Theorem isrhm2d 14310

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 2232	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
5	4	ringmgp 14146	. . . . 5 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
6	1, 5	syl 14	. . . 4 |- ( ph -> (mulGrp ` R ) e. Mnd )
7		eqid 2232	. . . . . 6 |- (mulGrp ` S ) = (mulGrp ` S )
8	7	ringmgp 14146	. . . . 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 2232	. . . . . . . 8 |- ( Base ` S ) = ( Base ` S )
12	10, 11	ghmf 13964	. . . . . . 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 14070	. . . . . . . 8 |- ( R e. Ring -> B = ( Base ` (mulGrp ` R ) ) )
15	1, 14	syl 14	. . . . . . 7 |- ( ph -> B = ( Base ` (mulGrp ` R ) ) )
16	7, 11	mgpbasg 14070	. . . . . . . 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 5504	. . . . . 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 2624	. . . . . 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 14068	. . . . . . . . . . . 12 |- ( R e. Ring -> .x. = ( +g ` (mulGrp ` R ) ) )
24	1, 23	syl 14	. . . . . . . . . . 11 |- ( ph -> .x. = ( +g ` (mulGrp ` R ) ) )
25	24	oveqd 6067	. . . . . . . . . 10 |- ( ph -> ( x .x. y ) = ( x ( +g ` (mulGrp ` R ) ) y ) )
26	25	fveq2d 5674	. . . . . . . . 9 |- ( ph -> ( F ` ( x .x. y ) ) = ( F ` ( x ( +g ` (mulGrp ` R ) ) y ) ) )
27		isrhmd.u	. . . . . . . . . . . 12 |- .X. = ( .r ` S )
28	7, 27	mgpplusgg 14068	. . . . . . . . . . 11 |- ( S e. Ring -> .X. = ( +g ` (mulGrp ` S ) ) )
29	2, 28	syl 14	. . . . . . . . . 10 |- ( ph -> .X. = ( +g ` (mulGrp ` S ) ) )
30	29	oveqd 6067	. . . . . . . . 9 |- ( ph -> ( ( F ` x ) .X. ( F ` y ) ) = ( ( F ` x ) ( +g ` (mulGrp ` S ) ) ( F ` y ) ) )
31	26, 30	eqeq12d 2247	. . . . . . . 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 2757	. . . . . . 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 2757	. . . . . 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 14105	. . . . . . . 8 |- ( R e. Ring -> .1. = ( 0g ` (mulGrp ` R ) ) )
38	1, 37	syl 14	. . . . . . 7 |- ( ph -> .1. = ( 0g ` (mulGrp ` R ) ) )
39	38	fveq2d 5674	. . . . . 6 |- ( ph -> ( F ` .1. ) = ( F ` ( 0g ` (mulGrp ` R ) ) ) )
40		isrhmd.n	. . . . . . . 8 |- N = ( 1r ` S )
41	7, 40	ringidvalg 14105	. . . . . . 7 |- ( S e. Ring -> N = ( 0g ` (mulGrp ` S ) ) )
42	2, 41	syl 14	. . . . . 6 |- ( ph -> N = ( 0g ` (mulGrp ` S ) ) )
43	35, 39, 42	3eqtr3d 2273	. . . . 5 |- ( ph -> ( F ` ( 0g ` (mulGrp ` R ) ) ) = ( 0g ` (mulGrp ` S ) ) )
44	19, 34, 43	3jca 1204	. . . 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 2232	. . . . 5 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
46		eqid 2232	. . . . 5 |- ( Base ` (mulGrp ` S ) ) = ( Base ` (mulGrp ` S ) )
47		eqid 2232	. . . . 5 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
48		eqid 2232	. . . . 5 |- ( +g ` (mulGrp ` S ) ) = ( +g ` (mulGrp ` S ) )
49		eqid 2232	. . . . 5 |- ( 0g ` (mulGrp ` R ) ) = ( 0g ` (mulGrp ` R ) )
50		eqid 2232	. . . . 5 |- ( 0g ` (mulGrp ` S ) ) = ( 0g ` (mulGrp ` S ) )
51	45, 46, 47, 48, 49, 50	ismhm 13674	. . . 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 1209	. . 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 14303	. 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 1209	1 |- ( ph -> F e. ( R RingHom S ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   A.wral 2520   -->wf 5348   ` cfv 5352  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   .rcmulr 13291   0gc0g 13469   Mndcmnd 13629   MndHom cmhm 13670   GrpHom cghm 13957  mulGrpcmgp 14064   1rcur 14103   Ringcrg 14140   RingHom crh 14295

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-ltadd 8243

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-map 6884  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-plusg 13303  df-mulr 13304  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-mhm 13672  df-grp 13716  df-ghm 13958  df-mgp 14065  df-ur 14104  df-ring 14142  df-rhm 14297

This theorem is referenced by:  isrhmd  14311  rhmopp  14321  qusrhm  14676  mulgrhm  14757