Theorem pwsco2rhm 18739

Description: Left composition with a ring homomorphism yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)

Hypotheses

Ref	Expression
pwsco2rhm.y	|- Y = ( R ^s A )
pwsco2rhm.z	|- Z = ( S ^s A )
pwsco2rhm.b	|- B = ( Base ` Y )
pwsco2rhm.a	|- ( ph -> A e. V )
pwsco2rhm.f	|- ( ph -> F e. ( R RingHom S ) )

Assertion

Ref	Expression
pwsco2rhm	|- ( ph -> ( g e. B |-> ( F o. g ) ) e. ( Y RingHom Z ) )

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

Allowed substitution hint:   V( g)

Proof of Theorem pwsco2rhm

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

Step	Hyp	Ref	Expression
1		pwsco2rhm.f	. . . . 5 |- ( ph -> F e. ( R RingHom S ) )
2		rhmrcl1 18719	. . . . 5 |- ( F e. ( R RingHom S ) -> R e. Ring )
3	1, 2	syl 17	. . . 4 |- ( ph -> R e. Ring )
4		pwsco2rhm.a	. . . 4 |- ( ph -> A e. V )
5		pwsco2rhm.y	. . . . 5 |- Y = ( R ^s A )
6	5	pwsring 18615	. . . 4 |- ( ( R e. Ring /\ A e. V ) -> Y e. Ring )
7	3, 4, 6	syl2anc 693	. . 3 |- ( ph -> Y e. Ring )
8		rhmrcl2 18720	. . . . 5 |- ( F e. ( R RingHom S ) -> S e. Ring )
9	1, 8	syl 17	. . . 4 |- ( ph -> S e. Ring )
10		pwsco2rhm.z	. . . . 5 |- Z = ( S ^s A )
11	10	pwsring 18615	. . . 4 |- ( ( S e. Ring /\ A e. V ) -> Z e. Ring )
12	9, 4, 11	syl2anc 693	. . 3 |- ( ph -> Z e. Ring )
13	7, 12	jca 554	. 2 |- ( ph -> ( Y e. Ring /\ Z e. Ring ) )
14		pwsco2rhm.b	. . . . 5 |- B = ( Base ` Y )
15		rhmghm 18725	. . . . . . 7 |- ( F e. ( R RingHom S ) -> F e. ( R GrpHom S ) )
16	1, 15	syl 17	. . . . . 6 |- ( ph -> F e. ( R GrpHom S ) )
17		ghmmhm 17670	. . . . . 6 |- ( F e. ( R GrpHom S ) -> F e. ( R MndHom S ) )
18	16, 17	syl 17	. . . . 5 |- ( ph -> F e. ( R MndHom S ) )
19	5, 10, 14, 4, 18	pwsco2mhm 17371	. . . 4 |- ( ph -> ( g e. B |-> ( F o. g ) ) e. ( Y MndHom Z ) )
20		ringgrp 18552	. . . . . 6 |- ( Y e. Ring -> Y e. Grp )
21	7, 20	syl 17	. . . . 5 |- ( ph -> Y e. Grp )
22		ringgrp 18552	. . . . . 6 |- ( Z e. Ring -> Z e. Grp )
23	12, 22	syl 17	. . . . 5 |- ( ph -> Z e. Grp )
24		ghmmhmb 17671	. . . . 5 |- ( ( Y e. Grp /\ Z e. Grp ) -> ( Y GrpHom Z ) = ( Y MndHom Z ) )
25	21, 23, 24	syl2anc 693	. . . 4 |- ( ph -> ( Y GrpHom Z ) = ( Y MndHom Z ) )
26	19, 25	eleqtrrd 2704	. . 3 |- ( ph -> ( g e. B |-> ( F o. g ) ) e. ( Y GrpHom Z ) )
27		eqid 2622	. . . . 5 |- ( (mulGrp ` R ) ^s A ) = ( (mulGrp ` R ) ^s A )
28		eqid 2622	. . . . 5 |- ( (mulGrp ` S ) ^s A ) = ( (mulGrp ` S ) ^s A )
