Theorem rhmpropd 13886

Description: Ring homomorphism depends only on the ring attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.)

Hypotheses

Ref	Expression
rhmpropd.a	|- ( ph -> B = ( Base ` J ) )
rhmpropd.b	|- ( ph -> C = ( Base ` K ) )
rhmpropd.c	|- ( ph -> B = ( Base ` L ) )
rhmpropd.d	|- ( ph -> C = ( Base ` M ) )
rhmpropd.e	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` J ) y ) = ( x ( +g ` L ) y ) )
rhmpropd.f	|- ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` M ) y ) )
rhmpropd.g	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` J ) y ) = ( x ( .r ` L ) y ) )
rhmpropd.h	|- ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` M ) y ) )

Assertion

Ref	Expression
rhmpropd	|- ( ph -> ( J RingHom K ) = ( L RingHom M ) )

Distinct variable groups:   x, y, J   x, K, y   x, L, y   x, M, y   ph, x, y   x, B, y   x, C, y

Proof of Theorem rhmpropd

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2196	. . . . . 6 |- (mulGrp ` J ) = (mulGrp ` J )
2		eqid 2196	. . . . . 6 |- (mulGrp ` K ) = (mulGrp ` K )
3	1, 2	isrhm 13790	. . . . 5 |- ( f e. ( J RingHom K ) <-> ( ( J e. Ring /\ K e. Ring ) /\ ( f e. ( J GrpHom K ) /\ f e. ( (mulGrp ` J ) MndHom (mulGrp ` K ) ) ) ) )
4	3	simplbi 274	. . . 4 |- ( f e. ( J RingHom K ) -> ( J e. Ring /\ K e. Ring ) )
5	4	a1i 9	. . 3 |- ( ph -> ( f e. ( J RingHom K ) -> ( J e. Ring /\ K e. Ring ) ) )
6		eqid 2196	. . . . . 6 |- (mulGrp ` L ) = (mulGrp ` L )
7		eqid 2196	. . . . . 6 |- (mulGrp ` M ) = (mulGrp ` M )
8	6, 7	isrhm 13790	. . . . 5 |- ( f e. ( L RingHom M ) <-> ( ( L e. Ring /\ M e. Ring ) /\ ( f e. ( L GrpHom M ) /\ f e. ( (mulGrp ` L ) MndHom (mulGrp ` M ) ) ) ) )
9	8	simplbi 274	. . . 4 |- ( f e. ( L RingHom M ) -> ( L e. Ring /\ M e. Ring ) )
10		rhmpropd.a	. . . . . 6 |- ( ph -> B = ( Base ` J ) )
11		rhmpropd.c	. . . . . 6 |- ( ph -> B = ( Base ` L ) )
12		rhmpropd.e	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` J ) y ) = ( x ( +g ` L ) y ) )
13		rhmpropd.g	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` J ) y ) = ( x ( .r ` L ) y ) )
14	10, 11, 12, 13	ringpropd 13670	. . . . 5 |- ( ph -> ( J e. Ring <-> L e. Ring ) )
15		rhmpropd.b	. . . . . 6 |- ( ph -> C = ( Base ` K ) )
16		rhmpropd.d	. . . . . 6 |- ( ph -> C = ( Base ` M ) )
17		rhmpropd.f	. . . . . 6 |- ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` M ) y ) )
18		rhmpropd.h	. . . . . 6 |- ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` M ) y ) )
19	15, 16, 17, 18	ringpropd 13670	. . . . 5 |- ( ph -> ( K e. Ring <-> M e. Ring ) )
20	14, 19	anbi12d 473	. . . 4 |- ( ph -> ( ( J e. Ring /\ K e. Ring ) <-> ( L e. Ring /\ M e. Ring ) ) )
21	9, 20	imbitrrid 156	. . 3 |- ( ph -> ( f e. ( L RingHom M ) -> ( J e. Ring /\ K e. Ring ) ) )
22	20	adantr 276	. . . . . 6 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> ( ( J e. Ring /\ K e. Ring ) <-> ( L e. Ring /\ M e. Ring ) ) )
23	10, 15, 11, 16, 12, 17	ghmpropd 13489	. . . . . . . . 9 |- ( ph -> ( J GrpHom K ) = ( L GrpHom M ) )
24	23	eleq2d 2266	. . . . . . . 8 |- ( ph -> ( f e. ( J GrpHom K ) <-> f e. ( L GrpHom M ) ) )
25	24	adantr 276	. . . . . . 7 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> ( f e. ( J GrpHom K ) <-> f e. ( L GrpHom M ) ) )
26	10	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> B = ( Base ` J ) )
27		simprl 529	. . . . . . . . . . 11 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> J e. Ring )
28		eqid 2196	. . . . . . . . . . . 12 |- ( Base ` J ) = ( Base ` J )
29	1, 28	mgpbasg 13558	. . . . . . . . . . 11 |- ( J e. Ring -> ( Base ` J ) = ( Base ` (mulGrp ` J ) ) )
