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Theorem pwssplit3 27293
Description: Splitting for structure powers, part 3: restriction is a module 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
pwssplit3  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y LMHom  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 pwssplit3
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwssplit1.b . 2  |-  B  =  ( Base `  Y
)
2 eqid 2296 . 2  |-  ( .s
`  Y )  =  ( .s `  Y
)
3 eqid 2296 . 2  |-  ( .s
`  Z )  =  ( .s `  Z
)
4 eqid 2296 . 2  |-  (Scalar `  Y )  =  (Scalar `  Y )
5 eqid 2296 . 2  |-  (Scalar `  Z )  =  (Scalar `  Z )
6 eqid 2296 . 2  |-  ( Base `  (Scalar `  Y )
)  =  ( Base `  (Scalar `  Y )
)
7 simp1 955 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  W  e.  LMod )
8 simp2 956 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  U  e.  X )
9 pwssplit1.y . . . 4  |-  Y  =  ( W  ^s  U )
109pwslmod 15743 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  X )  ->  Y  e.  LMod )
117, 8, 10syl2anc 642 . 2  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  Y  e.  LMod )
12 simp3 957 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  V  C_  U )
13 ssexg 4176 . . . 4  |-  ( ( V  C_  U  /\  U  e.  X )  ->  V  e.  _V )
1412, 8, 13syl2anc 642 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  V  e.  _V )
15 pwssplit1.z . . . 4  |-  Z  =  ( W  ^s  V )
1615pwslmod 15743 . . 3  |-  ( ( W  e.  LMod  /\  V  e.  _V )  ->  Z  e.  LMod )
177, 14, 16syl2anc 642 . 2  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  Z  e.  LMod )
18 eqid 2296 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
1915, 18pwssca 13411 . . . 4  |-  ( ( W  e.  LMod  /\  V  e.  _V )  ->  (Scalar `  W )  =  (Scalar `  Z ) )
207, 14, 19syl2anc 642 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  (Scalar `  W )  =  (Scalar `  Z ) )
219, 18pwssca 13411 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  X )  ->  (Scalar `  W )  =  (Scalar `  Y ) )
227, 8, 21syl2anc 642 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  (Scalar `  W )  =  (Scalar `  Y ) )
2320, 22eqtr3d 2330 . 2  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  (Scalar `  Z )  =  (Scalar `  Y ) )
24 lmodgrp 15650 . . 3  |-  ( W  e.  LMod  ->  W  e. 
Grp )
25 pwssplit1.c . . . 4  |-  C  =  ( Base `  Z
)
26 pwssplit1.f . . . 4  |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )
279, 15, 1, 25, 26pwssplit2 27292 . . 3  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y  GrpHom  Z ) )
2824, 27syl3an1 1215 . 2  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y  GrpHom  Z ) )
29 snex 4232 . . . . . . . 8  |-  { a }  e.  _V
30 xpexg 4816 . . . . . . . 8  |-  ( ( U  e.  X  /\  { a }  e.  _V )  ->  ( U  X.  { a } )  e.  _V )
318, 29, 30sylancl 643 . . . . . . 7  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  ( U  X.  { a } )  e.  _V )
32 vex 2804 . . . . . . 7  |-  b  e. 
_V
33 offres 6108 . . . . . . 7  |-  ( ( ( U  X.  {
a } )  e. 
_V  /\  b  e.  _V )  ->  ( ( ( U  X.  {
a } )  o F ( .s `  W ) b )  |`  V )  =  ( ( ( U  X.  { a } )  |`  V )  o F ( .s `  W
) ( b  |`  V ) ) )
3431, 32, 33sylancl 643 . . . . . 6  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  (
( ( U  X.  { a } )  o F ( .s
`  W ) b )  |`  V )  =  ( ( ( U  X.  { a } )  |`  V )  o F ( .s
`  W ) ( b  |`  V )
) )
3534adantr 451 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
( ( ( U  X.  { a } )  o F ( .s `  W ) b )  |`  V )  =  ( ( ( U  X.  { a } )  |`  V )  o F ( .s
`  W ) ( b  |`  V )
) )
36 xpssres 5005 . . . . . . . 8  |-  ( V 
C_  U  ->  (
( U  X.  {
a } )  |`  V )  =  ( V  X.  { a } ) )
37363ad2ant3 978 . . . . . . 7  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  (
( U  X.  {
a } )  |`  V )  =  ( V  X.  { a } ) )
3837adantr 451 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
( ( U  X.  { a } )  |`  V )  =  ( V  X.  { a } ) )
3938oveq1d 5889 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
( ( ( U  X.  { a } )  |`  V )  o F ( .s `  W ) ( b  |`  V ) )  =  ( ( V  X.  { a } )  o F ( .s
`  W ) ( b  |`  V )
) )
4035, 39eqtrd 2328 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
