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Theorem pwssplit2 27189
Description: Splitting for structure powers, part 2: restriction is a group homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Hypotheses
Ref Expression
pwssplit1.y  |-  Y  =  ( W  ^s  U )
pwssplit1.z  |-  Z  =  ( W  ^s  V )
pwssplit1.b  |-  B  =  ( Base `  Y
)
pwssplit1.c  |-  C  =  ( Base `  Z
)
pwssplit1.f  |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )
Assertion
Ref Expression
pwssplit2  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y  GrpHom  Z ) )
Distinct variable groups:    x, Y    x, W    x, U    x, Z    x, V    x, B    x, C    x, X
Allowed substitution hint:    F( x)

Proof of Theorem pwssplit2
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwssplit1.b . 2  |-  B  =  ( Base `  Y
)
2 pwssplit1.c . 2  |-  C  =  ( Base `  Z
)
3 eqid 2283 . 2  |-  ( +g  `  Y )  =  ( +g  `  Y )
4 eqid 2283 . 2  |-  ( +g  `  Z )  =  ( +g  `  Z )
5 simp1 955 . . 3  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  W  e.  Grp )
6 simp2 956 . . 3  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  U  e.  X )
7 pwssplit1.y . . . 4  |-  Y  =  ( W  ^s  U )
87pwsgrp 14606 . . 3  |-  ( ( W  e.  Grp  /\  U  e.  X )  ->  Y  e.  Grp )
95, 6, 8syl2anc 642 . 2  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  Y  e.  Grp )
10 simp3 957 . . . 4  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  V  C_  U )
11 ssexg 4160 . . . 4  |-  ( ( V  C_  U  /\  U  e.  X )  ->  V  e.  _V )
1210, 6, 11syl2anc 642 . . 3  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  V  e.  _V )
13 pwssplit1.z . . . 4  |-  Z  =  ( W  ^s  V )
1413pwsgrp 14606 . . 3  |-  ( ( W  e.  Grp  /\  V  e.  _V )  ->  Z  e.  Grp )
155, 12, 14syl2anc 642 . 2  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  Z  e.  Grp )
16 pwssplit1.f . . 3  |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )
177, 13, 1, 2, 16pwssplit0 27187 . 2  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  F : B --> C )
18 offres 6092 . . . . 5  |-  ( ( a  e.  B  /\  b  e.  B )  ->  ( ( a  o F ( +g  `  W
) b )  |`  V )  =  ( ( a  |`  V )  o F ( +g  `  W ) ( b  |`  V ) ) )
1918adantl 452 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
( a  o F ( +g  `  W
) b )  |`  V )  =  ( ( a  |`  V )  o F ( +g  `  W ) ( b  |`  V ) ) )
205adantr 451 . . . . . 6  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  W  e.  Grp )
21 simpl2 959 . . . . . 6  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  U  e.  X )
22 simprl 732 . . . . . 6  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  a  e.  B )
23 simprr 733 . . . . . 6  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  b  e.  B )
24 eqid 2283 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
257, 1, 20, 21, 22, 23, 24, 3pwsplusgval 13389 . . . . 5  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
a ( +g  `  Y
) b )  =  ( a  o F ( +g  `  W
) b ) )
2625reseq1d 4954 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
( a ( +g  `  Y ) b )  |`  V )  =  ( ( a  o F ( +g  `  W
) b )  |`  V ) )
2716fvtresfn 26763 . . . . . 6  |-  ( a  e.  B  ->  ( F `  a )  =  ( a  |`  V ) )
2816fvtresfn 26763 . . . . . 6  |-  ( b  e.  B  ->  ( F `  b )  =  ( b  |`  V ) )
2927, 28oveqan12d 5877 . . . . 5  |-  ( ( a  e.  B  /\  b  e.  B )  ->  ( ( F `  a )  o F ( +g  `  W
) ( F `  b ) )  =  ( ( a  |`  V )  o F ( +g  `  W
) ( b  |`  V ) ) )
3029adantl 452 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
( F `  a
)  o F ( +g  `  W ) ( F `  b
) )  =  ( ( a  |`  V )  o F ( +g  `  W ) ( b  |`  V ) ) )
3119, 26, 303eqtr4d 2325 . . 3  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
( a ( +g  `  Y ) b )  |`  V )  =  ( ( F `  a
)  o F ( +g  `  W ) ( F `  b
) ) )
321, 3grpcl 14495 . . . . . 6  |-  ( ( Y  e.  Grp  /\  a  e.  B  /\  b  e.  B )  ->  ( a ( +g  `  Y ) b )  e.  B )
33323expb 1152 . . . . 5  |-  ( ( Y  e.  Grp  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
a ( +g  `  Y
) b )  e.  B )
349, 33sylan 457 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
a ( +g  `  Y
) b )  e.  B )
3516fvtresfn 26763 . . . 4  |-  ( ( a ( +g  `  Y
) b )  e.  B  ->  ( F `  ( a ( +g  `  Y ) b ) )  =  ( ( a ( +g  `  Y
) b )  |`  V ) )
3634, 35syl 15 . . 3  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  ( F `  ( a
( +g  `  Y ) b ) )  =  ( ( a ( +g  `  Y ) b )  |`  V ) )
3712adantr 451 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  V  e.  _V )
38 ffvelrn 5663 . . . . . 6  |-  ( ( F : B --> C  /\  a  e.  B )  ->  ( F `  a
)  e.  C )
3917, 38sylan 457 . . . . 5  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  B )  ->  ( F `  a
)  e.  C )
4039adantrr 697 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  ( F `  a )  e.  C )
41 ffvelrn 5663 . . . . . 6  |-  ( ( F : B --> C  /\  b  e.  B )  ->  ( F `  b
)  e.  C )
4217, 41sylan 457 . . . . 5  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  b  e.  B )  ->  ( F `  b
)  e.  C )
4342adantrl 696 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  ( F `  b )  e.  C )
4413, 2, 20, 37, 40, 43, 24, 4pwsplusgval 13389 . . 3  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
( F `  a
) ( +g  `  Z
) ( F `  b ) )  =  ( ( F `  a )  o F ( +g  `  W
) ( F `  b ) ) )
4531, 36, 443eqtr4d 2325 . 2  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  ( F `  ( a
( +g  `  Y ) b ) )  =  ( ( F `  a ) ( +g  `  Z ) ( F `
 b ) ) )
461, 2, 3, 4, 9, 15, 17, 45isghmd 14692 1  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y  GrpHom  Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152    e. cmpt 4077    |` cres 4691   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   Basecbs 13148   +g cplusg 13208    ^s cpws 13347   Grpcgrp 14362    GrpHom cghm 14680
This theorem is referenced by:  pwssplit3  27190
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-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-grp 14489  df-minusg 14490  df-ghm 14681
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