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Theorem pwssplit3 27169
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 2438 . 2  |-  ( .s
`  Y )  =  ( .s `  Y
)
3 eqid 2438 . 2  |-  ( .s
`  Z )  =  ( .s `  Z
)
4 eqid 2438 . 2  |-  (Scalar `  Y )  =  (Scalar `  Y )
5 eqid 2438 . 2  |-  (Scalar `  Z )  =  (Scalar `  Z )
6 eqid 2438 . 2  |-  ( Base `  (Scalar `  Y )
)  =  ( Base `  (Scalar `  Y )
)
7 simp1 958 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  W  e.  LMod )
8 simp2 959 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  U  e.  X )
9 pwssplit1.y . . . 4  |-  Y  =  ( W  ^s  U )
109pwslmod 16048 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  X )  ->  Y  e.  LMod )
117, 8, 10syl2anc 644 . 2  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  Y  e.  LMod )
12 simp3 960 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  V  C_  U )
138, 12ssexd 4352 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  V  e.  _V )
14 pwssplit1.z . . . 4  |-  Z  =  ( W  ^s  V )
1514pwslmod 16048 . . 3  |-  ( ( W  e.  LMod  /\  V  e.  _V )  ->  Z  e.  LMod )
167, 13, 15syl2anc 644 . 2  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  Z  e.  LMod )
17 eqid 2438 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
1814, 17pwssca 13720 . . . 4  |-  ( ( W  e.  LMod  /\  V  e.  _V )  ->  (Scalar `  W )  =  (Scalar `  Z ) )
197, 13, 18syl2anc 644 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  (Scalar `  W )  =  (Scalar `  Z ) )
209, 17pwssca 13720 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  X )  ->  (Scalar `  W )  =  (Scalar `  Y ) )
217, 8, 20syl2anc 644 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  (Scalar `  W )  =  (Scalar `  Y ) )
2219, 21eqtr3d 2472 . 2  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  (Scalar `  Z )  =  (Scalar `  Y ) )
23 lmodgrp 15959 . . 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 27168 . . 3  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y  GrpHom  Z ) )
2723, 26syl3an1 1218 . 2  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y  GrpHom  Z ) )
28 snex 4407 . . . . . . . 8  |-  { a }  e.  _V
29 xpexg 4991 . . . . . . . 8  |-  ( ( U  e.  X  /\  { a }  e.  _V )  ->  ( U  X.  { a } )  e.  _V )
308, 28, 29sylancl 645 . . . . . . 7  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  ( U  X.  { a } )  e.  _V )
31 vex 2961 . . . . . . 7  |-  b  e. 
_V
32 offres 6321 . . . . . . 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 ) ) )
3330, 31, 32sylancl 645 . . . . . 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 )
) )
3433adantr 453 . . . . 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 )
) )
35 xpssres 5182 . . . . . . . 8  |-  ( V 
C_  U  ->  (
( U  X.  {
a } )  |`  V )  =  ( V  X.  { a } ) )
36353ad2ant3 981 . . . . . . 7  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  (
( U  X.  {
a } )  |`  V )  =  ( V  X.  { a } ) )
3736adantr 453 . . . . . 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 6098 . . . . 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 )
) )
3934, 38eqtrd 2470 . . . 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 ) ) )
40 eqid 2438 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
41 eqid 2438 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
42 simpl1 961 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  ->  W  e.  LMod )
43 simpl2 962 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  ->  U  e.  X )
4421fveq2d 5734 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  ( Base `  (Scalar `  W
) )  =  (
Base `  (Scalar `  Y
) ) )
4544eleq2d 2505 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  (
a  e.  ( Base `  (Scalar `  W )
)  <->  a  e.  (
Base `  (Scalar `  Y
) ) ) )
4645biimpar 473 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  ( Base `  (Scalar `  Y )
) )  ->  a  e.  ( Base `  (Scalar `  W ) ) )
4746adantrr 699 . . . . . 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 735 . . . . . 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 13718 . . . . 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 ) )
5049reseq1d 5147 . . . 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 ) )
5125fvtresfn 26746 . . . . . 6  |-  ( b  e.  B  ->  ( F `  b )  =  ( b  |`  V ) )
5251ad2antll 711 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
( F `  b
)  =  ( b  |`  V ) )
5352oveq2d 6099 . . . 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 ) ) )
5439, 50, 533eqtr4d 2480 . . 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
) ) )
551, 4, 2, 6lmodvscl 15969 . . . . . 6  |-  ( ( Y  e.  LMod  /\  a  e.  ( Base `  (Scalar `  Y ) )  /\  b  e.  B )  ->  ( a ( .s
`  Y ) b )  e.  B )
56553expb 1155 . . . . 5  |-  ( ( Y  e.  LMod  /\  (
a  e.  ( Base `  (Scalar `  Y )
)  /\  b  e.  B ) )  -> 
( a ( .s
`  Y ) b )  e.  B )
5711, 56sylan 459 . . . 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 26746 . . . 4  |-  ( ( a ( .s `  Y ) b )  e.  B  ->  ( F `  ( a
( .s `  Y
) b ) )  =  ( ( a ( .s `  Y
) b )  |`  V ) )
5957, 58syl 16 . . 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 453 . . . 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 27166 . . . . . 6  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  F : B --> C )
6261ffvelrnda 5872 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  b  e.  B )  ->  ( F `  b
)  e.  C )
6362adantrl 698 . . . 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 13718 . . 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 ) ) )
6554, 59, 643eqtr4d 2480 . 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 16117 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322   {csn 3816    e. cmpt 4268    X. cxp 4878    |` cres 4882   ` cfv 5456  (class class class)co 6083    o Fcof 6305   Basecbs 13471  Scalarcsca 13534   .scvsca 13535    ^s cpws 13672   Grpcgrp 14687    GrpHom cghm 15005   LModclmod 15952   LMHom clmhm 16097
This theorem is referenced by:  pwssplit4  27170  pwslnmlem2  27174  frlmsplit2  27222
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-map 7022  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-fz 11046  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-plusg 13544  df-mulr 13545  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-hom 13555  df-cco 13556  df-prds 13673  df-pws 13675  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-ghm 15006  df-mgp 15651  df-rng 15665  df-ur 15667  df-lmod 15954  df-lmhm 16100
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