Theorem rhmpropd 14422

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 2234	. . . . . 6 |- (mulGrp ` J ) = (mulGrp ` J )
2		eqid 2234	. . . . . 6 |- (mulGrp ` K ) = (mulGrp ` K )
3	1, 2	isrhm 14325	. . . . 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 2234	. . . . . 6 |- (mulGrp ` L ) = (mulGrp ` L )
7		eqid 2234	. . . . . 6 |- (mulGrp ` M ) = (mulGrp ` M )
8	6, 7	isrhm 14325	. . . . 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 14203	. . . . 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 14203	. . . . 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 14021	. . . . . . . . 9 |- ( ph -> ( J GrpHom K ) = ( L GrpHom M ) )
24	23	eleq2d 2304	. . . . . . . 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 531	. . . . . . . . . . 11 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> J e. Ring )
28		eqid 2234	. . . . . . . . . . . 12 |- ( Base ` J ) = ( Base ` J )
29	1, 28	mgpbasg 14091	. . . . . . . . . . 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 2267	. . . . . . . . 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 533	. . . . . . . . . . 11 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> K e. Ring )
34		eqid 2234	. . . . . . . . . . . 12 |- ( Base ` K ) = ( Base ` K )
35	2, 34	mgpbasg 14091	. . . . . . . . . . 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 2267	. . . . . . . . 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 2234	. . . . . . . . . . . 12 |- ( Base ` L ) = ( Base ` L )
41	6, 40	mgpbasg 14091	. . . . . . . . . . 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 2267	. . . . . . . . 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 2234	. . . . . . . . . . . 12 |- ( Base ` M ) = ( Base ` M )
47	7, 46	mgpbasg 14091	. . . . . . . . . . 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 2267	. . . . . . . . 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 2234	. . . . . . . . . . . . 13 |- ( .r ` J ) = ( .r ` J )
53	1, 52	mgpplusgg 14089	. . . . . . . . . . . 12 |- ( J e. Ring -> ( .r ` J ) = ( +g ` (mulGrp ` J ) ) )
54	53	oveqd 6069	. . . . . . . . . . 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 2234	. . . . . . . . . . . . 13 |- ( .r ` L ) = ( .r ` L )
58	6, 57	mgpplusgg 14089	. . . . . . . . . . . 12 |- ( L e. Ring -> ( .r ` L ) = ( +g ` (mulGrp ` L ) ) )
59	58	oveqd 6069	. . . . . . . . . . 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 2275	. . . . . . . . 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 2234	. . . . . . . . . . . . 13 |- ( .r ` K ) = ( .r ` K )
65	2, 64	mgpplusgg 14089	. . . . . . . . . . . 12 |- ( K e. Ring -> ( .r ` K ) = ( +g ` (mulGrp ` K ) ) )
66	65	oveqd 6069	. . . . . . . . . . 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 2234	. . . . . . . . . . . . 13 |- ( .r ` M ) = ( .r ` M )
70	7, 69	mgpplusgg 14089	. . . . . . . . . . . 12 |- ( M e. Ring -> ( .r ` M ) = ( +g ` (mulGrp ` M ) ) )
71	70	oveqd 6069	. . . . . . . . . . 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 2275	. . . . . . . . 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 13700	. . . . . . . 8 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> ( (mulGrp ` J ) MndHom (mulGrp ` K ) ) = ( (mulGrp ` L ) MndHom (mulGrp ` M ) ) )
75	74	eleq2d 2304	. . . . . . 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 713	. 2 |- ( ph -> ( f e. ( J RingHom K ) <-> f e. ( L RingHom M ) ) )
81	80	eqrdv 2232	1 |- ( ph -> ( J RingHom K ) = ( L RingHom M ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   ` cfv 5354  (class class class)co 6052   Basecbs 13233   +g cplusg 13311   .rcmulr 13312   MndHom cmhm 13691   GrpHom cghm 13978  mulGrpcmgp 14085   Ringcrg 14161   RingHom crh 14317

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-addcom 8232  ax-addass 8234  ax-i2m1 8237  ax-0lt1 8238  ax-0id 8240  ax-rnegex 8241  ax-pre-ltirr 8244  ax-pre-ltadd 8248

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-map 6886  df-pnf 8315  df-mnf 8316  df-ltxr 8318  df-inn 9243  df-2 9301  df-3 9302  df-ndx 13236  df-slot 13237  df-base 13239  df-sets 13240  df-plusg 13324  df-mulr 13325  df-0g 13492  df-mgm 13590  df-sgrp 13636  df-mnd 13651  df-mhm 13693  df-grp 13737  df-ghm 13979  df-mgp 14086  df-ur 14125  df-ring 14163  df-rhm 14319

This theorem is referenced by:  zrhpropd  14823