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Theorem pwsco1rhm 15526
Description: Right composition with a function on the index sets yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
Hypotheses
Ref Expression
pwsco1rhm.y  |-  Y  =  ( R  ^s  A )
pwsco1rhm.z  |-  Z  =  ( R  ^s  B )
pwsco1rhm.c  |-  C  =  ( Base `  Z
)
pwsco1rhm.r  |-  ( ph  ->  R  e.  Ring )
pwsco1rhm.a  |-  ( ph  ->  A  e.  V )
pwsco1rhm.b  |-  ( ph  ->  B  e.  W )
pwsco1rhm.f  |-  ( ph  ->  F : A --> B )
Assertion
Ref Expression
pwsco1rhm  |-  ( ph  ->  ( g  e.  C  |->  ( g  o.  F
) )  e.  ( Z RingHom  Y ) )
Distinct variable groups:    A, g    B, g    ph, g    R, g   
g, Y    C, g    g, F    g, Z
Allowed substitution hints:    V( g)    W( g)

Proof of Theorem pwsco1rhm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwsco1rhm.r . . . 4  |-  ( ph  ->  R  e.  Ring )
2 pwsco1rhm.b . . . 4  |-  ( ph  ->  B  e.  W )
3 pwsco1rhm.z . . . . 5  |-  Z  =  ( R  ^s  B )
43pwsrng 15414 . . . 4  |-  ( ( R  e.  Ring  /\  B  e.  W )  ->  Z  e.  Ring )
51, 2, 4syl2anc 642 . . 3  |-  ( ph  ->  Z  e.  Ring )
6 pwsco1rhm.a . . . 4  |-  ( ph  ->  A  e.  V )
7 pwsco1rhm.y . . . . 5  |-  Y  =  ( R  ^s  A )
87pwsrng 15414 . . . 4  |-  ( ( R  e.  Ring  /\  A  e.  V )  ->  Y  e.  Ring )
91, 6, 8syl2anc 642 . . 3  |-  ( ph  ->  Y  e.  Ring )
105, 9jca 518 . 2  |-  ( ph  ->  ( Z  e.  Ring  /\  Y  e.  Ring )
)
11 pwsco1rhm.c . . . . 5  |-  C  =  ( Base `  Z
)
12 rngmnd 15366 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
131, 12syl 15 . . . . 5  |-  ( ph  ->  R  e.  Mnd )
14 pwsco1rhm.f . . . . 5  |-  ( ph  ->  F : A --> B )
157, 3, 11, 13, 6, 2, 14pwsco1mhm 14462 . . . 4  |-  ( ph  ->  ( g  e.  C  |->  ( g  o.  F
) )  e.  ( Z MndHom  Y ) )
16 rnggrp 15362 . . . . . 6  |-  ( Z  e.  Ring  ->  Z  e. 
Grp )
175, 16syl 15 . . . . 5  |-  ( ph  ->  Z  e.  Grp )
18 rnggrp 15362 . . . . . 6  |-  ( Y  e.  Ring  ->  Y  e. 
