Theorem isrhm2d 14178

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 2231	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
5	4	ringmgp 14014	. . . . 5 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
6	1, 5	syl 14	. . . 4 |- ( ph -> (mulGrp ` R ) e. Mnd )
7		eqid 2231	. . . . . 6 |- (mulGrp ` S ) = (mulGrp ` S )
8	7	ringmgp 14014	. . . . 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 2231	. . . . . . . 8 |- ( Base ` S ) = ( Base ` S )
12	10, 11	ghmf 13833	. . . . . . 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 13938	. . . . . . . 8 |- ( R e. Ring -> B = ( Base ` (mulGrp ` R ) ) )
15	1, 14	syl 14	. . . . . . 7 |- ( ph -> B = ( Base ` (mulGrp ` R ) ) )
16	7, 11	mgpbasg 13938	. . . . . . . 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 5478	. . . . . 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 2614	. . . . . 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 13936	. . . . . . . . . . . 12 |- ( R e. Ring -> .x. = ( +g ` (mulGrp ` R ) ) )
24	1, 23	syl 14	. . . . . . . . . . 11 |- ( ph -> .x. = ( +g ` (mulGrp ` R ) ) )
25	24	oveqd 6034	. . . . . . . . . 10 |- ( ph -> ( x .x. y ) = ( x ( +g ` (mulGrp ` R ) ) y ) )
26	25	fveq2d 5643	. . . . . . . . 9 |- ( ph -> ( F ` ( x .x. y ) ) = ( F ` ( x ( +g ` (mulGrp ` R ) ) y ) ) )
27		isrhmd.u	. . . . . . . . . . . 12 |- .X. = ( .r ` S )
28	7, 27	mgpplusgg 13936	. . . . . . . . . . 11 |- ( S e. Ring -> .X. = ( +g ` (mulGrp ` S ) ) )
29	2, 28	syl 14	. . . . . . . . . 10 |- ( ph -> .X. = ( +g ` (mulGrp ` S ) ) )
30	29	oveqd 6034	. . . . . . . . 9 |- ( ph -> ( ( F ` x ) .X. ( F ` y ) ) = ( ( F ` x ) ( +g ` (mulGrp ` S ) ) ( F ` y ) ) )
31	26, 30	eqeq12d 2246	. . . . . . . 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 2746	. . . . . . 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 2746	. . . . . 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 13973	. . . . . . . 8 |- ( R e. Ring -> .1. = ( 0g ` (mulGrp ` R ) ) )
38	1, 37	syl 14	. . . . . . 7 |- ( ph -> .1. = ( 0g ` (mulGrp ` R ) ) )
39	38	fveq2d 5643	. . . . . 6 |- ( ph -> ( F ` .1. ) = ( F ` ( 0g ` (mulGrp ` R ) ) ) )
40		isrhmd.n	. . . . . . . 8 |- N = ( 1r ` S )
41	7, 40	ringidvalg 13973	. . . . . . 7 |- ( S e. Ring -> N = ( 0g ` (mulGrp ` S ) ) )
42	2, 41	syl 14	. . . . . 6 |- ( ph -> N = ( 0g ` (mulGrp ` S ) ) )
43	35, 39, 42	3eqtr3d 2272	. . . . 5 |- ( ph -> ( F ` ( 0g ` (mulGrp ` R ) ) ) = ( 0g ` (mulGrp ` S ) ) )
44	19, 34, 43	3jca 1203	. . . 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 2231	. . . . 5 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
46		eqid 2231	. . . . 5 |- ( Base ` (mulGrp ` S ) ) = ( Base ` (mulGrp ` S ) )
47		eqid 2231	. . . . 5 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
48		eqid 2231	. . . . 5 |- ( +g ` (mulGrp ` S ) ) = ( +g ` (mulGrp ` S ) )
49		eqid 2231	. . . . 5 |- ( 0g ` (mulGrp ` R ) ) = ( 0g ` (mulGrp ` R ) )
50		eqid 2231	. . . . 5 |- ( 0g ` (mulGrp ` S ) ) = ( 0g ` (mulGrp ` S ) )
51	45, 46, 47, 48, 49, 50	ismhm 13543	. . . 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 1208	. . 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 14171	. 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 1208	1 |- ( ph -> F e. ( R RingHom S ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   A.wral 2510   -->wf 5322   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   .rcmulr 13160   0gc0g 13338   Mndcmnd 13498   MndHom cmhm 13539   GrpHom cghm 13826  mulGrpcmgp 13932   1rcur 13971   Ringcrg 14008   RingHom crh 14163

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-map 6818  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-mhm 13541  df-grp 13585  df-ghm 13827  df-mgp 13933  df-ur 13972  df-ring 14010  df-rhm 14165

This theorem is referenced by:  isrhmd  14179  rhmopp  14189  qusrhm  14541  mulgrhm  14622