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Theorem pwssplit3 26602
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 2283 . 2  |-  ( .s
`  Y )  =  ( .s `  Y
)
3 eqid 2283 . 2  |-  ( .s
`  Z )  =  ( .s `  Z
)
4 eqid 2283 . 2  |-  (Scalar `  Y )  =  (Scalar `  Y )
5 eqid 2283 . 2  |-  (Scalar `  Z )  =  (Scalar `  Z )
6 eqid 2283 . 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 15727 . . 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 4160 . . . 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 15727 . . 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 2283 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
1915, 18pwssca 13395 . . . 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 13395 . . . 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 2317 . 2  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  (Scalar `  Z )  =  (Scalar `  Y ) )
24 lmodgrp 15634 . . 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 26601 . . 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 4216 . . . . . . . 8  |-  { a }  e.  _V
30 xpexg 4800 . . . . . . . 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 2791 . . . . . . 7  |-  b  e. 
_V
33 offres 6092 . . . . . . 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 4989 . . . . . . . 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 5873 . . . . 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 2315 . . . 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 2283 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
42 eqid 2283 . . . . . 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 5529 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  ( Base `  (Scalar `  W
) )  =  (
Base `  (Scalar `  Y
) ) )
4645eleq2d 2350 . . . . . . . 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 13393 . . . . 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 4954 . . . 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 26175 . . . . . 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 5874 . . . 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 2325 . . 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 15644 . . . . . 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 26175 . . . 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 26599 . . . . . 6  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  F : B --> C )
63 ffvelrn 5663 . . . . . 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 13393 . . 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 2325 . 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 15796 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 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   {csn 3640    e. cmpt 4077    X. cxp 4687    |` cres 4691   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   Basecbs 13148  Scalarcsca 13211   .scvsca 13212    ^s cpws 13347   Grpcgrp 14362    GrpHom cghm 14680   LModclmod 15627   LMHom clmhm 15776
This theorem is referenced by:  pwssplit4  26603  pwslnmlem2  26607  frlmsplit2  26655
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-grp 14489  df-minusg 14490  df-ghm 14681  df-mgp 15326  df-rng 15340  df-ur 15342  df-lmod 15629  df-lmhm 15779
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