Theorem isrhm2d 13664

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 2193	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
5	4	ringmgp 13501	. . . . 5 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
6	1, 5	syl 14	. . . 4 |- ( ph -> (mulGrp ` R ) e. Mnd )
7		eqid 2193	. . . . . 6 |- (mulGrp ` S ) = (mulGrp ` S )
8	7	ringmgp 13501	. . . . 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 2193	. . . . . . . 8 |- ( Base ` S ) = ( Base ` S )
12	10, 11	ghmf 13320	. . . . . . 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 13425	. . . . . . . 8 |- ( R e. Ring -> B = ( Base ` (mulGrp ` R ) ) )
15	1, 14	syl 14	. . . . . . 7 |- ( ph -> B = ( Base ` (mulGrp ` R ) ) )
16	7, 11	mgpbasg 13425	. . . . . . . 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 5400	. . . . . 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 2576	. . . . . 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 13423	. . . . . . . . . . . 12 |- ( R e. Ring -> .x. = ( +g ` (mulGrp ` R ) ) )
24	1, 23	syl 14	. . . . . . . . . . 11 |- ( ph -> .x. = ( +g ` (mulGrp ` R ) ) )
25	24	oveqd 5936	. . . . . . . . . 10 |- ( ph -> ( x .x. y ) = ( x ( +g ` (mulGrp ` R ) ) y ) )
26	25	fveq2d 5559	. . . . . . . . 9 |- ( ph -> ( F ` ( x .x. y ) ) = ( F ` ( x ( +g ` (mulGrp ` R ) ) y ) ) )
27		isrhmd.u	. . . . . . . . . . . 12 |- .X. = ( .r ` S )
28	7, 27	mgpplusgg 13423	. . . . . . . . . . 11 |- ( S e. Ring -> .X. = ( +g ` (mulGrp ` S ) ) )
29	2, 28	syl 14	. . . . . . . . . 10 |- ( ph -> .X. = ( +g ` (mulGrp ` S ) ) )
30	29	oveqd 5936	. . . . . . . . 9 |- ( ph -> ( ( F ` x ) .X. ( F ` y ) ) = ( ( F ` x ) ( +g ` (mulGrp ` S ) ) ( F ` y ) ) )
31	26, 30	eqeq12d 2208	. . . . . . . 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 2706	. . . . . . 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 2706	. . . . . 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 13460	. . . . . . . 8 |- ( R e. Ring -> .1. = ( 0g ` (mulGrp ` R ) ) )
38	1, 37	syl 14	. . . . . . 7 |- ( ph -> .1. = ( 0g ` (mulGrp ` R ) ) )
39	38	fveq2d 5559	. . . . . 6 |- ( ph -> ( F ` .1. ) = ( F ` ( 0g ` (mulGrp ` R ) ) ) )
40		isrhmd.n	. . . . . . . 8 |- N = ( 1r ` S )
41	7, 40	ringidvalg 13460	. . . . . . 7 |- ( S e. Ring -> N = ( 0g ` (mulGrp ` S ) ) )
42	2, 41	syl 14	. . . . . 6 |- ( ph -> N = ( 0g ` (mulGrp ` S ) ) )
43	35, 39, 42	3eqtr3d 2234	. . . . 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 2193	. . . . 5 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
46		eqid 2193	. . . . 5 |- ( Base ` (mulGrp ` S ) ) = ( Base ` (mulGrp ` S ) )
47		eqid 2193	. . . . 5 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
48		eqid 2193	. . . . 5 |- ( +g ` (mulGrp ` S ) ) = ( +g ` (mulGrp ` S ) )
49		eqid 2193	. . . . 5 |- ( 0g ` (mulGrp ` R ) ) = ( 0g ` (mulGrp ` R ) )
50		eqid 2193	. . . . 5 |- ( 0g ` (mulGrp ` S ) ) = ( 0g ` (mulGrp ` S ) )
51	45, 46, 47, 48, 49, 50	ismhm 13036	. . . 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 13657	. 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 2164   A.wral 2472   -->wf 5251   ` cfv 5255  (class class class)co 5919   Basecbs 12621   +g cplusg 12698   .rcmulr 12699   0gc0g 12870   Mndcmnd 13000   MndHom cmhm 13032   GrpHom cghm 13313  mulGrpcmgp 13419   1rcur 13458   Ringcrg 13495   RingHom crh 13649

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-map 6706  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-plusg 12711  df-mulr 12712  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-mhm 13034  df-grp 13078  df-ghm 13314  df-mgp 13420  df-ur 13459  df-ring 13497  df-rhm 13651

This theorem is referenced by:  isrhmd  13665  rhmopp  13675  qusrhm  14027  mulgrhm  14108