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Theorem pwsco2rhm 16849
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 16831 . . . . 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 16729 . . . 4  |-  ( ( R  e.  Ring  /\  A  e.  V )  ->  Y  e.  Ring )
73, 4, 6syl2anc 661 . . 3  |-  ( ph  ->  Y  e.  Ring )
8 rhmrcl2 16832 . . . . 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 16729 . . . 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 16837 . . . . . . 7  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
161, 15syl 16 . . . . . 6  |-  ( ph  ->  F  e.  ( R 
GrpHom  S ) )
17 ghmmhm 15778 . . . . . 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 15520 . . . 4  |-  ( ph  ->  ( g  e.  B  |->  ( F  o.  g
) )  e.  ( Y MndHom  Z ) )
20 rnggrp 16672 . . . . . 6  |-  ( Y  e.  Ring  ->  Y  e. 
Grp )
217, 20syl 16 . . . . 5  |-  ( ph  ->  Y  e.  Grp )
22 rnggrp 16672 . . . . . 6  |-  ( Z  e.  Ring  ->  Z  e. 
Grp )
2312, 22syl 16 . . . . 5  |-  ( ph  ->  Z  e.  Grp )
24 ghmmhmb 15779 . . . . 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 2520 . . 3  |-  ( ph  ->  ( g  e.  B  |->  ( F  o.  g
) )  e.  ( Y  GrpHom  Z ) )
27 eqid 2443 . . . . 5  |-  ( (mulGrp `  R )  ^s  A )  =  ( (mulGrp `  R )  ^s  A )
28 eqid 2443 . . . . 5  |-  ( (mulGrp `  S )  ^s  A )  =  ( (mulGrp `  S )  ^s  A )
29 eqid 2443 . . . . 5  |-  ( Base `  ( (mulGrp `  R
)  ^s  A ) )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) )
30 eqid 2443 . . . . . . 7  |-  (mulGrp `  R )  =  (mulGrp `  R )
31 eqid 2443 . . . . . . 7  |-  (mulGrp `  S )  =  (mulGrp `  S )
3230, 31rhmmhm 16834 . . . . . 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 15520 . . . 4  |-  ( ph  ->  ( g  e.  (
Base `  ( (mulGrp `  R )  ^s  A ) )  |->  ( F  o.  g ) )  e.  ( ( (mulGrp `  R )  ^s  A ) MndHom 
( (mulGrp `  S
)  ^s  A ) ) )
35 eqid 2443 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  R )
365, 35pwsbas 14446 . . . . . . . 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 2493 . . . . . 6  |-  ( ph  ->  ( ( Base `  R
)  ^m  A )  =  B )
3930rngmgp 16673 . . . . . . . 8  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
403, 39syl 16 . . . . . . 7  |-  ( ph  ->  (mulGrp `  R )  e.  Mnd )
4130, 35mgpbas 16619 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
4227, 41pwsbas 14446 . . . . . . 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 2477 . . . . 5  |-  ( ph  ->  B  =  ( Base `  ( (mulGrp `  R
)  ^s  A ) ) )
4544mpteq1d 4394 . . . 4  |-  ( ph  ->  ( g  e.  B  |->  ( F  o.  g
) )  =  ( g  e.  ( Base `  ( (mulGrp `  R
)  ^s  A ) )  |->  ( F  o.  g ) ) )
46 eqidd 2444 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Y ) )  =  ( Base `  (mulGrp `  Y ) ) )
47 eqidd 2444 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Z ) )  =  ( Base `  (mulGrp `  Z ) ) )
48 eqid 2443 . . . . . . . 8  |-  (mulGrp `  Y )  =  (mulGrp `  Y )
49 eqid 2443 . . . . . . . 8  |-  ( Base `  (mulGrp `  Y )
)  =  ( Base `  (mulGrp `  Y )
)
50 eqid 2443 . . . . . . . 8  |-  ( +g  `  (mulGrp `  Y )
)  =  ( +g  `  (mulGrp `  Y )
)
51 eqid 2443 . . . . . . . 8  |-  ( +g  `  ( (mulGrp `  R
)  ^s  A ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  A ) )
525, 30, 27, 48, 49, 29, 50, 51pwsmgp 16732 . . . . . . 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 2443 . . . . . . . 8  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
56 eqid 2443 . . . . . . . 8  |-  ( Base `  (mulGrp `  Z )
)  =  ( Base `  (mulGrp `  Z )
)
57 eqid 2443 . . . . . . . 8  |-  ( Base `  ( (mulGrp `  S
)  ^s  A ) )  =  ( Base `  (
(mulGrp `  S )  ^s  A ) )
58 eqid 2443 . . . . . . . 8  |-  ( +g  `  (mulGrp `  Z )
)  =  ( +g  `  (mulGrp `  Z )
)
59 eqid 2443 . . . . . . . 8  |-  ( +g  `  ( (mulGrp `  S
)  ^s  A ) )  =  ( +g  `  (
(mulGrp `  S )  ^s  A ) )
6010, 31, 28, 55, 56, 57, 58, 59pwsmgp 16732 . . . . . . 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 14650 . . . . 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 14650 . . . . 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 15491 . . . 4  |-  ( ph  ->  ( (mulGrp `  Y
) MndHom  (mulGrp `  Z )
)  =  ( ( (mulGrp `  R )  ^s  A ) MndHom  ( (mulGrp `  S )  ^s  A ) ) )
6834, 45, 673eltr4d 2524 . . 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 16833 . 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 1369    e. wcel 1756    e. cmpt 4371    o. ccom 4865   ` cfv 5439  (class class class)co 6112    ^m cmap 7235   Basecbs 14195   +g cplusg 14259    ^s cpws 14406   Mndcmnd 15430   Grpcgrp 15431   MndHom cmhm 15483    GrpHom cghm 15765  mulGrpcmgp 16613   Ringcrg 16667   RingHom crh 16826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-of 6341  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-ixp 7285  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-sup 7712  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-fz 11459  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-plusg 14272  df-mulr 14273  df-sca 14275  df-vsca 14276  df-ip 14277  df-tset 14278  df-ple 14279  df-ds 14281  df-hom 14283  df-cco 14284  df-0g 14401  df-prds 14407  df-pws 14409  df-mnd 15436  df-mhm 15485  df-grp 15566  df-minusg 15567  df-ghm 15766  df-mgp 16614  df-ur 16626  df-rng 16669  df-rnghom 16828
This theorem is referenced by: (None)
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