Theorem rhmpropd 18815

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		rhmpropd.a	. . . . . 6 |- ( ph -> B = ( Base ` J ) )
2		rhmpropd.c	. . . . . 6 |- ( ph -> B = ( Base ` L ) )
3		rhmpropd.e	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` J ) y ) = ( x ( +g ` L ) y ) )
4		rhmpropd.g	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` J ) y ) = ( x ( .r ` L ) y ) )
5	1, 2, 3, 4	ringpropd 18582	. . . . 5 |- ( ph -> ( J e. Ring <-> L e. Ring ) )
6		rhmpropd.b	. . . . . 6 |- ( ph -> C = ( Base ` K ) )
7		rhmpropd.d	. . . . . 6 |- ( ph -> C = ( Base ` M ) )
8		rhmpropd.f	. . . . . 6 |- ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` M ) y ) )
9		rhmpropd.h	. . . . . 6 |- ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` M ) y ) )
10	6, 7, 8, 9	ringpropd 18582	. . . . 5 |- ( ph -> ( K e. Ring <-> M e. Ring ) )
11	5, 10	anbi12d 747	. . . 4 |- ( ph -> ( ( J e. Ring /\ K e. Ring ) <-> ( L e. Ring /\ M e. Ring ) ) )
12	1, 6, 2, 7, 3, 8	ghmpropd 17698	. . . . . 6 |- ( ph -> ( J GrpHom K ) = ( L GrpHom M ) )
13	12	eleq2d 2687	. . . . 5 |- ( ph -> ( f e. ( J GrpHom K ) <-> f e. ( L GrpHom M ) ) )
14		eqid 2622	. . . . . . . . 9 |- (mulGrp ` J ) = (mulGrp ` J )
15		eqid 2622	. . . . . . . . 9 |- ( Base ` J ) = ( Base ` J )
16	14, 15	mgpbas 18495	. . . . . . . 8 |- ( Base ` J ) = ( Base ` (mulGrp ` J ) )
17	1, 16	syl6eq 2672	. . . . . . 7 |- ( ph -> B = ( Base ` (mulGrp ` J ) ) )
18		eqid 2622	. . . . . . . . 9 |- (mulGrp ` K ) = (mulGrp ` K )
19		eqid 2622	. . . . . . . . 9 |- ( Base ` K ) = ( Base ` K )
20	18, 19	mgpbas 18495	. . . . . . . 8 |- ( Base ` K ) = ( Base ` (mulGrp ` K ) )
21	6, 20	syl6eq 2672	. . . . . . 7 |- ( ph -> C = ( Base ` (mulGrp ` K ) ) )
22		eqid 2622	. . . . . . . . 9 |- (mulGrp ` L ) = (mulGrp ` L )
23		eqid 2622	. . . . . . . . 9 |- ( Base ` L ) = ( Base ` L )
24	22, 23	mgpbas 18495	. . . . . . . 8 |- ( Base ` L ) = ( Base ` (mulGrp ` L ) )
25	2, 24	syl6eq 2672	. . . . . . 7 |- ( ph -> B = ( Base ` (mulGrp ` L ) ) )
26		eqid 2622	. . . . . . . . 9 |- (mulGrp ` M ) = (mulGrp ` M )
27		eqid 2622	. . . . . . . . 9 |- ( Base ` M ) = ( Base ` M )
28	26, 27	mgpbas 18495	. . . . . . . 8 |- ( Base ` M ) = ( Base ` (mulGrp ` M ) )
29	7, 28	syl6eq 2672	. . . . . . 7 |- ( ph -> C = ( Base ` (mulGrp ` M ) ) )
30		eqid 2622	. . . . . . . . . 10 |- ( .r ` J ) = ( .r ` J )
31	14, 30	mgpplusg 18493	. . . . . . . . 9 |- ( .r ` J ) = ( +g ` (mulGrp ` J ) )
32	31	oveqi 6663	. . . . . . . 8 |- ( x ( .r ` J ) y ) = ( x ( +g ` (mulGrp ` J ) ) y )
