Theorem isrhm2d 13797

Description: Demonstration of ring homomorphism. (Contributed by Mario Carneiro, 13-Jun-2015.)

Hypotheses

Ref	Expression
isrhmd.b	|- B = ( Base ` R )
isrhmd.o	|- .1. = ( 1r ` R )
isrhmd.n	|- N = ( 1r ` S )
isrhmd.t	|- .x. = ( .r ` R )
isrhmd.u	|- .X. = ( .r ` S )
isrhmd.r	|- ( ph -> R e. Ring )
isrhmd.s	|- ( ph -> S e. Ring )
isrhmd.ho	|- ( ph -> ( F ` .1. ) = N )
isrhmd.ht	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( F ` ( x .x. y ) ) = ( ( F ` x ) .X. ( F ` y ) ) )
isrhm2d.f	|- ( ph -> F e. ( R GrpHom S ) )

Assertion

Ref	Expression
isrhm2d	|- ( ph -> F e. ( R RingHom S ) )

Distinct variable groups:   ph, x, y   x, B, y   x, F, y   x, R, y   x, S, y

Allowed substitution hints:   .x. ( x, y)   .X. ( x, y)   .1. ( x, y)   N( x, y)

Proof of Theorem isrhm2d

Step	Hyp	Ref	Expression
1		isrhmd.r	. 2 |- ( ph -> R e. Ring )
2		isrhmd.s	. 2 |- ( ph -> S e. Ring )
3		isrhm2d.f	. . 3 |- ( ph -> F e. ( R GrpHom S ) )
4		eqid 2196	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
5	4	ringmgp 13634	. . . . 5 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
6	1, 5	syl 14	. . . 4 |- ( ph -> (mulGrp ` R ) e. Mnd )
7		eqid 2196	. . . . . 6 |- (mulGrp ` S ) = (mulGrp ` S )
8	7	ringmgp 13634	. . . . 5 |- ( S e. Ring -> (mulGrp ` S ) e. Mnd )
9	2, 8	syl 14	. . . 4 |- ( ph -> (mulGrp ` S ) e. Mnd )
10		isrhmd.b	. . . . . . . 8 |- B = ( Base ` R )
11		eqid 2196	. . . . . . . 8 |- ( Base ` S ) = ( Base ` S )
12	10, 11	ghmf 13453	. . . . . . 7 |- ( F e. ( R GrpHom S ) -> F : B --> ( Base ` S ) )
13	3, 12	syl 14	. . . . . 6 |- ( ph -> F : B --> ( Base ` S ) )
14	4, 10	mgpbasg 13558	. . . . . . . 8 |- ( R e. Ring -> B = ( Base ` (mulGrp ` R ) ) )
15	1, 14	syl 14	. . . . . . 7 |- ( ph -> B = ( Base ` (mulGrp ` R ) ) )
16	7, 11	mgpbasg 13558	. . . . . . . 8 |- ( S e. Ring -> ( Base ` S ) = ( Base ` (mulGrp ` S ) ) )
17	2, 16	syl 14	. . . . . . 7 |- ( ph -> ( Base ` S ) = ( Base ` (mulGrp ` S ) ) )
18	15, 17	feq23d 5406	. . . . . 6 |- ( ph -> ( F : B --> ( Base ` S ) <-> F : ( Base ` (mulGrp ` R ) ) --> ( Base ` (mulGrp ` S ) ) ) )
19	13, 18	mpbid 147	. . . . 5 |- ( ph -> F : ( Base ` (mulGrp ` R ) ) --> ( Base ` (mulGrp ` S ) ) )
20		isrhmd.ht	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( F ` ( x .x. y ) ) = ( ( F ` x ) .X. ( F ` y ) ) )
21	20	ralrimivva 2579	. . . . . 6 |- ( ph -> A. x e. B A. y e. B ( F ` ( x .x. y ) ) = ( ( F ` x ) .X. ( F ` y ) ) )
22		isrhmd.t	. . . . . . . . . . . . 13 |- .x. = ( .r ` R )
23	4, 22	mgpplusgg 13556	. . . . . . . . . . . 12 |- ( R e. Ring -> .x. = ( +g ` (mulGrp ` R ) ) )
24	1, 23	syl 14	. . . . . . . . . . 11 |- ( ph -> .x. = ( +g ` (mulGrp ` R ) ) )
25	24	oveqd 5942	. . . . . . . . . 10 |- ( ph -> ( x .x. y ) = ( x ( +g ` (mulGrp ` R ) ) y ) )
26	25	fveq2d 5565	. . . . . . . . 9 |- ( ph -> ( F ` ( x .x. y ) ) = ( F ` ( x ( +g ` (mulGrp ` R ) ) y ) ) )
27		isrhmd.u	. . . . . . . . . . . 12 |- .X. = ( .r ` S )
28	7, 27	mgpplusgg 13556	. . . . . . . . . . 11 |- ( S e. Ring -> .X. = ( +g ` (mulGrp ` S ) ) )
29	2, 28	syl 14	. . . . . . . . . 10 |- ( ph -> .X. = ( +g ` (mulGrp ` S ) ) )
30	29	oveqd 5942	. . . . . . . . 9 |- ( ph -> ( ( F ` x ) .X. ( F ` y ) ) = ( ( F ` x ) ( +g ` (mulGrp ` S ) ) ( F ` y ) ) )
31	26, 30	eqeq12d 2211	. . . . . . . 8 |- ( ph -> ( ( F ` ( x .x. y ) ) = ( ( F ` x ) .X. ( F ` y ) ) <-> ( F ` ( x ( +g ` (mulGrp ` R ) ) y ) ) = ( ( F ` x ) ( +g ` (mulGrp ` S ) ) ( F ` y ) ) ) )
