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Theorem pwsco2rhm 15824
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.
StepHypRef Expression
1 pwsco2rhm.f . . . . 5  |-  ( ph  ->  F  e.  ( R RingHom  S ) )
2 rhmrcl1 15812 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  R  e.  Ring )
31, 2syl 16 . . . 4  |-  ( ph  ->  R  e.  Ring )
4 pwsco2rhm.a . . . 4  |-  ( ph  ->  A  e.  V )
5 pwsco2rhm.y . . . . 5  |-  Y  =  ( R  ^s  A )
65pwsrng 15711 . . . 4  |-  ( ( R  e.  Ring  /\  A  e.  V )  ->  Y  e.  Ring )
73, 4, 6syl2anc 643 . . 3  |-  ( ph  ->  Y  e.  Ring )
8 rhmrcl2 15813 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  S  e.  Ring )
91, 8syl 16 . . . 4  |-  ( ph  ->  S  e.  Ring )
10 pwsco2rhm.z . . . . 5  |-  Z  =  ( S  ^s  A )
1110pwsrng 15711 . . . 4  |-  ( ( S  e.  Ring  /\  A  e.  V )  ->  Z  e.  Ring )
129, 4, 11syl2anc 643 . . 3  |-  ( ph  ->  Z  e.  Ring )
137, 12jca 519 . 2  |-  ( ph  ->  ( Y  e.  Ring  /\  Z  e.  Ring )
)
14 pwsco2rhm.b . . . . 5  |-  B  =  ( Base `  Y
)
15 rhmghm 15816 . . . . . . 7  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
161, 15syl 16 . . . . . 6  |-  ( ph  ->  F  e.  ( R 
GrpHom  S ) )
17 ghmmhm 15006 . . . . . 6  |-  ( F  e.  ( R  GrpHom  S )  ->  F  e.  ( R MndHom  S ) )
1816, 17syl 16 . . . . 5  |-  ( ph  ->  F  e.  ( R MndHom  S ) )
195, 10, 14, 4, 18pwsco2mhm 14760 . . . 4  |-  ( ph  ->  ( g  e.  B  |->  ( F  o.  g
) )  e.  ( Y MndHom  Z ) )
20 rnggrp 15659 . . . . . 6  |-  ( Y  e.  Ring  ->  Y  e. 
Grp )
217, 20syl 16 . . . . 5  |-  ( ph  ->  Y  e.  Grp )
22 rnggrp 15659 . . . . . 6  |-  ( Z  e.  Ring  ->  Z  e. 
Grp )
2312, 22syl 16 . . . . 5  |-  ( ph  ->  Z  e.  Grp )
24 ghmmhmb 15007 . . . . 5  |-  ( ( Y  e.  Grp  /\  Z  e.  Grp )  ->  ( Y  GrpHom  Z )  =  ( Y MndHom  Z
) )
2521, 23, 24syl2anc 643 . . . 4  |-  ( ph  ->  ( Y  GrpHom  Z )  =  ( Y MndHom  Z
) )
2619, 25eleqtrrd 2512 . . 3  |-  ( ph  ->  ( g  e.  B  |->  ( F  o.  g
) )  e.  ( Y  GrpHom  Z ) )
27 eqid 2435 . . . . 5  |-  ( (mulGrp `  R )  ^s  A )  =  ( (mulGrp `  R )  ^s  A )
28 eqid 2435 . . . . 5  |-  ( (mulGrp `  S )  ^s  A )  =  ( (mulGrp `  S )  ^s  A )
29 eqid 2435 . . . . 5  |-  ( Base `  ( (mulGrp `  R
)  ^s  A ) )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) )
30 eqid 2435 . . . . . . 7  |-  (mulGrp `  R )  =  (mulGrp `  R )
31 eqid 2435 . . . . . . 7  |-  (mulGrp `  S )  =  (mulGrp `  S )
3230, 31rhmmhm 15815 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) ) )
331, 32syl 16 . . . . 5  |-  ( ph  ->  F  e.  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) ) )
3427, 28, 29, 4, 33pwsco2mhm 14760 . . . 4  |-  ( ph  ->  ( g  e.  (
Base `  ( (mulGrp `  R )  ^s  A ) )  |->  ( F  o.  g ) )  e.  ( ( (mulGrp `  R )  ^s  A ) MndHom 
( (mulGrp `  S
)  ^s  A ) ) )
35 eqid 2435 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  R )
