Theorem pwsco1rhm 18738

Description: Right composition with a function on the index sets yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)

Hypotheses

Ref	Expression
pwsco1rhm.y	|- Y = ( R ^s A )
pwsco1rhm.z	|- Z = ( R ^s B )
pwsco1rhm.c	|- C = ( Base ` Z )
pwsco1rhm.r	|- ( ph -> R e. Ring )
pwsco1rhm.a	|- ( ph -> A e. V )
pwsco1rhm.b	|- ( ph -> B e. W )
pwsco1rhm.f	|- ( ph -> F : A --> B )

Assertion

Ref	Expression
pwsco1rhm	|- ( ph -> ( g e. C |-> ( g o. F ) ) e. ( Z RingHom Y ) )

Distinct variable groups:   A, g   B, g   ph, g   R, g   g, Y   C, g   g, F   g, Z

Allowed substitution hints:   V( g)   W( g)

Proof of Theorem pwsco1rhm

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwsco1rhm.r	. . . 4 |- ( ph -> R e. Ring )
2		pwsco1rhm.b	. . . 4 |- ( ph -> B e. W )
3		pwsco1rhm.z	. . . . 5 |- Z = ( R ^s B )
4	3	pwsring 18615	. . . 4 |- ( ( R e. Ring /\ B e. W ) -> Z e. Ring )
5	1, 2, 4	syl2anc 693	. . 3 |- ( ph -> Z e. Ring )
6		pwsco1rhm.a	. . . 4 |- ( ph -> A e. V )
7		pwsco1rhm.y	. . . . 5 |- Y = ( R ^s A )
8	7	pwsring 18615	. . . 4 |- ( ( R e. Ring /\ A e. V ) -> Y e. Ring )
9	1, 6, 8	syl2anc 693	. . 3 |- ( ph -> Y e. Ring )
10	5, 9	jca 554	. 2 |- ( ph -> ( Z e. Ring /\ Y e. Ring ) )
11		pwsco1rhm.c	. . . . 5 |- C = ( Base ` Z )
12		ringmnd 18556	. . . . . 6 |- ( R e. Ring -> R e. Mnd )
13	1, 12	syl 17	. . . . 5 |- ( ph -> R e. Mnd )
14		pwsco1rhm.f	. . . . 5 |- ( ph -> F : A --> B )
15	7, 3, 11, 13, 6, 2, 14	pwsco1mhm 17370	. . . 4 |- ( ph -> ( g e. C |-> ( g o. F ) ) e. ( Z MndHom Y ) )
16		ringgrp 18552	. . . . . 6 |- ( Z e. Ring -> Z e. Grp )
17	5, 16	syl 17	. . . . 5 |- ( ph -> Z e. Grp )
18		ringgrp 18552	. . . . . 6 |- ( Y e. Ring -> Y e. Grp )
19	9, 18	syl 17	. . . . 5 |- ( ph -> Y e. Grp )
20		ghmmhmb 17671	. . . . 5 |- ( ( Z e. Grp /\ Y e. Grp ) -> ( Z GrpHom Y ) = ( Z MndHom Y ) )
21	17, 19, 20	syl2anc 693	. . . 4 |- ( ph -> ( Z GrpHom Y ) = ( Z MndHom Y ) )
22	15, 21	eleqtrrd 2704	. . 3 |- ( ph -> ( g e. C |-> ( g o. F ) ) e. ( Z GrpHom Y ) )
23		eqid 2622	. . . . 5 |- ( (mulGrp ` R ) ^s A ) = ( (mulGrp ` R ) ^s A )
24		eqid 2622	. . . . 5 |- ( (mulGrp ` R ) ^s B ) = ( (mulGrp ` R ) ^s B )
25		eqid 2622	. . . . 5 |- ( Base ` ( (mulGrp ` R ) ^s B ) ) = ( Base ` ( (mulGrp ` R ) ^s B ) )
26		eqid 2622	. . . . . . 7 |- (mulGrp ` R ) = (mulGrp ` R )
27	26	ringmgp 18553	. . . . . 6 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
28	1, 27	syl 17	. . . . 5 |- ( ph -> (mulGrp ` R ) e. Mnd )