30	27, 29	syl 14	. . . . . . . . . 10 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> ( Base ` J ) = ( Base ` (mulGrp ` J ) ) )
31	26, 30	eqtrd 2229	. . . . . . . . 9 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> B = ( Base ` (mulGrp ` J ) ) )
32	15	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> C = ( Base ` K ) )
33		simprr 531	. . . . . . . . . . 11 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> K e. Ring )
34		eqid 2196	. . . . . . . . . . . 12 |- ( Base ` K ) = ( Base ` K )
35	2, 34	mgpbasg 13558	. . . . . . . . . . 11 |- ( K e. Ring -> ( Base ` K ) = ( Base ` (mulGrp ` K ) ) )
36	33, 35	syl 14	. . . . . . . . . 10 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> ( Base ` K ) = ( Base ` (mulGrp ` K ) ) )
37	32, 36	eqtrd 2229	. . . . . . . . 9 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> C = ( Base ` (mulGrp ` K ) ) )
38	11	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> B = ( Base ` L ) )
39	20	simprbda 383	. . . . . . . . . . 11 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> L e. Ring )
40		eqid 2196	. . . . . . . . . . . 12 |- ( Base ` L ) = ( Base ` L )
41	6, 40	mgpbasg 13558	. . . . . . . . . . 11 |- ( L e. Ring -> ( Base ` L ) = ( Base ` (mulGrp ` L ) ) )
42	39, 41	syl 14	. . . . . . . . . 10 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> ( Base ` L ) = ( Base ` (mulGrp ` L ) ) )
43	38, 42	eqtrd 2229	. . . . . . . . 9 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> B = ( Base ` (mulGrp ` L ) ) )
44	16	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> C = ( Base ` M ) )
45	20	simplbda 384	. . . . . . . . . . 11 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> M e. Ring )
46		eqid 2196	. . . . . . . . . . . 12 |- ( Base ` M ) = ( Base ` M )
47	7, 46	mgpbasg 13558	. . . . . . . . . . 11 |- ( M e. Ring -> ( Base ` M ) = ( Base ` (mulGrp ` M ) ) )
48	45, 47	syl 14	. . . . . . . . . 10 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> ( Base ` M ) = ( Base ` (mulGrp ` M ) ) )
49	44, 48	eqtrd 2229	. . . . . . . . 9 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> C = ( Base ` (mulGrp ` M ) ) )
50	13	adantlr 477	. . . . . . . . . 10 |- ( ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` J ) y ) = ( x ( .r ` L ) y ) )
51	27	adantr 276	. . . . . . . . . . 11 |- ( ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) /\ ( x e. B /\ y e. B ) ) -> J e. Ring )
52		eqid 2196	. . . . . . . . . . . . 13 |- ( .r ` J ) = ( .r ` J )
53	1, 52	mgpplusgg 13556	. . . . . . . . . . . 12 |- ( J e. Ring -> ( .r ` J ) = ( +g ` (mulGrp ` J ) ) )
54	53	oveqd 5942	. . . . . . . . . . 11 |- ( J e. Ring -> ( x ( .r ` J ) y ) = ( x ( +g ` (mulGrp ` J ) ) y ) )
55	51, 54	syl 14	. . . . . . . . . 10 |- ( ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` J ) y ) = ( x ( +g ` (mulGrp ` J ) ) y ) )
56	39	adantr 276	. . . . . . . . . . 11 |- ( ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) /\ ( x e. B /\ y e. B ) ) -> L e. Ring )
57		eqid 2196	. . . . . . . . . . . . 13 |- ( .r ` L ) = ( .r ` L )
58	6, 57	mgpplusgg 13556	. . . . . . . . . . . 12 |- ( L e. Ring -> ( .r ` L ) = ( +g ` (mulGrp ` L ) ) )
59	58	oveqd 5942	. . . . . . . . . . 11 |- ( L e. Ring -> ( x ( .r ` L ) y ) = ( x ( +g ` (mulGrp ` L ) ) y ) )
60	56, 59	syl 14	. . . . . . . . . 10 |- ( ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` L ) y ) = ( x ( +g ` (mulGrp ` L ) ) y ) )
61	50, 55, 60	3eqtr3d 2237	. . . . . . . . 9 |- ( ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` (mulGrp ` J ) ) y ) = ( x ( +g ` (mulGrp ` L ) ) y ) )