Grp )
199, 18syl 15 . . . . 5  |-  ( ph  ->  Y  e.  Grp )
20 ghmmhmb 14710 . . . . 5  |-  ( ( Z  e.  Grp  /\  Y  e.  Grp )  ->  ( Z  GrpHom  Y )  =  ( Z MndHom  Y
) )
2117, 19, 20syl2anc 642 . . . 4  |-  ( ph  ->  ( Z  GrpHom  Y )  =  ( Z MndHom  Y
) )
2215, 21eleqtrrd 2373 . . 3  |-  ( ph  ->  ( g  e.  C  |->  ( g  o.  F
) )  e.  ( Z  GrpHom  Y ) )
23 eqid 2296 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
243, 23pwsbas 13402 . . . . . . . . 9  |-  ( ( R  e.  Mnd  /\  B  e.  W )  ->  ( ( Base `  R
)  ^m  B )  =  ( Base `  Z
) )
2513, 2, 24syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( ( Base `  R
)  ^m  B )  =  ( Base `  Z
) )
2625, 11syl6eqr 2346 . . . . . . 7  |-  ( ph  ->  ( ( Base `  R
)  ^m  B )  =  C )
27 eqid 2296 . . . . . . . . . 10  |-  (mulGrp `  R )  =  (mulGrp `  R )
2827rngmgp 15363 . . . . . . . . 9  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
291, 28syl 15 . . . . . . . 8  |-  ( ph  ->  (mulGrp `  R )  e.  Mnd )
30 eqid 2296 . . . . . . . . 9  |-  ( (mulGrp `  R )  ^s  B )  =  ( (mulGrp `  R )  ^s  B )
3127, 23mgpbas 15347 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
3230, 31pwsbas 13402 . . . . . . . 8  |-  ( ( (mulGrp `  R )  e.  Mnd  /\  B  e.  W )  ->  (
( Base `  R )  ^m  B )  =  (
Base `  ( (mulGrp `  R )  ^s  B ) ) )
3329, 2, 32syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( ( Base `  R
)  ^m  B )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) ) )
3426, 33eqtr3d 2330 . . . . . 6  |-  ( ph  ->  C  =  ( Base `  ( (mulGrp `  R
)  ^s  B ) ) )
35 mpteq1 4116 . . . . . 6  |-  ( C  =  ( Base `  (
(mulGrp `  R )  ^s  B ) )  -> 
( g  e.  C  |->  ( g  o.  F
) )  =  ( g  e.  ( Base `  ( (mulGrp `  R
)  ^s  B ) )  |->  ( g  o.  F ) ) )
3634, 35syl 15 . . . . 5  |-  ( ph  ->  ( g  e.  C  |->  ( g  o.  F
) )  =  ( g  e.  ( Base `  ( (mulGrp `  R
)  ^s  B ) )  |->  ( g  o.  F ) ) )
37 eqid 2296 . . . . . 6  |-  ( (mulGrp `  R )  ^s  A )  =  ( (mulGrp `  R )  ^s  A )
38 eqid 2296 . . . . . 6  |-  ( Base `  ( (mulGrp `  R
)  ^s  B ) )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) )
3937, 30, 38, 29, 6, 2, 14pwsco1mhm 14462 . . . . 5  |-  ( ph  ->  ( g  e.  (
Base `  ( (mulGrp `  R )  ^s  B ) )  |->  ( g  o.  F ) )  e.  ( ( (mulGrp `  R )  ^s  B ) MndHom 
( (mulGrp `  R
)  ^s  A ) ) )
4036, 39eqeltrd 2370 . . . 4  |-  ( ph  ->  ( g  e.  C  |->  ( g  o.  F
) )  e.  ( ( (mulGrp `  R
)  ^s  B ) MndHom  ( (mulGrp `  R )  ^s  A ) ) )
41 eqidd 2297 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Z ) )  =  ( Base `  (mulGrp `  Z ) ) )
42 eqidd 2297 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Y ) )  =  ( Base `  (mulGrp `  Y ) ) )
43 eqid 2296 . . . . . . . 8  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
44 eqid 2296 . . . . . . . 8  |-  ( Base `  (mulGrp `  Z )
)  =  ( Base `  (mulGrp `  Z )
)
45 eqid 2296 . . . . . . . 8  |-  ( +g  `  (mulGrp `  Z )
)  =  ( +g  `  (mulGrp `  Z )
)
46 eqid 2296 . . . . . . . 8  |-  ( +g  `  ( (mulGrp `  R
)  ^s  B ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  B ) )
473, 27, 30, 43, 44, 38, 45, 46pwsmgp 15417 . . . . . . 7  |-  ( ( R  e.  Ring  /\  B  e.  W )  ->  (
( Base `  (mulGrp `  Z
) )  =  (
Base `  ( (mulGrp `  R )  ^s  B ) )  /\  ( +g  `  (mulGrp `  Z )
)  =  ( +g  `  ( (mulGrp `  R
)  ^s  B ) ) ) )
481, 2, 47syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( Base `  (mulGrp `  Z ) )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) )  /\  ( +g  `  (mulGrp `  Z ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  B ) ) ) )