32	15, 31	raleqbidv 2709	. . . . . . 7 |- ( ph -> ( A. y e. B ( F ` ( x .x. y ) ) = ( ( F ` x ) .X. ( F ` y ) ) <-> A. y e. ( Base ` (mulGrp ` R ) ) ( F ` ( x ( +g ` (mulGrp ` R ) ) y ) ) = ( ( F ` x ) ( +g ` (mulGrp ` S ) ) ( F ` y ) ) ) )
33	15, 32	raleqbidv 2709	. . . . . 6 |- ( ph -> ( A. x e. B A. y e. B ( F ` ( x .x. y ) ) = ( ( F ` x ) .X. ( F ` y ) ) <-> A. x e. ( Base ` (mulGrp ` R ) ) A. y e. ( Base ` (mulGrp ` R ) ) ( F ` ( x ( +g ` (mulGrp ` R ) ) y ) ) = ( ( F ` x ) ( +g ` (mulGrp ` S ) ) ( F ` y ) ) ) )
34	21, 33	mpbid 147	. . . . 5 |- ( ph -> A. x e. ( Base ` (mulGrp ` R ) ) A. y e. ( Base ` (mulGrp ` R ) ) ( F ` ( x ( +g ` (mulGrp ` R ) ) y ) ) = ( ( F ` x ) ( +g ` (mulGrp ` S ) ) ( F ` y ) ) )
35		isrhmd.ho	. . . . . 6 |- ( ph -> ( F ` .1. ) = N )
36		isrhmd.o	. . . . . . . . 9 |- .1. = ( 1r ` R )
37	4, 36	ringidvalg 13593	. . . . . . . 8 |- ( R e. Ring -> .1. = ( 0g ` (mulGrp ` R ) ) )
38	1, 37	syl 14	. . . . . . 7 |- ( ph -> .1. = ( 0g ` (mulGrp ` R ) ) )
39	38	fveq2d 5565	. . . . . 6 |- ( ph -> ( F ` .1. ) = ( F ` ( 0g ` (mulGrp ` R ) ) ) )
40		isrhmd.n	. . . . . . . 8 |- N = ( 1r ` S )
41	7, 40	ringidvalg 13593	. . . . . . 7 |- ( S e. Ring -> N = ( 0g ` (mulGrp ` S ) ) )
42	2, 41	syl 14	. . . . . 6 |- ( ph -> N = ( 0g ` (mulGrp ` S ) ) )
43	35, 39, 42	3eqtr3d 2237	. . . . 5 |- ( ph -> ( F ` ( 0g ` (mulGrp ` R ) ) ) = ( 0g ` (mulGrp ` S ) ) )
44	19, 34, 43	3jca 1179	. . . 4 |- ( ph -> ( F : ( Base ` (mulGrp ` R ) ) --> ( Base ` (mulGrp ` S ) ) /\ A. x e. ( Base ` (mulGrp ` R ) ) A. y e. ( Base ` (mulGrp ` R ) ) ( F ` ( x ( +g ` (mulGrp ` R ) ) y ) ) = ( ( F ` x ) ( +g ` (mulGrp ` S ) ) ( F ` y ) ) /\ ( F ` ( 0g ` (mulGrp ` R ) ) ) = ( 0g ` (mulGrp ` S ) ) ) )
45		eqid 2196	. . . . 5 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
46		eqid 2196	. . . . 5 |- ( Base ` (mulGrp ` S ) ) = ( Base ` (mulGrp ` S ) )
47		eqid 2196	. . . . 5 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
48		eqid 2196	. . . . 5 |- ( +g ` (mulGrp ` S ) ) = ( +g ` (mulGrp ` S ) )
49		eqid 2196	. . . . 5 |- ( 0g ` (mulGrp ` R ) ) = ( 0g ` (mulGrp ` R ) )
50		eqid 2196	. . . . 5 |- ( 0g ` (mulGrp ` S ) ) = ( 0g ` (mulGrp ` S ) )
51	45, 46, 47, 48, 49, 50	ismhm 13163	. . . 4 |- ( F e. ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) <-> ( ( (mulGrp ` R ) e. Mnd /\ (mulGrp ` S ) e. Mnd ) /\ ( F : ( Base ` (mulGrp ` R ) ) --> ( Base ` (mulGrp ` S ) ) /\ A. x e. ( Base ` (mulGrp ` R ) ) A. y e. ( Base ` (mulGrp ` R ) ) ( F ` ( x ( +g ` (mulGrp ` R ) ) y ) ) = ( ( F ` x ) ( +g ` (mulGrp ` S ) ) ( F ` y ) ) /\ ( F ` ( 0g ` (mulGrp ` R ) ) ) = ( 0g ` (mulGrp ` S ) ) ) ) )
52	6, 9, 44, 51	syl21anbrc 1184	. . 3 |- ( ph -> F e. ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) )
53	3, 52	jca 306	. 2 |- ( ph -> ( F e. ( R GrpHom S ) /\ F e. ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) ) )
54	4, 7	isrhm 13790	. 2 |- ( F e. ( R RingHom S ) <-> ( ( R e. Ring /\ S e. Ring ) /\ ( F e. ( R GrpHom S ) /\ F e. ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) ) ) )
55	1, 2, 53, 54	syl21anbrc 1184	1 |- ( ph -> F e. ( R RingHom S ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167   A.wral 2475   -->wf 5255   ` cfv 5259  (class class class)co 5925   Basecbs 12703   +g cplusg 12780   .rcmulr 12781   0gc0g 12958   Mndcmnd 13118   MndHom cmhm 13159   GrpHom cghm 13446  mulGrpcmgp 13552   1rcur 13591   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:  isrhmd  13798  rhmopp  13808  qusrhm  14160  mulgrhm  14241