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Theorem pwsco2rhm 17200
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 17181 . . . . 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 17077 . . . 4  |-  ( ( R  e.  Ring  /\  A  e.  V )  ->  Y  e.  Ring )
73, 4, 6syl2anc 661 . . 3  |-  ( ph  ->  Y  e.  Ring )
8 rhmrcl2 17182 . . . . 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 17077 . . . 4  |-  ( ( S  e.  Ring  /\  A  e.  V )  ->  Z  e.  Ring )
129, 4, 11syl2anc 661 . . 3  |-  ( ph  ->  Z  e.  Ring )
137, 12jca 532 . 2  |-  ( ph  ->  ( Y  e.  Ring  /\  Z  e.  Ring )
)
14 pwsco2rhm.b . . . . 5  |-  B  =  ( Base `  Y
)
15 rhmghm 17187 . . . . . . 7  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
161, 15syl 16 . . . . . 6  |-  ( ph  ->  F  e.  ( R 
GrpHom  S ) )
17 ghmmhm 16091 . . . . . 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 15824 . . . 4  |-  ( ph  ->  ( g  e.  B  |->  ( F  o.  g
) )  e.  ( Y MndHom  Z ) )
20 rnggrp 17017 . . . . . 6  |-  ( Y  e.  Ring  ->  Y  e. 
Grp )
217, 20syl 16 . . . . 5  |-  ( ph  ->  Y  e.  Grp )
22 rnggrp 17017 . . . . . 6  |-  ( Z  e.  Ring  ->  Z  e. 
Grp )
2312, 22syl 16 . . . . 5  |-  ( ph  ->  Z  e.  Grp )
24 ghmmhmb 16092 . . . . 5  |-  ( ( Y  e.  Grp  /\  Z  e.  Grp )  ->  ( Y  GrpHom  Z )  =  ( Y MndHom  Z
) )
2521, 23, 24syl2anc 661 . . . 4  |-  ( ph  ->  ( Y  GrpHom  Z )  =  ( Y MndHom  Z
) )
2619, 25eleqtrrd 2558 . . 3  |-  ( ph  ->  ( g  e.  B  |->  ( F  o.  g
) )  e.  ( Y  GrpHom  Z ) )
27 eqid 2467 . . . . 5  |-  ( (mulGrp `  R )  ^s  A )  =  ( (mulGrp `  R )  ^s  A )
28 eqid 2467 . . . . 5  |-  ( (mulGrp `  S )  ^s  A )  =  ( (mulGrp `  S )  ^s  A )
29 eqid 2467 . . . . 5  |-  ( Base `  ( (mulGrp `  R
)  ^s  A ) )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) )
30 eqid 2467 . . . . . . 7  |-  (mulGrp `  R )  =  (mulGrp `  R )
31 eqid 2467 . . . . . . 7  |-  (mulGrp `  S )  =  (mulGrp `  S )
3230, 31rhmmhm 17184 . . . . . 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 15824 . . . 4  |-  ( ph  ->  ( g  e.  (
Base `  ( (mulGrp `  R )  ^s  A ) )  |->  ( F  o.  g ) )  e.  ( ( (mulGrp `  R )  ^s  A ) MndHom 
( (mulGrp `  S
)  ^s  A ) ) )
35 eqid 2467 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  R )
365, 35pwsbas 14745 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  A  e.  V )  ->  (
( Base `  R )  ^m  A )  =  (
Base `  Y )
)
373, 4, 36syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( ( Base `  R
)  ^m  A )  =  ( Base `  Y
) )
3837, 14syl6eqr 2526 . . . . . 6  |-  ( ph  ->  ( ( Base `  R
)  ^m  A )  =  B )
3930rngmgp 17018 . . . . . . . 8  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
403, 39syl 16 . . . . . . 7  |-  ( ph  ->  (mulGrp `  R )  e.  Mnd )
4130, 35mgpbas 16961 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
4227, 41pwsbas 14745 . . . . . . 7  |-  ( ( (mulGrp `  R )  e.  Mnd  /\  A  e.  V )  ->  (
( Base `  R )  ^m  A )  =  (
Base `  ( (mulGrp `  R )  ^s  A ) ) )
4340, 4, 42syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( Base `  R
)  ^m  A )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) ) )
4438, 43eqtr3d 2510 . . . . 5  |-  ( ph  ->  B  =  ( Base `  ( (mulGrp `  R
)  ^s  A ) ) )
4544mpteq1d 4528 . . . 4  |-  ( ph  ->  ( g  e.  B  |->  ( F  o.  g
) )  =  ( g  e.  ( Base `  ( (mulGrp `  R
)  ^s  A ) )  |->  ( F  o.  g ) ) )
46 eqidd 2468 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Y ) )  =  ( Base `  (mulGrp `  Y ) ) )
47 eqidd 2468 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Z ) )  =  ( Base `  (mulGrp `  Z ) ) )