365, 35pwsbas 13699 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  A  e.  V )  ->  (
( Base `  R )  ^m  A )  =  (
Base `  Y )
)
373, 4, 36syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( ( Base `  R
)  ^m  A )  =  ( Base `  Y
) )
3837, 14syl6eqr 2485 . . . . . 6  |-  ( ph  ->  ( ( Base `  R
)  ^m  A )  =  B )
3930rngmgp 15660 . . . . . . . 8  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
403, 39syl 16 . . . . . . 7  |-  ( ph  ->  (mulGrp `  R )  e.  Mnd )
4130, 35mgpbas 15644 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
4227, 41pwsbas 13699 . . . . . . 7  |-  ( ( (mulGrp `  R )  e.  Mnd  /\  A  e.  V )  ->  (
( Base `  R )  ^m  A )  =  (
Base `  ( (mulGrp `  R )  ^s  A ) ) )
4340, 4, 42syl2anc 643 . . . . . 6  |-  ( ph  ->  ( ( Base `  R
)  ^m  A )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) ) )
4438, 43eqtr3d 2469 . . . . 5  |-  ( ph  ->  B  =  ( Base `  ( (mulGrp `  R
)  ^s  A ) ) )
4544mpteq1d 4282 . . . 4  |-  ( ph  ->  ( g  e.  B  |->  ( F  o.  g
) )  =  ( g  e.  ( Base `  ( (mulGrp `  R
)  ^s  A ) )  |->  ( F  o.  g ) ) )
46 eqidd 2436 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Y ) )  =  ( Base `  (mulGrp `  Y ) ) )
47 eqidd 2436 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Z ) )  =  ( Base `  (mulGrp `  Z ) ) )
48 eqid 2435 . . . . . . . 8  |-  (mulGrp `  Y )  =  (mulGrp `  Y )
49 eqid 2435 . . . . . . . 8  |-  ( Base `  (mulGrp `  Y )
)  =  ( Base `  (mulGrp `  Y )
)
50 eqid 2435 . . . . . . . 8  |-  ( +g  `  (mulGrp `  Y )
)  =  ( +g  `  (mulGrp `  Y )
)
51 eqid 2435 . . . . . . . 8  |-  ( +g  `  ( (mulGrp `  R
)  ^s  A ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  A ) )
525, 30, 27, 48, 49, 29, 50, 51pwsmgp 15714 . . . . . . 7  |-  ( ( R  e.  Ring  /\  A  e.  V )  ->  (
( Base `  (mulGrp `  Y
) )  =  (
Base `  ( (mulGrp `  R )  ^s  A ) )  /\  ( +g  `  (mulGrp `  Y )
)  =  ( +g  `  ( (mulGrp `  R
)  ^s  A ) ) ) )
533, 4, 52syl2anc 643 . . . . . 6  |-  ( ph  ->  ( ( Base `  (mulGrp `  Y ) )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) )  /\  ( +g  `  (mulGrp `  Y ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  A ) ) ) )
5453simpld 446 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Y ) )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) ) )
55 eqid 2435 . . . . . . . 8  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
56 eqid 2435 . . . . . . . 8  |-  ( Base `  (mulGrp `  Z )
)  =  ( Base `  (mulGrp `  Z )
)
57 eqid 2435 . . . . . . . 8  |-  ( Base `  ( (mulGrp `  S
)  ^s  A ) )  =  ( Base `  (
(mulGrp `  S )  ^s  A ) )
58 eqid 2435 . . . . . . . 8  |-  ( +g  `  (mulGrp `  Z )
)  =  ( +g  `  (mulGrp `  Z )
)
59 eqid 2435 . . . . . . . 8  |-  ( +g  `  ( (mulGrp `  S
)  ^s  A ) )  =  ( +g  `  (
(mulGrp `  S )  ^s  A ) )
