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Theorem pwsco1rhm 15510
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 15398 . . . 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 15398 . . . 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 15350 . . . . . 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 14446 . . . 4  |-  ( ph  ->  ( g  e.  C  |->  ( g  o.  F
) )  e.  ( Z MndHom  Y ) )
16 rnggrp 15346 . . . . . 6  |-  ( Z  e.  Ring  ->  Z  e. 
Grp )
175, 16syl 15 . . . . 5  |-  ( ph  ->  Z  e.  Grp )
18 rnggrp 15346 . . . . . 6  |-  ( Y  e.  Ring  ->  Y  e. 
Grp )
199, 18syl 15 . . . . 5  |-  ( ph  ->  Y  e.  Grp )
20 ghmmhmb 14694 . . . . 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 2360 . . 3  |-  ( ph  ->  ( g  e.  C  |->  ( g  o.  F
) )  e.  ( Z  GrpHom  Y ) )
23 eqid 2283 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
243, 23pwsbas 13386 . . . . . . . . 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 2333 . . . . . . 7  |-  ( ph  ->  ( ( Base `  R
)  ^m  B )  =  C )
27 eqid 2283 . . . . . . . . . 10  |-  (mulGrp `  R )  =  (mulGrp `  R )
2827rngmgp 15347 . . . . . . . . 9  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
291, 28syl 15 . . . . . . . 8  |-  ( ph  ->  (mulGrp `  R )  e.  Mnd )
30 eqid 2283 . . . . . . . . 9  |-  ( (mulGrp `  R )  ^s  B )  =  ( (mulGrp `  R )  ^s  B )
3127, 23mgpbas 15331 . . . . . . . . 9  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
3230, 31pwsbas 13386 . . . . . . . 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 2317 . . . . . 6  |-  ( ph  ->  C  =  ( Base `  ( (mulGrp `  R
)  ^s  B ) ) )
35 mpteq1 4100 . . . . . 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 2283 . . . . . 6  |-  ( (mulGrp `  R )  ^s  A )  =  ( (mulGrp `  R )  ^s  A )
38 eqid 2283 . . . . . 6  |-  ( Base `  ( (mulGrp `  R
)  ^s  B ) )  =  ( Base `  (
(mulGrp `  R )  ^s  B ) )
3937, 30, 38, 29, 6, 2, 14pwsco1mhm 14446 . . . . 5  |-  ( ph  ->  ( g  e.  (
Base `  ( (mulGrp `  R )  ^s  B ) )  |->  ( g  o.  F ) )  e.  ( ( (mulGrp `  R )  ^s  B ) MndHom 
( (mulGrp `  R
)  ^s  A ) ) )
4036, 39eqeltrd 2357 . . . 4  |-  ( ph  ->  ( g  e.  C  |->  ( g  o.  F
) )  e.  ( ( (mulGrp `  R
)  ^s  B ) MndHom  ( (mulGrp `  R )  ^s  A ) ) )
41 eqidd 2284 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Z ) )  =  ( Base `  (mulGrp `  Z ) ) )
42 eqidd 2284 . . . . 5  |-  ( ph  ->  ( Base `  (mulGrp `  Y ) )  =  ( Base `  (mulGrp `  Y ) ) )
43 eqid 2283 . . . . . . . 8  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
44 eqid 2283 . . . . . . . 8  |-  ( Base `  (mulGrp `  Z )
)  =  ( Base `  (mulGrp `  Z )
)
45 eqid 2283 . . . . . . . 8  |-  ( +g  `  (mulGrp `  Z )
)  =  ( +g  `  (mulGrp `  Z )
)
46 eqid 2283 . . . . . . . 8  |-  ( +g  `  ( (mulGrp `  R
)  ^s  B ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  B ) )
473, 27, 30, 43, 44, 38, 45, 46pwsmgp 15401 . . . . . . 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 2283 . . . . . . . 8  |-  (mulGrp `  Y )  =  (mulGrp `  Y )
51 eqid 2283 . . . . . . . 8  |-  ( Base `  (mulGrp `  Y )
)  =  ( Base `  (mulGrp `  Y )
)
52 eqid 2283 . . . . . . . 8  |-  ( Base `  ( (mulGrp `  R
)  ^s  A ) )  =  ( Base `  (
(mulGrp `  R )  ^s  A ) )
53 eqid 2283 . . . . . . . 8  |-  ( +g  `  (mulGrp `  Y )
)  =  ( +g  `  (mulGrp `  Y )
)
54 eqid 2283 . . . . . . . 8  |-  ( +g  `  ( (mulGrp `  R
)  ^s  A ) )  =  ( +g  `  (
(mulGrp `  R )  ^s  A ) )
557, 27, 37, 50, 51, 52, 53, 54pwsmgp 15401 . . . . . . 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 13593 . . . . 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 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 ) )
6241, 42, 49, 57, 59, 61mhmpropd 14421 . . . 4  |-  ( ph  ->  ( (mulGrp `  Z
) MndHom  (mulGrp `  Y )
)  =  ( ( (mulGrp `  R )  ^s  B ) MndHom  ( (mulGrp `  R )  ^s  A ) ) )
6340, 62eleqtrrd 2360 . . 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 15501 . 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 1623    e. wcel 1684    e. cmpt 4077    o. ccom 4693   -->wf 5251   ` 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 is referenced by:  evl1rhm  19412
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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