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Theorem pwsco2rhm 15511
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 15499 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  R  e.  Ring )
31, 2syl 15 . . . 4  |-  ( ph  ->  R  e.  Ring )
4 pwsco2rhm.a . . . 4  |-  ( ph  ->  A  e.  V )
5 pwsco2rhm.y . . . . 5  |-  Y  =  ( R  ^s  A )
65pwsrng 15398 . . . 4  |-  ( ( R  e.  Ring  /\  A  e.  V )  ->  Y  e.  Ring )
73, 4, 6syl2anc 642 . . 3  |-  ( ph  ->  Y  e.  Ring )
8 rhmrcl2 15500 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  S  e.  Ring )
91, 8syl 15 . . . 4  |-  ( ph  ->  S  e.  Ring )
10 pwsco2rhm.z . . . . 5  |-  Z  =  ( S  ^s  A )
1110pwsrng 15398 . . . 4  |-  ( ( S  e.  Ring  /\  A  e.  V )  ->  Z  e.  Ring )
129, 4, 11syl2anc 642 . . 3  |-  ( ph  ->  Z  e.  Ring )
137, 12jca 518 . 2  |-  ( ph  ->  ( Y  e.  Ring  /\  Z  e.  Ring )
)
14 pwsco2rhm.b . . . . 5  |-  B  =  ( Base `  Y
)
15 rhmghm 15503 . . . . . . 7  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
161, 15syl 15 . . . . . 6  |-  ( ph  ->  F  e.  ( R 
GrpHom  S ) )
17 ghmmhm 14693 . . . . . 6  |-  ( F  e.  ( R  GrpHom  S )  ->  F  e.  ( R MndHom  S ) )
1816, 17syl 15 . . . . 5  |-  ( ph  ->  F  e.  ( R MndHom  S ) )
195, 10, 14, 4, 18pwsco2mhm 14447 . . . 4  |-  ( ph  ->  ( g  e.  B  |->  ( F  o.  g
) )  e.  ( Y MndHom  Z ) )
20 rnggrp 15346 . . . . . 6  |-  ( Y  e.  Ring  ->  Y  e. 
Grp )
217, 20syl 15 . . . . 5  |-  ( ph  ->  Y  e.  Grp )
22 rnggrp 15346 . . . . . 6  |-  ( Z  e.  Ring  ->  Z  e. 
Grp )
2312, 22syl 15 . . . . 5  |-  ( ph  ->  Z  e.  Grp )
24 ghmmhmb 14694 . . . . 5  |-  ( ( Y  e.  Grp  /\  Z  e.  Grp )  ->  ( Y  GrpHom  Z )  =  ( Y MndHom  Z
) )
2521, 23, 24syl2anc 642 . . . 4  |-  ( ph  ->  ( Y  GrpHom  Z )  =  ( Y MndHom  Z
) )
2619, 25eleqtrrd 2360 . . 3  |-  ( ph  ->  ( g  e.  B  |->  ( F  o.  g
) )  e.  ( Y  GrpHom  Z ) )
27 eqid 2283 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
285, 27pwsbas 13386 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  A  e.  V )  ->  (
( Base `  R )  ^m  A )  =  (
Base `  Y )
)
293, 4, 28syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( ( Base `  R
)  ^m  A )  =  ( Base `  Y
) )
3029, 14syl6eqr 2333 . . . . . . 7  |-  ( ph  ->  ( ( Base `  R
)  ^m  A )  =  B )
31 eqid 2283 . . . . . . . . . 10  |-  (mulGrp `  R )  =  (mulGrp `  R )
3231rngmgp 15347 . . . . . . . . 9  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
333, 32syl 15 . . . . . . . 8  |-  ( ph  ->  (mulGrp `  R )  e.  Mnd )
34 eqid 2283 . . . . . . . . 9  |-  ( (mulGrp `  R )  ^s  A )  =  ( (mulGrp `  R )  ^s  A )
3531, 27mgpbas 15331 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
3634, 35pwsbas 13386 . . . . . . . 8  |-  ( ( (mulGrp `  R )  e.  Mnd  /\  A  e.  V )  ->  (
( Base `  R )  ^m  A )  =  (