33		eqid 2622	. . . . . . . . . 10 |- ( .r ` L ) = ( .r ` L )
34	22, 33	mgpplusg 18493	. . . . . . . . 9 |- ( .r ` L ) = ( +g ` (mulGrp ` L ) )
35	34	oveqi 6663	. . . . . . . 8 |- ( x ( .r ` L ) y ) = ( x ( +g ` (mulGrp ` L ) ) y )
36	4, 32, 35	3eqtr3g 2679	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` (mulGrp ` J ) ) y ) = ( x ( +g ` (mulGrp ` L ) ) y ) )
37		eqid 2622	. . . . . . . . . 10 |- ( .r ` K ) = ( .r ` K )
38	18, 37	mgpplusg 18493	. . . . . . . . 9 |- ( .r ` K ) = ( +g ` (mulGrp ` K ) )
39	38	oveqi 6663	. . . . . . . 8 |- ( x ( .r ` K ) y ) = ( x ( +g ` (mulGrp ` K ) ) y )
40		eqid 2622	. . . . . . . . . 10 |- ( .r ` M ) = ( .r ` M )
41	26, 40	mgpplusg 18493	. . . . . . . . 9 |- ( .r ` M ) = ( +g ` (mulGrp ` M ) )
42	41	oveqi 6663	. . . . . . . 8 |- ( x ( .r ` M ) y ) = ( x ( +g ` (mulGrp ` M ) ) y )
43	9, 39, 42	3eqtr3g 2679	. . . . . . 7 |- ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x ( +g ` (mulGrp ` K ) ) y ) = ( x ( +g ` (mulGrp ` M ) ) y ) )
44	17, 21, 25, 29, 36, 43	mhmpropd 17341	. . . . . 6 |- ( ph -> ( (mulGrp ` J ) MndHom (mulGrp ` K ) ) = ( (mulGrp ` L ) MndHom (mulGrp ` M ) ) )
45	44	eleq2d 2687	. . . . 5 |- ( ph -> ( f e. ( (mulGrp ` J ) MndHom (mulGrp ` K ) ) <-> f e. ( (mulGrp ` L ) MndHom (mulGrp ` M ) ) ) )
46	13, 45	anbi12d 747	. . . 4 |- ( ph -> ( ( f e. ( J GrpHom K ) /\ f e. ( (mulGrp ` J ) MndHom (mulGrp ` K ) ) ) <-> ( f e. ( L GrpHom M ) /\ f e. ( (mulGrp ` L ) MndHom (mulGrp ` M ) ) ) ) )
47	11, 46	anbi12d 747	. . 3 |- ( ph -> ( ( ( 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 ) ) ) ) ) )
48	14, 18	isrhm 18721	. . 3 |- ( f e. ( J RingHom K ) <-> ( ( J e. Ring /\ K e. Ring ) /\ ( f e. ( J GrpHom K ) /\ f e. ( (mulGrp ` J ) MndHom (mulGrp ` K ) ) ) ) )
49	22, 26	isrhm 18721	. . 3 |- ( f e. ( L RingHom M ) <-> ( ( L e. Ring /\ M e. Ring ) /\ ( f e. ( L GrpHom M ) /\ f e. ( (mulGrp ` L ) MndHom (mulGrp ` M ) ) ) ) )
50	47, 48, 49	3bitr4g 303	. 2 |- ( ph -> ( f e. ( J RingHom K ) <-> f e. ( L RingHom M ) ) )
51	50	eqrdv 2620	1 |- ( ph -> ( J RingHom K ) = ( L RingHom M ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   .rcmulr 15942   MndHom cmhm 17333   GrpHom cghm 17657  mulGrpcmgp 18489   Ringcrg 18547   RingHom crh 18712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-ghm 17658  df-mgp 18490  df-ur 18502  df-ring 18549  df-rnghom 18715

This theorem is referenced by:  evls1rhm  19687  evl1rhm  19696  zrhpropd  19863