Theorem rhmpropd 14545

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 2238	. . . . . 6 |- (mulGrp ` J ) = (mulGrp ` J )
2		eqid 2238	. . . . . 6 |- (mulGrp ` K ) = (mulGrp ` K )
3	1, 2	isrhm 14448	. . . . 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 2238	. . . . . 6 |- (mulGrp ` L ) = (mulGrp ` L )
7		eqid 2238	. . . . . 6 |- (mulGrp ` M ) = (mulGrp ` M )
8	6, 7	isrhm 14448	. . . . 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 14326	. . . . 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 14326	. . . . 5 |- ( ph -> ( K e. Ring <-> M e. Ring ) )
20	14, 19	anbi12d 477	. . . 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 14069	. . . . . . . . 9 |- ( ph -> ( J GrpHom K ) = ( L GrpHom M ) )
24	23	eleq2d 2308	. . . . . . . 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 535	. . . . . . . . . . 11 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> J e. Ring )
28		eqid 2238	. . . . . . . . . . . 12 |- ( Base ` J ) = ( Base ` J )
29	1, 28	mgpbasg 14207	. . . . . . . . . . 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 2271	. . . . . . . . 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 537	. . . . . . . . . . 11 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> K e. Ring )
34		eqid 2238	. . . . . . . . . . . 12 |- ( Base ` K ) = ( Base ` K )
35	2, 34	mgpbasg 14207	. . . . . . . . . . 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 2271	. . . . . . . . 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 2238	. . . . . . . . . . . 12 |- ( Base ` L ) = ( Base ` L )
41	6, 40	mgpbasg 14207	. . . . . . . . . . 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 2271	. . . . . . . . 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 2238	. . . . . . . . . . . 12 |- ( Base ` M ) = ( Base ` M )
47	7, 46	mgpbasg 14207	. . . . . . . . . . 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 2271	. . . . . . . . 9 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> C = ( Base ` (mulGrp ` M ) ) )
50	13	adantlr 481	. . . . . . . . . 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 2238	. . . . . . . . . . . . 13 |- ( .r ` J ) = ( .r ` J )
53	1, 52	mgpplusgg 14204	. . . . . . . . . . . 12 |- ( J e. Ring -> ( .r ` J ) = ( +g ` (mulGrp ` J ) ) )
54	53	oveqd 6096	. . . . . . . . . . 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 2238	. . . . . . . . . . . . 13 |- ( .r ` L ) = ( .r ` L )
58	6, 57	mgpplusgg 14204	. . . . . . . . . . . 12 |- ( L e. Ring -> ( .r ` L ) = ( +g ` (mulGrp ` L ) ) )
59	58	oveqd 6096	. . . . . . . . . . 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 2279	. . . . . . . . 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 481	. . . . . . . . . 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 2238	. . . . . . . . . . . . 13 |- ( .r ` K ) = ( .r ` K )
65	2, 64	mgpplusgg 14204	. . . . . . . . . . . 12 |- ( K e. Ring -> ( .r ` K ) = ( +g ` (mulGrp ` K ) ) )
66	65	oveqd 6096	. . . . . . . . . . 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 2238	. . . . . . . . . . . . 13 |- ( .r ` M ) = ( .r ` M )
70	7, 69	mgpplusgg 14204	. . . . . . . . . . . 12 |- ( M e. Ring -> ( .r ` M ) = ( +g ` (mulGrp ` M ) ) )
71	70	oveqd 6096	. . . . . . . . . . 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 2279	. . . . . . . . 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 13756	. . . . . . . 8 |- ( ( ph /\ ( J e. Ring /\ K e. Ring ) ) -> ( (mulGrp ` J ) MndHom (mulGrp ` K ) ) = ( (mulGrp ` L ) MndHom (mulGrp ` M ) ) )
75	74	eleq2d 2308	. . . . . . 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 477	. . . . . 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 477	. . . . 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 717	. 2 |- ( ph -> ( f e. ( J RingHom K ) <-> f e. ( L RingHom M ) ) )
81	80	eqrdv 2236	1 |- ( ph -> ( J RingHom K ) = ( L RingHom M ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   ` cfv 5375  (class class class)co 6079   Basecbs 13335   +g cplusg 13414   .rcmulr 13415   MndHom cmhm 13747   GrpHom cghm 14026  mulGrpcmgp 14200   Ringcrg 14283   RingHom crh 14440

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-pre-ltirr 8285  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-map 6918  df-pnf 8356  df-mnf 8357  df-ltxr 8359  df-inn 9288  df-2 9346  df-3 9347  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-plusg 13427  df-mulr 13428  df-0g 13595  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-mhm 13749  df-grp 13791  df-ghm 14027  df-mgp 14201  df-ur 14246  df-ring 14285  df-rhm 14442

This theorem is referenced by:  zrhpropd  14944