62	18	adantlr 477	. . . . . . . . . 10 |- ( ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) /\ ( x e. C /\ y e. C ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` M ) y ) )
63	33	adantr 276	. . . . . . . . . . 11 |- ( ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) /\ ( x e. C /\ y e. C ) ) -> K e. Ring )
64		eqid 2196	. . . . . . . . . . . . 13 |- ( .r ` K ) = ( .r ` K )
65	2, 64	mgpplusgg 13556	. . . . . . . . . . . 12 |- ( K e. Ring -> ( .r ` K ) = ( +g ` (mulGrp ` K ) ) )
66	65	oveqd 5942	. . . . . . . . . . 11 |- ( K e. Ring -> ( x ( .r ` K ) y ) = ( x ( +g ` (mulGrp ` K ) ) y ) )
67	63, 66	syl 14	. . . . . . . . . 10 |- ( ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) /\ ( x e. C /\ y e. C ) ) -> ( x ( .r ` K ) y ) = ( x ( +g ` (mulGrp ` K ) ) y ) )
68	45	adantr 276	. . . . . . . . . . 11 |- ( ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) /\ ( x e. C /\ y e. C ) ) -> M e. Ring )
69		eqid 2196	. . . . . . . . . . . . 13 |- ( .r ` M ) = ( .r ` M )
70	7, 69	mgpplusgg 13556	. . . . . . . . . . . 12 |- ( M e. Ring -> ( .r ` M ) = ( +g ` (mulGrp ` M ) ) )
71	70	oveqd 5942	. . . . . . . . . . 11 |- ( M e. Ring -> ( x ( .r ` M ) y ) = ( x ( +g ` (mulGrp ` M ) ) y ) )
72	68, 71	syl 14	. . . . . . . . . 10 |- ( ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) /\ ( x e. C /\ y e. C ) ) -> ( x ( .r ` M ) y ) = ( x ( +g ` (mulGrp ` M ) ) y ) )
73	62, 67, 72	3eqtr3d 2237	. . . . . . . . 9 |- ( ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) /\ ( x e. C /\ y e. C ) ) -> ( x ( +g ` (mulGrp ` K ) ) y ) = ( x ( +g ` (mulGrp ` M ) ) y ) )
74	31, 37, 43, 49, 61, 73	mhmpropd 13168	. . . . . . . 8 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> ( (mulGrp ` J ) MndHom (mulGrp ` K ) ) = ( (mulGrp ` L ) MndHom (mulGrp ` M ) ) )
75	74	eleq2d 2266	. . . . . . 7 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> ( f e. ( (mulGrp ` J ) MndHom (mulGrp ` K ) ) <-> f e. ( (mulGrp ` L ) MndHom (mulGrp ` M ) ) ) )
76	25, 75	anbi12d 473	. . . . . 6 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> ( ( f e. ( J GrpHom K ) /\ f e. ( (mulGrp ` J ) MndHom (mulGrp ` K ) ) ) <-> ( f e. ( L GrpHom M ) /\ f e. ( (mulGrp ` L ) MndHom (mulGrp ` M ) ) ) ) )
77	22, 76	anbi12d 473	. . . . 5 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> ( ( ( J e. Ring /\ K e. Ring ) /\ ( f e. ( J GrpHom K ) /\ f e. ( (mulGrp ` J ) MndHom (mulGrp ` K ) ) ) ) <-> ( ( L e. Ring /\ M e. Ring ) /\ ( f e. ( L GrpHom M ) /\ f e. ( (mulGrp ` L ) MndHom (mulGrp ` M ) ) ) ) ) )
78	77, 3, 8	3bitr4g 223	. . . 4 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> ( f e. ( J RingHom K ) <-> f e. ( L RingHom M ) ) )
79	78	ex 115	. . 3 |- ( ph -> ( ( J e. Ring /\ K e. Ring ) -> ( f e. ( J RingHom K ) <-> f e. ( L RingHom M ) ) ) )
80	5, 21, 79	pm5.21ndd 706	. 2 |- ( ph -> ( f e. ( J RingHom K ) <-> f e. ( L RingHom M ) ) )
81	80	eqrdv 2194	1 |- ( ph -> ( J RingHom K ) = ( L RingHom M ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   ` cfv 5259  (class class class)co 5925   Basecbs 12703   +g cplusg 12780   .rcmulr 12781   MndHom cmhm 13159   GrpHom cghm 13446  mulGrpcmgp 13552   Ringcrg 13628   RingHom crh 13782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-map 6718  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-3 9067  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-plusg 12793  df-mulr 12794  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-mhm 13161  df-grp 13205  df-ghm 13447  df-mgp 13553  df-ur 13592  df-ring 13630  df-rhm 13784

This theorem is referenced by:  zrhpropd  14258