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Theorem pwssplit2 27166
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 2436 . 2  |-  ( +g  `  Y )  =  ( +g  `  Y )
4 eqid 2436 . 2  |-  ( +g  `  Z )  =  ( +g  `  Z )
5 simp1 957 . . 3  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  W  e.  Grp )
6 simp2 958 . . 3  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  U  e.  X )
7 pwssplit1.y . . . 4  |-  Y  =  ( W  ^s  U )
87pwsgrp 14929 . . 3  |-  ( ( W  e.  Grp  /\  U  e.  X )  ->  Y  e.  Grp )
95, 6, 8syl2anc 643 . 2  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  Y  e.  Grp )
10 simp3 959 . . . 4  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  V  C_  U )
116, 10ssexd 4350 . . 3  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  V  e.  _V )
12 pwssplit1.z . . . 4  |-  Z  =  ( W  ^s  V )
1312pwsgrp 14929 . . 3  |-  ( ( W  e.  Grp  /\  V  e.  _V )  ->  Z  e.  Grp )
145, 11, 13syl2anc 643 . 2  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  Z  e.  Grp )
15 pwssplit1.f . . 3  |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )
167, 12, 1, 2, 15pwssplit0 27164 . 2  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  F : B --> C )
17 offres 6319 . . . . 5  |-  ( ( a  e.  B  /\  b  e.  B )  ->  ( ( a  o F ( +g  `  W
) b )  |`  V )  =  ( ( a  |`  V )  o F ( +g  `  W ) ( b  |`  V ) ) )
1817adantl 453 . . . 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 ) ) )
195adantr 452 . . . . . 6  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  W  e.  Grp )
20 simpl2 961 . . . . . 6  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  U  e.  X )
21 simprl 733 . . . . . 6  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  a  e.  B )
22 simprr 734 . . . . . 6  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  b  e.  B )
23 eqid 2436 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
247, 1, 19, 20, 21, 22, 23, 3pwsplusgval 13712 . . . . 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 ) )
2524reseq1d 5145 . . . 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 ) )
2615fvtresfn 26744 . . . . . 6  |-  ( a  e.  B  ->  ( F `  a )  =  ( a  |`  V ) )
2715fvtresfn 26744 . . . . . 6  |-  ( b  e.  B  ->  ( F `  b )  =  ( b  |`  V ) )
2826, 27oveqan12d 6100 . . . . 5  |-  ( ( a  e.  B  /\  b  e.  B )  ->  ( ( F `  a )  o F ( +g  `  W
) ( F `  b ) )  =  ( ( a  |`  V )  o F ( +g  `  W
) ( b  |`  V ) ) )
2928adantl 453 . . . 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 ) ) )
3018, 25, 293eqtr4d 2478 . . 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
) ) )
311, 3grpcl 14818 . . . . . 6  |-  ( ( Y  e.  Grp  /\  a  e.  B  /\  b  e.  B )  ->  ( a ( +g  `  Y ) b )  e.  B )
32313expb 1154 . . . . 5  |-  ( ( Y  e.  Grp  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
a ( +g  `  Y
) b )  e.  B )
339, 32sylan 458 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
a ( +g  `  Y
) b )  e.  B )
3415fvtresfn 26744 . . . 4  |-  ( ( a ( +g  `  Y
) b )  e.  B  ->  ( F `  ( a ( +g  `  Y ) b ) )  =  ( ( a ( +g  `  Y
) b )  |`  V ) )
3533, 34syl 16 . . 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 ) )
3611adantr 452 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  V  e.  _V )
3716ffvelrnda 5870 . . . . 5  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  B )  ->  ( F `  a
)  e.  C )
3837adantrr 698 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  ( F `  a )  e.  C )
3916ffvelrnda 5870 . . . . 5  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  b  e.  B )  ->  ( F `  b
)  e.  C )
4039adantrl 697 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  ( F `  b )  e.  C )
4112, 2, 19, 36, 38, 40, 23, 4pwsplusgval 13712 . . 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 ) ) )
4230, 35, 413eqtr4d 2478 . 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 ) ) )
431, 2, 3, 4, 9, 14, 16, 42isghmd 15015 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2956    C_ wss 3320    e. cmpt 4266    |` cres 4880   ` cfv 5454  (class class class)co 6081    o Fcof 6303   Basecbs 13469   +g cplusg 13529    ^s cpws 13670   Grpcgrp 14685    GrpHom cghm 15003
This theorem is referenced by:  pwssplit3  27167
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-hom 13553  df-cco 13554  df-prds 13671  df-pws 13673  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-ghm 15004
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