6010, 31, 28, 55, 56, 57, 58, 59pwsmgp 15714 . . . . . . 7  |-  ( ( S  e.  Ring  /\  A  e.  V )  ->  (
( Base `  (mulGrp `  Z
) )  =  (
Base `  ( (mulGrp `  S )  ^s  A ) )  /\  ( +g  `  (mulGrp `  Z )
)  =  ( +g  `  ( (mulGrp `  S
)  ^s  A ) ) ) )
619, 4, 60syl2anc 643 . . . . . 6  |-  ( ph  ->  ( ( Base `  (mulGrp `  Z ) )  =  ( Base `  (
(mulGrp `  S )  ^s  A ) )  /\  ( +g  `  (mulGrp `  Z ) )  =  ( +g  `  (
(mulGrp `  S )  ^s  A ) ) ) )
6261simpld 446 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Z ) )  =  ( Base `  (
(mulGrp `  S )  ^s  A ) ) )
6353simprd 450 . . . . . 6  |-  ( ph  ->  ( +g  `  (mulGrp `  Y ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  A ) ) )
6463proplem3 13906 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  (mulGrp `  Y ) )  /\  y  e.  ( Base `  (mulGrp `  Y )
) ) )  -> 
( x ( +g  `  (mulGrp `  Y )
) y )  =  ( x ( +g  `  ( (mulGrp `  R
)  ^s  A ) ) y ) )
6561simprd 450 . . . . . 6  |-  ( ph  ->  ( +g  `  (mulGrp `  Z ) )  =  ( +g  `  (
(mulGrp `  S )  ^s  A ) ) )
6665proplem3 13906 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  (mulGrp `  Z ) )  /\  y  e.  ( Base `  (mulGrp `  Z )
) ) )  -> 
( x ( +g  `  (mulGrp `  Z )
) y )  =  ( x ( +g  `  ( (mulGrp `  S
)  ^s  A ) ) y ) )
6746, 47, 54, 62, 64, 66mhmpropd 14734 . . . 4  |-  ( ph  ->  ( (mulGrp `  Y
) MndHom  (mulGrp `  Z )
)  =  ( ( (mulGrp `  R )  ^s  A ) MndHom  ( (mulGrp `  S )  ^s  A ) ) )
6834, 45, 673eltr4d 2516 . . 3  |-  ( ph  ->  ( g  e.  B  |->  ( F  o.  g
) )  e.  ( (mulGrp `  Y ) MndHom  (mulGrp `  Z ) ) )
6926, 68jca 519 . 2  |-  ( ph  ->  ( ( g  e.  B  |->  ( F  o.  g ) )  e.  ( Y  GrpHom  Z )  /\  ( g  e.  B  |->  ( F  o.  g ) )  e.  ( (mulGrp `  Y
) MndHom  (mulGrp `  Z )
) ) )
7048, 55isrhm 15814 . 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 ) ) ) ) )
7113, 69, 70sylanbrc 646 1  |-  ( ph  ->  ( g  e.  B  |->  ( F  o.  g
) )  e.  ( Y RingHom  Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    e. cmpt 4258    o. ccom 4874   ` cfv 5446  (class class class)co 6073    ^m cmap 7010   Basecbs 13459   +g cplusg 13519    ^s cpws 13660   Mndcmnd 14674   Grpcgrp 14675   MndHom cmhm 14726    GrpHom cghm 14993  mulGrpcmgp 15638   Ringcrg 15650   RingHom crh 15807
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-7 10053  df-8 10054  df-9 10055  df-10 10056  df-n0 10212  df-z 10273  df-dec 10373  df-uz 10479  df-fz 11034  df-struct 13461  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-plusg 13532  df-mulr 13533  df-sca 13535  df-vsca 13536  df-tset 13538  df-ple 13539  df-ds 13541  df-hom 13543  df-cco 13544  df-prds 13661  df-pws 13663  df-0g 13717  df-mnd 14680  df-mhm 14728  df-grp 14802  df-minusg 14803  df-ghm 14994  df-mgp 15639  df-rng 15653  df-ur 15655  df-rnghom 15809
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