48 eqid 2467 . . . . . . . 8  |-  (mulGrp `  Y )  =  (mulGrp `  Y )
49 eqid 2467 . . . . . . . 8  |-  ( Base `  (mulGrp `  Y )
)  =  ( Base `  (mulGrp `  Y )
)
50 eqid 2467 . . . . . . . 8  |-  ( +g  `  (mulGrp `  Y )
)  =  ( +g  `  (mulGrp `  Y )
)
51 eqid 2467 . . . . . . . 8  |-  ( +g  `  ( (mulGrp `  R
)  ^s  A ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  A ) )
525, 30, 27, 48, 49, 29, 50, 51pwsmgp 17080 . . . . . . 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 661 . . . . . 6  |-  ( ph  ->  ( ( Base `  (mulGrp `  Y ) )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) )  /\  ( +g  `  (mulGrp `  Y ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  A ) ) ) )
5453simpld 459 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Y ) )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) ) )
55 eqid 2467 . . . . . . . 8  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
56 eqid 2467 . . . . . . . 8  |-  ( Base `  (mulGrp `  Z )
)  =  ( Base `  (mulGrp `  Z )
)
57 eqid 2467 . . . . . . . 8  |-  ( Base `  ( (mulGrp `  S
)  ^s  A ) )  =  ( Base `  (
(mulGrp `  S )  ^s  A ) )
58 eqid 2467 . . . . . . . 8  |-  ( +g  `  (mulGrp `  Z )
)  =  ( +g  `  (mulGrp `  Z )
)
59 eqid 2467 . . . . . . . 8  |-  ( +g  `  ( (mulGrp `  S
)  ^s  A ) )  =  ( +g  `  (
(mulGrp `  S )  ^s  A ) )
6010, 31, 28, 55, 56, 57, 58, 59pwsmgp 17080 . . . . . . 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 661 . . . . . 6  |-  ( ph  ->  ( ( Base `  (mulGrp `  Z ) )  =  ( Base `  (
(mulGrp `  S )  ^s  A ) )  /\  ( +g  `  (mulGrp `  Z ) )  =  ( +g  `  (
(mulGrp `  S )  ^s  A ) ) ) )
6261simpld 459 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Z ) )  =  ( Base `  (
(mulGrp `  S )  ^s  A ) ) )
6353simprd 463 . . . . . 6  |-  ( ph  ->  ( +g  `  (mulGrp `  Y ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  A ) ) )
6463proplem3 14949 . . . . 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 463 . . . . . 6  |-  ( ph  ->  ( +g  `  (mulGrp `  Z ) )  =  ( +g  `  (
(mulGrp `  S )  ^s  A ) ) )
6665proplem3 14949 . . . . 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 15795 . . . 4  |-  ( ph  ->  ( (mulGrp `  Y
) MndHom  (mulGrp `  Z )
)  =  ( ( (mulGrp `  R )  ^s  A ) MndHom  ( (mulGrp `  S )  ^s  A ) ) )
6834, 45, 673eltr4d 2570 . . 3  |-  ( ph  ->  ( g  e.  B  |->  ( F  o.  g
) )  e.  ( (mulGrp `  Y ) MndHom  (mulGrp `  Z ) ) )
6926, 68jca 532 . 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 17183 . 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 664 1  |-  ( ph  ->  ( g  e.  B  |->  ( F  o.  g
) )  e.  ( Y RingHom  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    |-> cmpt 4505    o. ccom 5003   ` cfv 5588  (class class class)co 6285    ^m cmap 7421   Basecbs 14493   +g cplusg 14558    ^s cpws 14705   Mndcmnd 15729   Grpcgrp 15730   MndHom cmhm 15787    GrpHom cghm 16078  mulGrpcmgp 16955   Ringcrg 17012   RingHom crh 17174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-ixp 7471  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-uz 11084  df-fz 11674  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-plusg 14571  df-mulr 14572  df-sca 14574  df-vsca 14575  df-ip 14576  df-tset 14577  df-ple 14578  df-ds 14580  df-hom 14582  df-cco 14583  df-0g 14700  df-prds 14706  df-pws 14708  df-mnd 15735  df-mhm 15789  df-grp 15871  df-minusg 15872  df-ghm 16079  df-mgp 16956  df-ur 16968  df-rng 17014  df-rnghom 17177
This theorem is referenced by: (None)
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