29		eqid 2622	. . . . 5 |- ( Base ` ( (mulGrp ` R ) ^s A ) ) = ( Base ` ( (mulGrp ` R ) ^s A ) )
30		eqid 2622	. . . . . . 7 |- (mulGrp ` R ) = (mulGrp ` R )
31		eqid 2622	. . . . . . 7 |- (mulGrp ` S ) = (mulGrp ` S )
32	30, 31	rhmmhm 18722	. . . . . 6 |- ( F e. ( R RingHom S ) -> F e. ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) )
33	1, 32	syl 17	. . . . 5 |- ( ph -> F e. ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) )
34	27, 28, 29, 4, 33	pwsco2mhm 17371	. . . 4 |- ( ph -> ( g e. ( Base ` ( (mulGrp ` R ) ^s A ) ) |-> ( F o. g ) ) e. ( ( (mulGrp ` R ) ^s A ) MndHom ( (mulGrp ` S ) ^s A ) ) )
35		eqid 2622	. . . . . . . . 9 |- ( Base ` R ) = ( Base ` R )
36	5, 35	pwsbas 16147	. . . . . . . 8 |- ( ( R e. Ring /\ A e. V ) -> ( ( Base ` R ) ^m A ) = ( Base ` Y ) )
37	3, 4, 36	syl2anc 693	. . . . . . 7 |- ( ph -> ( ( Base ` R ) ^m A ) = ( Base ` Y ) )
38	37, 14	syl6eqr 2674	. . . . . 6 |- ( ph -> ( ( Base ` R ) ^m A ) = B )
39	30	ringmgp 18553	. . . . . . . 8 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
40	3, 39	syl 17	. . . . . . 7 |- ( ph -> (mulGrp ` R ) e. Mnd )
41	30, 35	mgpbas 18495	. . . . . . . 8 |- ( Base ` R ) = ( Base ` (mulGrp ` R ) )
42	27, 41	pwsbas 16147	. . . . . . 7 |- ( ( (mulGrp ` R ) e. Mnd /\ A e. V ) -> ( ( Base ` R ) ^m A ) = ( Base ` ( (mulGrp ` R ) ^s A ) ) )
43	40, 4, 42	syl2anc 693	. . . . . 6 |- ( ph -> ( ( Base ` R ) ^m A ) = ( Base ` ( (mulGrp ` R ) ^s A ) ) )
44	38, 43	eqtr3d 2658	. . . . 5 |- ( ph -> B = ( Base ` ( (mulGrp ` R ) ^s A ) ) )
45	44	mpteq1d 4738	. . . 4 |- ( ph -> ( g e. B |-> ( F o. g ) ) = ( g e. ( Base ` ( (mulGrp ` R ) ^s A ) ) |-> ( F o. g ) ) )
46		eqidd 2623	. . . . 5 |- ( ph -> ( Base ` (mulGrp ` Y ) ) = ( Base ` (mulGrp ` Y ) ) )
47		eqidd 2623	. . . . 5 |- ( ph -> ( Base ` (mulGrp ` Z ) ) = ( Base ` (mulGrp ` Z ) ) )
48		eqid 2622	. . . . . . . 8 |- (mulGrp ` Y ) = (mulGrp ` Y )
49		eqid 2622	. . . . . . . 8 |- ( Base ` (mulGrp ` Y ) ) = ( Base ` (mulGrp ` Y ) )
50		eqid 2622	. . . . . . . 8 |- ( +g ` (mulGrp ` Y ) ) = ( +g ` (mulGrp ` Y ) )
51		eqid 2622	. . . . . . . 8 |- ( +g ` ( (mulGrp ` R ) ^s A ) ) = ( +g ` ( (mulGrp ` R ) ^s A ) )
52	5, 30, 27, 48, 49, 29, 50, 51	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 ) ) ) )
53	3, 4, 52	syl2anc 693	. . . . . 6 |- ( ph -> ( ( Base ` (mulGrp ` Y ) ) = ( Base ` ( (mulGrp ` R ) ^s A ) ) /\ ( +g ` (mulGrp ` Y ) ) = ( +g ` ( (mulGrp ` R ) ^s A ) ) ) )