4948simpld 445 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Z ) )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) ) )
50 eqid 2296 . . . . . . . 8  |-  (mulGrp `  Y )  =  (mulGrp `  Y )
51 eqid 2296 . . . . . . . 8  |-  ( Base `  (mulGrp `  Y )
)  =  ( Base `  (mulGrp `  Y )
)
52 eqid 2296 . . . . . . . 8  |-  ( Base `  ( (mulGrp `  R
)  ^s  A ) )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) )
53 eqid 2296 . . . . . . . 8  |-  ( +g  `  (mulGrp `  Y )
)  =  ( +g  `  (mulGrp `  Y )
)
54 eqid 2296 . . . . . . . 8  |-  ( +g  `  ( (mulGrp `  R
)  ^s  A ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  A ) )
557, 27, 37, 50, 51, 52, 53, 54pwsmgp 15417 . . . . . . 7  |-  ( ( R  e.  Ring  /\  A  e.  V )  ->  (
( Base `  (mulGrp `  Y
) )  =  (
Base `  ( (mulGrp `  R )  ^s  A ) )  /\  ( +g  `  (mulGrp `  Y )
)  =  ( +g  `  ( (mulGrp `  R
)  ^s  A ) ) ) )
561, 6, 55syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( Base `  (mulGrp `  Y ) )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) )  /\  ( +g  `  (mulGrp `  Y ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  A ) ) ) )
5756simpld 445 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Y ) )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) ) )
5848simprd 449 . . . . . 6  |-  ( ph  ->  ( +g  `  (mulGrp `  Z ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  B ) ) )
5958proplem3 13609 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  (mulGrp `  Z ) )  /\  y  e.  ( Base `  (mulGrp `  Z )
) ) )  -> 
( x ( +g  `  (mulGrp `  Z )
) y )  =  ( x ( +g  `  ( (mulGrp `  R
)  ^s  B ) ) y ) )
6056simprd 449 . . . . . 6  |-  ( ph  ->  ( +g  `  (mulGrp `  Y ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  A ) ) )
6160proplem3 13609 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  (mulGrp `  Y ) )  /\  y  e.  ( Base `  (mulGrp `  Y )
) ) )  -> 
( x ( +g  `  (mulGrp `  Y )
) y )  =  ( x ( +g  `  ( (mulGrp `  R
)  ^s  A ) ) y ) )
6241, 42, 49, 57, 59, 61mhmpropd 14437 . . . 4  |-  ( ph  ->  ( (mulGrp `  Z
) MndHom  (mulGrp `  Y )
)  =  ( ( (mulGrp `  R )  ^s  B ) MndHom  ( (mulGrp `  R )  ^s  A ) ) )
6340, 62eleqtrrd 2373 . . 3  |-  ( ph  ->  ( g  e.  C  |->  ( g  o.  F
) )  e.  ( (mulGrp `  Z ) MndHom  (mulGrp `  Y ) ) )
6422, 63jca 518 . 2  |-  ( ph  ->  ( ( g  e.  C  |->  ( g  o.  F ) )  e.  ( Z  GrpHom  Y )  /\  ( g  e.  C  |->  ( g  o.  F ) )  e.  ( (mulGrp `  Z
) MndHom  (mulGrp `  Y )
) ) )
6543, 50isrhm 15517 . 2  |-  ( ( g  e.  C  |->  ( g  o.  F ) )  e.  ( Z RingHom  Y )  <->  ( ( Z  e.  Ring  /\  Y  e.  Ring )  /\  (
( g  e.  C  |->  ( g  o.  F
) )  e.  ( Z  GrpHom  Y )  /\  ( g  e.  C  |->  ( g  o.  F
) )  e.  ( (mulGrp `  Z ) MndHom  (mulGrp `  Y ) ) ) ) )
6610, 64, 65sylanbrc 645 1  |-  ( ph  ->  ( g  e.  C  |->  ( g  o.  F
) )  e.  ( Z RingHom  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    e. cmpt 4093    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   Basecbs 13164   +g cplusg 13224    ^s cpws 13363   Mndcmnd 14377   Grpcgrp 14378   MndHom cmhm 14429    GrpHom cghm 14696  mulGrpcmgp 15341   Ringcrg 15353   RingHom crh 15510
This theorem is referenced by:  evl1rhm  19428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-prds 13364  df-pws 13366  df-0g 13420  df-mnd 14383  df-mhm 14431  df-grp 14505  df-minusg 14506  df-ghm 14697  df-mgp 15342  df-rng 15356  df-ur 15358  df-rnghom 15512
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