	|- ( F e. ( R RngHom P ) <-> ( ( R e. Rng /\ P e. Rng ) /\ ( F e. ( R GrpHom P ) /\ A. a e. B A. b e. B ( F ` ( a .x. b ) ) = ( ( F ` a ) ( .r ` P ) ( F ` b ) ) ) ) )
19	1 13 16 18	syl21anbrc	|- ( ph -> F e. ( R RngHom P ) )