29	23, 24, 25, 28, 6, 2, 14	pwsco1mhm 17370	. . . 4 |- ( ph -> ( g e. ( Base ` ( (mulGrp ` R ) ^s B ) ) |-> ( g o. F ) ) e. ( ( (mulGrp ` R ) ^s B ) MndHom ( (mulGrp ` R ) ^s A ) ) )
30		eqid 2622	. . . . . . . . 9 |- ( Base ` R ) = ( Base ` R )
31	3, 30	pwsbas 16147	. . . . . . . 8 |- ( ( R e. Mnd /\ B e. W ) -> ( ( Base ` R ) ^m B ) = ( Base ` Z ) )
32	13, 2, 31	syl2anc 693	. . . . . . 7 |- ( ph -> ( ( Base ` R ) ^m B ) = ( Base ` Z ) )
33	32, 11	syl6eqr 2674	. . . . . 6 |- ( ph -> ( ( Base ` R ) ^m B ) = C )
34	26, 30	mgpbas 18495	. . . . . . . 8 |- ( Base ` R ) = ( Base ` (mulGrp ` R ) )
35	24, 34	pwsbas 16147	. . . . . . 7 |- ( ( (mulGrp ` R ) e. Mnd /\ B e. W ) -> ( ( Base ` R ) ^m B ) = ( Base ` ( (mulGrp ` R ) ^s B ) ) )
36	28, 2, 35	syl2anc 693	. . . . . 6 |- ( ph -> ( ( Base ` R ) ^m B ) = ( Base ` ( (mulGrp ` R ) ^s B ) ) )
37	33, 36	eqtr3d 2658	. . . . 5 |- ( ph -> C = ( Base ` ( (mulGrp ` R ) ^s B ) ) )
38	37	mpteq1d 4738	. . . 4 |- ( ph -> ( g e. C |-> ( g o. F ) ) = ( g e. ( Base ` ( (mulGrp ` R ) ^s B ) ) |-> ( g o. F ) ) )
39		eqidd 2623	. . . . 5 |- ( ph -> ( Base ` (mulGrp ` Z ) ) = ( Base ` (mulGrp ` Z ) ) )
40		eqidd 2623	. . . . 5 |- ( ph -> ( Base ` (mulGrp ` Y ) ) = ( Base ` (mulGrp ` Y ) ) )
41		eqid 2622	. . . . . . . 8 |- (mulGrp ` Z ) = (mulGrp ` Z )
42		eqid 2622	. . . . . . . 8 |- ( Base ` (mulGrp ` Z ) ) = ( Base ` (mulGrp ` Z ) )
43		eqid 2622	. . . . . . . 8 |- ( +g ` (mulGrp ` Z ) ) = ( +g ` (mulGrp ` Z ) )
44		eqid 2622	. . . . . . . 8 |- ( +g ` ( (mulGrp ` R ) ^s B ) ) = ( +g ` ( (mulGrp ` R ) ^s B ) )
45	3, 26, 24, 41, 42, 25, 43, 44	pwsmgp 18618	. . . . . . 7 |- ( ( R e. Ring /\ B e. W ) -> ( ( Base ` (mulGrp ` Z ) ) = ( Base ` ( (mulGrp ` R ) ^s B ) ) /\ ( +g ` (mulGrp ` Z ) ) = ( +g ` ( (mulGrp ` R ) ^s B ) ) ) )
46	1, 2, 45	syl2anc 693	. . . . . 6 |- ( ph -> ( ( Base ` (mulGrp ` Z ) ) = ( Base ` ( (mulGrp ` R ) ^s B ) ) /\ ( +g ` (mulGrp ` Z ) ) = ( +g ` ( (mulGrp ` R ) ^s B ) ) ) )
47	46	simpld 475	. . . . 5 |- ( ph -> ( Base ` (mulGrp ` Z ) ) = ( Base ` ( (mulGrp ` R ) ^s B ) ) )
48		eqid 2622	. . . . . . . 8 |- (mulGrp ` Y ) = (mulGrp ` Y )
49		eqid 2622	. . . . . . . 8 |- ( Base ` (mulGrp ` Y ) ) = ( Base ` (mulGrp ` Y ) )
50		eqid 2622	. . . . . . . 8 |- ( Base ` ( (mulGrp ` R ) ^s A ) ) = ( Base ` ( (mulGrp ` R ) ^s A ) )
51		eqid 2622	. . . . . . . 8 |- ( +g ` (mulGrp ` Y ) ) = ( +g ` (mulGrp ` Y ) )
52		eqid 2622	. . . . . . . 8 |- ( +g ` ( (mulGrp ` R ) ^s A ) ) = ( +g ` ( (mulGrp ` R ) ^s A ) )
