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Theorem pwssplit3 18283
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 2422 . 2  |-  ( .s
`  Y )  =  ( .s `  Y
)
3 eqid 2422 . 2  |-  ( .s
`  Z )  =  ( .s `  Z
)
4 eqid 2422 . 2  |-  (Scalar `  Y )  =  (Scalar `  Y )
5 eqid 2422 . 2  |-  (Scalar `  Z )  =  (Scalar `  Z )
6 eqid 2422 . 2  |-  ( Base `  (Scalar `  Y )
)  =  ( Base `  (Scalar `  Y )
)
7 simp1 1005 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  W  e.  LMod )
8 simp2 1006 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  U  e.  X )
9 pwssplit1.y . . . 4  |-  Y  =  ( W  ^s  U )
109pwslmod 18192 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  X )  ->  Y  e.  LMod )
117, 8, 10syl2anc 665 . 2  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  Y  e.  LMod )
12 simp3 1007 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  V  C_  U )
138, 12ssexd 4571 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  V  e.  _V )
14 pwssplit1.z . . . 4  |-  Z  =  ( W  ^s  V )
1514pwslmod 18192 . . 3  |-  ( ( W  e.  LMod  /\  V  e.  _V )  ->  Z  e.  LMod )
167, 13, 15syl2anc 665 . 2  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  Z  e.  LMod )
17 eqid 2422 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
1814, 17pwssca 15393 . . . 4  |-  ( ( W  e.  LMod  /\  V  e.  _V )  ->  (Scalar `  W )  =  (Scalar `  Z ) )
197, 13, 18syl2anc 665 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  (Scalar `  W )  =  (Scalar `  Z ) )
209, 17pwssca 15393 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  X )  ->  (Scalar `  W )  =  (Scalar `  Y ) )
217, 8, 20syl2anc 665 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  (Scalar `  W )  =  (Scalar `  Y ) )
2219, 21eqtr3d 2465 . 2  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  (Scalar `  Z )  =  (Scalar `  Y ) )
23 lmodgrp 18097 . . 3  |-  ( W  e.  LMod  ->  W  e. 
Grp )
24 pwssplit1.c . . . 4  |-  C  =  ( Base `  Z
)
25 pwssplit1.f . . . 4  |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )
269, 14, 1, 24, 25pwssplit2 18282 . . 3  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y  GrpHom  Z ) )
2723, 26syl3an1 1297 . 2  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y  GrpHom  Z ) )
28 snex 4662 . . . . . . . 8  |-  { a }  e.  _V
29 xpexg 6607 . . . . . . . 8  |-  ( ( U  e.  X  /\  { a }  e.  _V )  ->  ( U  X.  { a } )  e.  _V )
308, 28, 29sylancl 666 . . . . . . 7  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  ( U  X.  { a } )  e.  _V )
31 vex 3083 . . . . . . 7  |-  b  e. 
_V
32 offres 6802 . . . . . . 7  |-  ( ( ( U  X.  {
a } )  e. 
_V  /\  b  e.  _V )  ->  ( ( ( U  X.  {
a } )  oF ( .s `  W ) b )  |`  V )  =  ( ( ( U  X.  { a } )  |`  V )  oF ( .s `  W
) ( b  |`  V ) ) )
3330, 31, 32sylancl 666 . . . . . 6  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  (
( ( U  X.  { a } )  oF ( .s
`  W ) b )  |`  V )  =  ( ( ( U  X.  { a } )  |`  V )  oF ( .s
`  W ) ( b  |`  V )
) )
3433adantr 466 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
( ( ( U  X.  { a } )  oF ( .s `  W ) b )  |`  V )  =  ( ( ( U  X.  { a } )  |`  V )  oF ( .s
`  W ) ( b  |`  V )
) )
35 xpssres 5158 . . . . . . . 8  |-  ( V 
C_  U  ->  (
( U  X.  {
a } )  |`  V )  =  ( V  X.  { a } ) )
36353ad2ant3 1028 . . . . . . 7  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  (
( U  X.  {
a } )  |`  V )  =  ( V  X.  { a } ) )
3736adantr 466 . . . . . 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 } ) )
3837oveq1d 6320 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
( ( ( U  X.  { a } )  |`  V )  oF ( .s
`  W ) ( b  |`  V )
)  =  ( ( V  X.  { a } )  oF ( .s `  W
) ( b  |`  V ) ) )
3934, 38eqtrd 2463 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
( ( ( U  X.  { a } )  oF ( .s `  W ) b )  |`  V )  =  ( ( V  X.  { a } )  oF ( .s `  W ) ( b  |`  V ) ) )
40 eqid 2422 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
41 eqid 2422 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
