Theorem isrhm2d 14445

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 2238	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
5	4	ringmgp 14280	. . . . 5 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
6	1, 5	syl 14	. . . 4 |- ( ph -> (mulGrp ` R ) e. Mnd )
7		eqid 2238	. . . . . 6 |- (mulGrp ` S ) = (mulGrp ` S )
8	7	ringmgp 14280	. . . . 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 2238	. . . . . . . 8 |- ( Base ` S ) = ( Base ` S )
12	10, 11	ghmf 14027	. . . . . . 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 14200	. . . . . . . 8 |- ( R e. Ring -> B = ( Base ` (mulGrp ` R ) ) )
15	1, 14	syl 14	. . . . . . 7 |- ( ph -> B = ( Base ` (mulGrp ` R ) ) )
16	7, 11	mgpbasg 14200	. . . . . . . 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 5524	. . . . . 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 2632	. . . . . 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 14198	. . . . . . . . . . . 12 |- ( R e. Ring -> .x. = ( +g ` (mulGrp ` R ) ) )
24	1, 23	syl 14	. . . . . . . . . . 11 |- ( ph -> .x. = ( +g ` (mulGrp ` R ) ) )
25	24	oveqd 6092	. . . . . . . . . 10 |- ( ph -> ( x .x. y ) = ( x ( +g ` (mulGrp ` R ) ) y ) )
26	25	fveq2d 5694	. . . . . . . . 9 |- ( ph -> ( F ` ( x .x. y ) ) = ( F ` ( x ( +g ` (mulGrp ` R ) ) y ) ) )
27		isrhmd.u	. . . . . . . . . . . 12 |- .X. = ( .r ` S )
28	7, 27	mgpplusgg 14198	. . . . . . . . . . 11 |- ( S e. Ring -> .X. = ( +g ` (mulGrp ` S ) ) )
29	2, 28	syl 14	. . . . . . . . . 10 |- ( ph -> .X. = ( +g ` (mulGrp ` S ) ) )
30	29	oveqd 6092	. . . . . . . . 9 |- ( ph -> ( ( F ` x ) .X. ( F ` y ) ) = ( ( F ` x ) ( +g ` (mulGrp ` S ) ) ( F ` y ) ) )
31	26, 30	eqeq12d 2253	. . . . . . . 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 2765	. . . . . . 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 2765	. . . . . 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 14239	. . . . . . . 8 |- ( R e. Ring -> .1. = ( 0g ` (mulGrp ` R ) ) )
38	1, 37	syl 14	. . . . . . 7 |- ( ph -> .1. = ( 0g ` (mulGrp ` R ) ) )
39	38	fveq2d 5694	. . . . . 6 |- ( ph -> ( F ` .1. ) = ( F ` ( 0g ` (mulGrp ` R ) ) ) )
40		isrhmd.n	. . . . . . . 8 |- N = ( 1r ` S )
41	7, 40	ringidvalg 14239	. . . . . . 7 |- ( S e. Ring -> N = ( 0g ` (mulGrp ` S ) ) )
42	2, 41	syl 14	. . . . . 6 |- ( ph -> N = ( 0g ` (mulGrp ` S ) ) )
43	35, 39, 42	3eqtr3d 2279	. . . . 5 |- ( ph -> ( F ` ( 0g ` (mulGrp ` R ) ) ) = ( 0g ` (mulGrp ` S ) ) )
44	19, 34, 43	3jca 1208	. . . 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 2238	. . . . 5 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
46		eqid 2238	. . . . 5 |- ( Base ` (mulGrp ` S ) ) = ( Base ` (mulGrp ` S ) )
47		eqid 2238	. . . . 5 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
48		eqid 2238	. . . . 5 |- ( +g ` (mulGrp ` S ) ) = ( +g ` (mulGrp ` S ) )
49		eqid 2238	. . . . 5 |- ( 0g ` (mulGrp ` R ) ) = ( 0g ` (mulGrp ` R ) )
50		eqid 2238	. . . . 5 |- ( 0g ` (mulGrp ` S ) ) = ( 0g ` (mulGrp ` S ) )
51	45, 46, 47, 48, 49, 50	ismhm 13745	. . . 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 1213	. . 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 14438	. 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 1213	1 |- ( ph -> F e. ( R RingHom S ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   -->wf 5368   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408   .rcmulr 13409   0gc0g 13587   Mndcmnd 13706   MndHom cmhm 13741   GrpHom cghm 14020  mulGrpcmgp 14194   1rcur 14237   Ringcrg 14274   RingHom crh 14430

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-map 6914  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-plusg 13421  df-mulr 13422  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-mhm 13743  df-grp 13785  df-ghm 14021  df-mgp 14195  df-ur 14238  df-ring 14276  df-rhm 14432

This theorem is referenced by:  isrhmd  14446  rhmopp  14456  qusrhm  14837  mulgrhm  14916