( ( ( U  X.  { a } )  o F ( .s `  W ) b )  |`  V )  =  ( ( V  X.  { a } )  o F ( .s `  W ) ( b  |`  V ) ) )
41 eqid 2296 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
42 eqid 2296 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
43 simpl1 958 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  ->  W  e.  LMod )
44 simpl2 959 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  ->  U  e.  X )
4522fveq2d 5545 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  ( Base `  (Scalar `  W
) )  =  (
Base `  (Scalar `  Y
) ) )
4645eleq2d 2363 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  (
a  e.  ( Base `  (Scalar `  W )
)  <->  a  e.  (
Base `  (Scalar `  Y
) ) ) )
4746biimpar 471 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  ( Base `  (Scalar `  Y )
) )  ->  a  e.  ( Base `  (Scalar `  W ) ) )
4847adantrr 697 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
a  e.  ( Base `  (Scalar `  W )
) )
49 simprr 733 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
b  e.  B )
509, 1, 41, 2, 18, 42, 43, 44, 48, 49pwsvscafval 13409 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
( a ( .s
`  Y ) b )  =  ( ( U  X.  { a } )  o F ( .s `  W
) b ) )
5150reseq1d 4970 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
( ( a ( .s `  Y ) b )  |`  V )  =  ( ( ( U  X.  { a } )  o F ( .s `  W
) b )  |`  V ) )
5226fvtresfn 26866 . . . . . 6  |-  ( b  e.  B  ->  ( F `  b )  =  ( b  |`  V ) )
5352ad2antll 709 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
( F `  b
)  =  ( b  |`  V ) )
5453oveq2d 5890 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
( ( V  X.  { a } )  o F ( .s
`  W ) ( F `  b ) )  =  ( ( V  X.  { a } )  o F ( .s `  W
) ( b  |`  V ) ) )
5540, 51, 543eqtr4d 2338 . . 3  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
( ( a ( .s `  Y ) b )  |`  V )  =  ( ( V  X.  { a } )  o F ( .s `  W ) ( F `  b
) ) )
561, 4, 2, 6lmodvscl 15660 . . . . . 6  |-  ( ( Y  e.  LMod  /\  a  e.  ( Base `  (Scalar `  Y ) )  /\  b  e.  B )  ->  ( a ( .s
`  Y ) b )  e.  B )
57563expb 1152 . . . . 5  |-  ( ( Y  e.  LMod  /\  (
a  e.  ( Base `  (Scalar `  Y )
)  /\  b  e.  B ) )  -> 
( a ( .s
`  Y ) b )  e.  B )
5811, 57sylan 457 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
( a ( .s
`  Y ) b )  e.  B )
5926fvtresfn 26866 . . . 4  |-  ( ( a ( .s `  Y ) b )  e.  B  ->  ( F `  ( a
( .s `  Y
) b ) )  =  ( ( a ( .s `  Y
) b )  |`  V ) )
6058, 59syl 15 . . 3  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
( F `  (
a ( .s `  Y ) b ) )  =  ( ( a ( .s `  Y ) b )  |`  V ) )
6114adantr 451 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  ->  V  e.  _V )
629, 15, 1, 25, 26pwssplit0 27290 . . . . . 6  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  F : B --> C )
63 ffvelrn 5679 . . . . . 6  |-  ( ( F : B --> C  /\  b  e.  B )  ->  ( F `  b
)  e.  C )
6462, 63sylan 457 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  b  e.  B )  ->  ( F `  b
)  e.  C )
6564adantrl 696 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
( F `  b
)  e.  C )
6615, 25, 41, 3, 18, 42, 43, 61, 48, 65pwsvscafval 13409 . . 3  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
( a ( .s
`  Z ) ( F `  b ) )  =  ( ( V  X.  { a } )  o F ( .s `  W
) ( F `  b ) ) )
6755, 60, 663eqtr4d 2338 . 2  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
( F `  (
a ( .s `  Y ) b ) )  =  ( a ( .s `  Z
) ( F `  b ) ) )
681, 2, 3, 4, 5, 6, 11, 17, 23, 28, 67islmhmd 15812 1  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y LMHom  Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   {csn 3653    e. cmpt 4093    X. cxp 4703    |` cres 4707   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   Basecbs 13164  Scalarcsca 13227   .scvsca 13228    ^s cpws 13363   Grpcgrp 14378    GrpHom cghm 14696   LModclmod 15643   LMHom clmhm 15792
This theorem is referenced by:  pwssplit4  27294  pwslnmlem2  27298  frlmsplit2  27346
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-grp 14505  df-minusg 14506  df-ghm 14697  df-mgp 15342  df-rng 15356  df-ur 15358  df-lmod 15645  df-lmhm 15795
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