Base `  ( (mulGrp `  R )  ^s  A ) ) )
3733, 4, 36syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( ( Base `  R
)  ^m  A )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) ) )
3830, 37eqtr3d 2317 . . . . . 6  |-  ( ph  ->  B  =  ( Base `  ( (mulGrp `  R
)  ^s  A ) ) )
39 mpteq1 4100 . . . . . 6  |-  ( B  =  ( Base `  (
(mulGrp `  R )  ^s  A ) )  -> 
( g  e.  B  |->  ( F  o.  g
) )  =  ( g  e.  ( Base `  ( (mulGrp `  R
)  ^s  A ) )  |->  ( F  o.  g ) ) )
4038, 39syl 15 . . . . 5  |-  ( ph  ->  ( g  e.  B  |->  ( F  o.  g
) )  =  ( g  e.  ( Base `  ( (mulGrp `  R
)  ^s  A ) )  |->  ( F  o.  g ) ) )
41 eqid 2283 . . . . . 6  |-  ( (mulGrp `  S )  ^s  A )  =  ( (mulGrp `  S )  ^s  A )
42 eqid 2283 . . . . . 6  |-  ( Base `  ( (mulGrp `  R
)  ^s  A ) )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) )
43 eqid 2283 . . . . . . . 8  |-  (mulGrp `  S )  =  (mulGrp `  S )
4431, 43rhmmhm 15502 . . . . . . 7  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) ) )
451, 44syl 15 . . . . . 6  |-  ( ph  ->  F  e.  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) ) )
4634, 41, 42, 4, 45pwsco2mhm 14447 . . . . 5  |-  ( ph  ->  ( g  e.  (
Base `  ( (mulGrp `  R )  ^s  A ) )  |->  ( F  o.  g ) )  e.  ( ( (mulGrp `  R )  ^s  A ) MndHom 
( (mulGrp `  S
)  ^s  A ) ) )
4740, 46eqeltrd 2357 . . . 4  |-  ( ph  ->  ( g  e.  B  |->  ( F  o.  g
) )  e.  ( ( (mulGrp `  R
)  ^s  A ) MndHom  ( (mulGrp `  S )  ^s  A ) ) )
48 eqidd 2284 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Y ) )  =  ( Base `  (mulGrp `  Y ) ) )
49 eqidd 2284 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Z ) )  =  ( Base `  (mulGrp `  Z ) ) )
50 eqid 2283 . . . . . . . 8  |-  (mulGrp `  Y )  =  (mulGrp `  Y )
51 eqid 2283 . . . . . . . 8  |-  ( Base `  (mulGrp `  Y )
)  =  ( Base `  (mulGrp `  Y )
)
52 eqid 2283 . . . . . . . 8  |-  ( +g  `  (mulGrp `  Y )
)  =  ( +g  `  (mulGrp `  Y )
)
53 eqid 2283 . . . . . . . 8  |-  ( +g  `  ( (mulGrp `  R
)  ^s  A ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  A ) )
545, 31, 34, 50, 51, 42, 52, 53pwsmgp 15401 . . . . . . 7  |-  ( ( R  e.  Ring  /\  A  e.  V )  ->  (
( Base `  (mulGrp `  Y
) )  =  (
Base `  ( (mulGrp `  R )  ^s  A ) )  /\  ( +g  `  (mulGrp `  Y )
)  =  ( +g  `  ( (mulGrp `  R
)  ^s  A ) ) ) )
553, 4, 54syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( Base `  (mulGrp `  Y ) )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) )  /\  ( +g  `  (mulGrp `  Y ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  A ) ) ) )
5655simpld 445 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Y ) )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) ) )
57 eqid 2283 . . . . . . . 8  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
58 eqid 2283 . . . . . . . 8  |-  ( Base `  (mulGrp `  Z )
)  =  ( Base `  (mulGrp `  Z )
)