42 simpl1 1008 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  ->  W  e.  LMod )
43 simpl2 1009 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  ->  U  e.  X )
4421fveq2d 5885 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  ( Base `  (Scalar `  W
) )  =  (
Base `  (Scalar `  Y
) ) )
4544eleq2d 2492 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  (
a  e.  ( Base `  (Scalar `  W )
)  <->  a  e.  (
Base `  (Scalar `  Y
) ) ) )
4645biimpar 487 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  ( Base `  (Scalar `  Y )
) )  ->  a  e.  ( Base `  (Scalar `  W ) ) )
4746adantrr 721 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
a  e.  ( Base `  (Scalar `  W )
) )
48 simprr 764 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
b  e.  B )
499, 1, 40, 2, 17, 41, 42, 43, 47, 48pwsvscafval 15391 . . . . 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 } )  oF ( .s `  W
) b ) )
5049reseq1d 5123 . . . 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 } )  oF ( .s `  W
) b )  |`  V ) )
5125fvtresfn 5966 . . . . . 6  |-  ( b  e.  B  ->  ( F `  b )  =  ( b  |`  V ) )
5251ad2antll 733 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
( F `  b
)  =  ( b  |`  V ) )
5352oveq2d 6321 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
( ( V  X.  { a } )  oF ( .s
`  W ) ( F `  b ) )  =  ( ( V  X.  { a } )  oF ( .s `  W
) ( b  |`  V ) ) )
5439, 50, 533eqtr4d 2473 . . 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 } )  oF ( .s `  W ) ( F `  b
) ) )
551, 4, 2, 6lmodvscl 18107 . . . . . 6  |-  ( ( Y  e.  LMod  /\  a  e.  ( Base `  (Scalar `  Y ) )  /\  b  e.  B )  ->  ( a ( .s
`  Y ) b )  e.  B )
56553expb 1206 . . . . 5  |-  ( ( Y  e.  LMod  /\  (
a  e.  ( Base `  (Scalar `  Y )
)  /\  b  e.  B ) )  -> 
( a ( .s
`  Y ) b )  e.  B )
5711, 56sylan 473 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
( a ( .s
`  Y ) b )  e.  B )
5825fvtresfn 5966 . . . 4  |-  ( ( a ( .s `  Y ) b )  e.  B  ->  ( F `  ( a
( .s `  Y
) b ) )  =  ( ( a ( .s `  Y
) b )  |`  V ) )
5957, 58syl 17 . . 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 ) )
6013adantr 466 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  ->  V  e.  _V )
619, 14, 1, 24, 25pwssplit0 18280 . . . . . 6  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  F : B --> C )
6261ffvelrnda 6037 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  b  e.  B )  ->  ( F `  b
)  e.  C )
6362adantrl 720 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
( F `  b
)  e.  C )
6414, 24, 40, 3, 17, 41, 42, 60, 47, 63pwsvscafval 15391 . . 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 } )  oF ( .s `  W
) ( F `  b ) ) )
6554, 59, 643eqtr4d 2473 . 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 ) ) )
661, 2, 3, 4, 5, 6, 11, 16, 22, 27, 65islmhmd 18261 1  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y LMHom  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   _Vcvv 3080    C_ wss 3436   {csn 3998    |-> cmpt 4482    X. cxp 4851    |` cres 4855   ` cfv 5601  (class class class)co 6305    oFcof 6543   Basecbs 15120  Scalarcsca 15192   .scvsca 15193    ^s cpws 15344   Grpcgrp 16668    GrpHom cghm 16879   LModclmod 18090   LMHom clmhm 18241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-map 7485  df-ixp 7534  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-sup 7965  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-fz 11792  df-struct 15122  df-ndx 15123  df-slot 15124  df-base 15125  df-sets 15126  df-plusg 15202  df-mulr 15203  df-sca 15205  df-vsca 15206  df-ip 15207  df-tset 15208  df-ple 15209  df-ds 15211  df-hom 15213  df-cco 15214  df-0g 15339  df-prds 15345  df-pws 15347  df-mgm 16487  df-sgrp 16526  df-mnd 16536  df-grp 16672  df-minusg 16673  df-ghm 16880  df-mgp 17723  df-ur 17735  df-ring 17781  df-lmod 18092  df-lmhm 18244
This theorem is referenced by:  frlmsplit2  19329  pwssplit4  35917  pwslnmlem2  35921
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