54	53	simpld 475	. . . . 5 |- ( ph -> ( Base ` (mulGrp ` Y ) ) = ( Base ` ( (mulGrp ` R ) ^s A ) ) )
55		eqid 2622	. . . . . . . 8 |- (mulGrp ` Z ) = (mulGrp ` Z )
56		eqid 2622	. . . . . . . 8 |- ( Base ` (mulGrp ` Z ) ) = ( Base ` (mulGrp ` Z ) )
57		eqid 2622	. . . . . . . 8 |- ( Base ` ( (mulGrp ` S ) ^s A ) ) = ( Base ` ( (mulGrp ` S ) ^s A ) )
58		eqid 2622	. . . . . . . 8 |- ( +g ` (mulGrp ` Z ) ) = ( +g ` (mulGrp ` Z ) )
59		eqid 2622	. . . . . . . 8 |- ( +g ` ( (mulGrp ` S ) ^s A ) ) = ( +g ` ( (mulGrp ` S ) ^s A ) )
60	10, 31, 28, 55, 56, 57, 58, 59	pwsmgp 18618	. . . . . . 7 |- ( ( S e. Ring /\ A e. V ) -> ( ( Base ` (mulGrp ` Z ) ) = ( Base ` ( (mulGrp ` S ) ^s A ) ) /\ ( +g ` (mulGrp ` Z ) ) = ( +g ` ( (mulGrp ` S ) ^s A ) ) ) )
61	9, 4, 60	syl2anc 693	. . . . . 6 |- ( ph -> ( ( Base ` (mulGrp ` Z ) ) = ( Base ` ( (mulGrp ` S ) ^s A ) ) /\ ( +g ` (mulGrp ` Z ) ) = ( +g ` ( (mulGrp ` S ) ^s A ) ) ) )
62	61	simpld 475	. . . . 5 |- ( ph -> ( Base ` (mulGrp ` Z ) ) = ( Base ` ( (mulGrp ` S ) ^s A ) ) )
63	53	simprd 479	. . . . . 6 |- ( ph -> ( +g ` (mulGrp ` Y ) ) = ( +g ` ( (mulGrp ` R ) ^s A ) ) )
64	63	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 ) )
65	61	simprd 479	. . . . . 6 |- ( ph -> ( +g ` (mulGrp ` Z ) ) = ( +g ` ( (mulGrp ` S ) ^s A ) ) )
66	65	oveqdr 6674	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` (mulGrp ` Z ) ) /\ y e. ( Base ` (mulGrp ` Z ) ) ) ) -> ( x ( +g ` (mulGrp ` Z ) ) y ) = ( x ( +g ` ( (mulGrp ` S ) ^s A ) ) y ) )
67	46, 47, 54, 62, 64, 66	mhmpropd 17341	. . . 4 |- ( ph -> ( (mulGrp ` Y ) MndHom (mulGrp ` Z ) ) = ( ( (mulGrp ` R ) ^s A ) MndHom ( (mulGrp ` S ) ^s A ) ) )
68	34, 45, 67	3eltr4d 2716	. . 3 |- ( ph -> ( g e. B |-> ( F o. g ) ) e. ( (mulGrp ` Y ) MndHom (mulGrp ` Z ) ) )
69	26, 68	jca 554	. 2 |- ( ph -> ( ( g e. B |-> ( F o. g ) ) e. ( Y GrpHom Z ) /\ ( g e. B |-> ( F o. g ) ) e. ( (mulGrp ` Y ) MndHom (mulGrp ` Z ) ) ) )
70	48, 55	isrhm 18721	. 2 |- ( ( g e. B |-> ( F o. g ) ) e. ( Y RingHom Z ) <-> ( ( Y e. Ring /\ Z e. Ring ) /\ ( ( g e. B |-> ( F o. g ) ) e. ( Y GrpHom Z ) /\ ( g e. B |-> ( F o. g ) ) e. ( (mulGrp ` Y ) MndHom (mulGrp ` Z ) ) ) ) )
71	13, 69, 70	sylanbrc 698	1 |- ( ph -> ( g e. B |-> ( F o. g ) ) e. ( Y RingHom Z ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   |-> cmpt 4729   o. ccom 5118   ` 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: (None)