59 eqid 2283 . . . . . . . 8  |-  ( Base `  ( (mulGrp `  S
)  ^s  A ) )  =  ( Base `  (
(mulGrp `  S )  ^s  A ) )
60 eqid 2283 . . . . . . . 8  |-  ( +g  `  (mulGrp `  Z )
)  =  ( +g  `  (mulGrp `  Z )
)
61 eqid 2283 . . . . . . . 8  |-  ( +g  `  ( (mulGrp `  S
)  ^s  A ) )  =  ( +g  `  (
(mulGrp `  S )  ^s  A ) )
6210, 43, 41, 57, 58, 59, 60, 61pwsmgp 15401 . . . . . . 7  |-  ( ( S  e.  Ring  /\  A  e.  V )  ->  (
( Base `  (mulGrp `  Z
) )  =  (
Base `  ( (mulGrp `  S )  ^s  A ) )  /\  ( +g  `  (mulGrp `  Z )
)  =  ( +g  `  ( (mulGrp `  S
)  ^s  A ) ) ) )
639, 4, 62syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( Base `  (mulGrp `  Z ) )  =  ( Base `  (
(mulGrp `  S )  ^s  A ) )  /\  ( +g  `  (mulGrp `  Z ) )  =  ( +g  `  (
(mulGrp `  S )  ^s  A ) ) ) )
6463simpld 445 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Z ) )  =  ( Base `  (
(mulGrp `  S )  ^s  A ) ) )
6555simprd 449 . . . . . 6  |-  ( ph  ->  ( +g  `  (mulGrp `  Y ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  A ) ) )
6665proplem3 13593 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  (mulGrp `  Y ) )  /\  y  e.  ( Base `  (mulGrp `  Y )
) ) )  -> 
( x ( +g  `  (mulGrp `  Y )
) y )  =  ( x ( +g  `  ( (mulGrp `  R
)  ^s  A ) ) y ) )
6763simprd 449 . . . . . 6  |-  ( ph  ->  ( +g  `  (mulGrp `  Z ) )  =  ( +g  `  (
(mulGrp `  S )  ^s  A ) ) )
6867proplem3 13593 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  (mulGrp `  Z ) )  /\  y  e.  ( Base `  (mulGrp `  Z )
) ) )  -> 
( x ( +g  `  (mulGrp `  Z )
) y )  =  ( x ( +g  `  ( (mulGrp `  S
)  ^s  A ) ) y ) )
6948, 49, 56, 64, 66, 68mhmpropd 14421 . . . 4  |-  ( ph  ->  ( (mulGrp `  Y
) MndHom  (mulGrp `  Z )
)  =  ( ( (mulGrp `  R )  ^s  A ) MndHom  ( (mulGrp `  S )  ^s  A ) ) )
7047, 69eleqtrrd 2360 . . 3  |-  ( ph  ->  ( g  e.  B  |->  ( F  o.  g
) )  e.  ( (mulGrp `  Y ) MndHom  (mulGrp `  Z ) ) )
7126, 70jca 518 . 2  |-  ( ph  ->  ( ( g  e.  B  |->  ( F  o.  g ) )  e.  ( Y  GrpHom  Z )  /\  ( g  e.  B  |->  ( F  o.  g ) )  e.  ( (mulGrp `  Y
) MndHom  (mulGrp `  Z )
) ) )
7250, 57isrhm 15501 . 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 ) ) ) ) )
7313, 71, 72sylanbrc 645 1  |-  ( ph  ->  ( g  e.  B  |->  ( F  o.  g
) )  e.  ( Y RingHom  Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    e. cmpt 4077    o. ccom 4693   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   Basecbs 13148   +g cplusg 13208    ^s cpws 13347   Mndcmnd 14361   Grpcgrp 14362   MndHom cmhm 14413    GrpHom cghm 14680  mulGrpcmgp 15325   Ringcrg 15337   RingHom crh 15494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-pws 13350  df-0g 13404  df-mnd 14367  df-mhm 14415  df-grp 14489  df-minusg 14490  df-ghm 14681  df-mgp 15326  df-rng 15340  df-ur 15342  df-rnghom 15496
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