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Theorem pwsco2rhm 15763
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 15751 . . . . 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 15650 . . . 4  |-  ( ( R  e.  Ring  /\  A  e.  V )  ->  Y  e.  Ring )
73, 4, 6syl2anc 643 . . 3  |-  ( ph  ->  Y  e.  Ring )
8 rhmrcl2 15752 . . . . 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 15650 . . . 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 15755 . . . . . . 7  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
161, 15syl 16 . . . . . 6  |-  ( ph  ->  F  e.  ( R 
GrpHom  S ) )
17 ghmmhm 14945 . . . . . 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 14699 . . . 4  |-  ( ph  ->  ( g  e.  B  |->  ( F  o.  g
) )  e.  ( Y MndHom  Z ) )
20 rnggrp 15598 . . . . . 6  |-  ( Y  e.  Ring  ->  Y  e. 
Grp )
217, 20syl 16 . . . . 5  |-  ( ph  ->  Y  e.  Grp )
22 rnggrp 15598 . . . . . 6  |-  ( Z  e.  Ring  ->  Z  e. 
Grp )
2312, 22syl 16 . . . . 5  |-  ( ph  ->  Z  e.  Grp )
24 ghmmhmb 14946 . . . . 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 2466 . . 3  |-  ( ph  ->  ( g  e.  B  |->  ( F  o.  g
) )  e.  ( Y  GrpHom  Z ) )
27 eqid 2389 . . . . 5  |-  ( (mulGrp `  R )  ^s  A )  =  ( (mulGrp `  R )  ^s  A )
28 eqid 2389 . . . . 5  |-  ( (mulGrp `  S )  ^s  A )  =  ( (mulGrp `  S )  ^s  A )
29 eqid 2389 . . . . 5  |-  ( Base `  ( (mulGrp `  R
)  ^s  A ) )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) )
30 eqid 2389 . . . . . . 7  |-  (mulGrp `  R )  =  (mulGrp `  R )
31 eqid 2389 . . . . . . 7  |-  (mulGrp `  S )  =  (mulGrp `  S )
3230, 31rhmmhm 15754 . . . . . 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 14699 . . . 4  |-  ( ph  ->  ( g  e.  (
Base `  ( (mulGrp `  R )  ^s  A ) )  |->  ( F  o.  g ) )  e.  ( ( (mulGrp `  R )  ^s  A ) MndHom 
( (mulGrp `  S
)  ^s  A ) ) )
35 eqid 2389 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  R )
365, 35pwsbas 13638 . . . . . . . 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 2439 . . . . . 6  |-  ( ph  ->  ( ( Base `  R
)  ^m  A )  =  B )
3930rngmgp 15599 . . . . . . . 8  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
403, 39syl 16 . . . . . . 7  |-  ( ph  ->  (mulGrp `  R )  e.  Mnd )
4130, 35mgpbas 15583 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
4227, 41pwsbas 13638 . . . . . . 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 2423 . . . . 5  |-  ( ph  ->  B  =  ( Base `  ( (mulGrp `  R
)  ^s  A ) ) )
4544mpteq1d 4233 . . . 4  |-  ( ph  ->  ( g  e.  B  |->  ( F  o.  g
) )  =  ( g  e.  ( Base `  ( (mulGrp `  R
)  ^s  A ) )  |->  ( F  o.  g ) ) )
46 eqidd 2390 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Y ) )  =  ( Base `  (mulGrp `  Y ) ) )
47 eqidd 2390 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Z ) )  =  ( Base `  (mulGrp `  Z ) ) )
48 eqid 2389 . . . . . . . 8  |-  (mulGrp `  Y )  =  (mulGrp `  Y )
49 eqid 2389 . . . . . . . 8  |-  ( Base `  (mulGrp `  Y )
)  =  ( Base `  (mulGrp `  Y )
)
50 eqid 2389 . . . . . . . 8  |-  ( +g  `  (mulGrp `  Y )
)  =  ( +g  `  (mulGrp `  Y )
)
51 eqid 2389 . . . . . . . 8  |-  ( +g  `  ( (mulGrp `  R
)  ^s  A ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  A ) )
525, 30, 27, 48, 49, 29, 50, 51pwsmgp 15653 . . . . . . 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 2389 . . . . . . . 8  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
56 eqid 2389 . . . . . . . 8  |-  ( Base `  (mulGrp `  Z )
)  =  ( Base `  (mulGrp `  Z )
)
57 eqid 2389 . . . . . . . 8  |-  ( Base `  ( (mulGrp `  S
)  ^s  A ) )  =  ( Base `  (
(mulGrp `  S )  ^s  A ) )
58 eqid 2389 . . . . . . . 8  |-  ( +g  `  (mulGrp `  Z )
)  =  ( +g  `  (mulGrp `  Z )
)
59 eqid 2389 . . . . . . . 8  |-  ( +g  `  ( (mulGrp `  S
)  ^s  A ) )  =  ( +g  `  (
(mulGrp `  S )  ^s  A ) )
6010, 31, 28, 55, 56, 57, 58, 59pwsmgp 15653 . . . . . . 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 13845 . . . . 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 13845 . . . . 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 14673 . . . 4  |-  ( ph  ->  ( (mulGrp `  Y
) MndHom  (mulGrp `  Z )
)  =  ( ( (mulGrp `  R )  ^s  A ) MndHom  ( (mulGrp `  S )  ^s  A ) ) )
6834, 45, 673eltr4d 2470 . . 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 15753 . 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 1649    e. wcel 1717    e. cmpt 4209    o. ccom 4824   ` cfv 5396  (class class class)co 6022    ^m cmap 6956   Basecbs 13398   +g cplusg 13458    ^s cpws 13599   Mndcmnd 14613   Grpcgrp 14614   MndHom cmhm 14665    GrpHom cghm 14932  mulGrpcmgp 15577   Ringcrg 15589   RingHom crh 15746
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-ixp 7002  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-7 9997  df-8 9998  df-9 9999  df-10 10000  df-n0 10156  df-z 10217  df-dec 10317  df-uz 10423  df-fz 10978  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-plusg 13471  df-mulr 13472  df-sca 13474  df-vsca 13475  df-tset 13477  df-ple 13478  df-ds 13480  df-hom 13482  df-cco 13483  df-prds 13600  df-pws 13602  df-0g 13656  df-mnd 14619  df-mhm 14667  df-grp 14741  df-minusg 14742  df-ghm 14933  df-mgp 15578  df-rng 15592  df-ur 15594  df-rnghom 15748
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