53	7, 26, 23, 48, 49, 50, 51, 52	pwsmgp 18618	. . . . . . 7 |- ( ( R e. Ring /\ A e. V ) -> ( ( Base ` (mulGrp ` Y ) ) = ( Base ` ( (mulGrp ` R ) ^s A ) ) /\ ( +g ` (mulGrp ` Y ) ) = ( +g ` ( (mulGrp ` R ) ^s A ) ) ) )
54	1, 6, 53	syl2anc 693	. . . . . 6 |- ( ph -> ( ( Base ` (mulGrp ` Y ) ) = ( Base ` ( (mulGrp ` R ) ^s A ) ) /\ ( +g ` (mulGrp ` Y ) ) = ( +g ` ( (mulGrp ` R ) ^s A ) ) ) )
55	54	simpld 475	. . . . 5 |- ( ph -> ( Base ` (mulGrp ` Y ) ) = ( Base ` ( (mulGrp ` R ) ^s A ) ) )
56	46	simprd 479	. . . . . 6 |- ( ph -> ( +g ` (mulGrp ` Z ) ) = ( +g ` ( (mulGrp ` R ) ^s B ) ) )
57	56	oveqdr 6674	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` (mulGrp ` Z ) ) /\ y e. ( Base ` (mulGrp ` Z ) ) ) ) -> ( x ( +g ` (mulGrp ` Z ) ) y ) = ( x ( +g ` ( (mulGrp ` R ) ^s B ) ) y ) )
58	54	simprd 479	. . . . . 6 |- ( ph -> ( +g ` (mulGrp ` Y ) ) = ( +g ` ( (mulGrp ` R ) ^s A ) ) )
59	58	oveqdr 6674	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` (mulGrp ` Y ) ) /\ y e. ( Base ` (mulGrp ` Y ) ) ) ) -> ( x ( +g ` (mulGrp ` Y ) ) y ) = ( x ( +g ` ( (mulGrp ` R ) ^s A ) ) y ) )
60	39, 40, 47, 55, 57, 59	mhmpropd 17341	. . . 4 |- ( ph -> ( (mulGrp ` Z ) MndHom (mulGrp ` Y ) ) = ( ( (mulGrp ` R ) ^s B ) MndHom ( (mulGrp ` R ) ^s A ) ) )
61	29, 38, 60	3eltr4d 2716	. . 3 |- ( ph -> ( g e. C |-> ( g o. F ) ) e. ( (mulGrp ` Z ) MndHom (mulGrp ` Y ) ) )
62	22, 61	jca 554	. 2 |- ( ph -> ( ( g e. C |-> ( g o. F ) ) e. ( Z GrpHom Y ) /\ ( g e. C |-> ( g o. F ) ) e. ( (mulGrp ` Z ) MndHom (mulGrp ` Y ) ) ) )
63	41, 48	isrhm 18721	. 2 |- ( ( g e. C |-> ( g o. F ) ) e. ( Z RingHom Y ) <-> ( ( Z e. Ring /\ Y e. Ring ) /\ ( ( g e. C |-> ( g o. F ) ) e. ( Z GrpHom Y ) /\ ( g e. C |-> ( g o. F ) ) e. ( (mulGrp ` Z ) MndHom (mulGrp ` Y ) ) ) ) )
64	10, 62, 63	sylanbrc 698	1 |- ( ph -> ( g e. C |-> ( g o. F ) ) e. ( Z RingHom Y ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   |-> cmpt 4729   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Basecbs 15857   +g cplusg 15941   ^_s cpws 16107   Mndcmnd 17294   MndHom cmhm 17333   Grpcgrp 17422   GrpHom cghm 17657  mulGrpcmgp 18489   Ringcrg 18547   RingHom crh 18712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-prds 16108  df-pws 16110  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-minusg 17426  df-ghm 17658  df-mgp 18490  df-ur 18502  df-ring 18549  df-rnghom 18715

This theorem is referenced by:  evls1rhmlem  19686