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Theorem pwssplit3 17142
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 2443 . 2  |-  ( .s
`  Y )  =  ( .s `  Y
)
3 eqid 2443 . 2  |-  ( .s
`  Z )  =  ( .s `  Z
)
4 eqid 2443 . 2  |-  (Scalar `  Y )  =  (Scalar `  Y )
5 eqid 2443 . 2  |-  (Scalar `  Z )  =  (Scalar `  Z )
6 eqid 2443 . 2  |-  ( Base `  (Scalar `  Y )
)  =  ( Base `  (Scalar `  Y )
)
7 simp1 988 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  W  e.  LMod )
8 simp2 989 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  U  e.  X )
9 pwssplit1.y . . . 4  |-  Y  =  ( W  ^s  U )
109pwslmod 17051 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  X )  ->  Y  e.  LMod )
117, 8, 10syl2anc 661 . 2  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  Y  e.  LMod )
12 simp3 990 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  V  C_  U )
138, 12ssexd 4439 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  V  e.  _V )
14 pwssplit1.z . . . 4  |-  Z  =  ( W  ^s  V )
1514pwslmod 17051 . . 3  |-  ( ( W  e.  LMod  /\  V  e.  _V )  ->  Z  e.  LMod )
167, 13, 15syl2anc 661 . 2  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  Z  e.  LMod )
17 eqid 2443 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
1814, 17pwssca 14434 . . . 4  |-  ( ( W  e.  LMod  /\  V  e.  _V )  ->  (Scalar `  W )  =  (Scalar `  Z ) )
197, 13, 18syl2anc 661 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  (Scalar `  W )  =  (Scalar `  Z ) )
209, 17pwssca 14434 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  X )  ->  (Scalar `  W )  =  (Scalar `  Y ) )
217, 8, 20syl2anc 661 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  (Scalar `  W )  =  (Scalar `  Y ) )
2219, 21eqtr3d 2477 . 2  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  (Scalar `  Z )  =  (Scalar `  Y ) )
23 lmodgrp 16955 . . 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 17141 . . 3  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y  GrpHom  Z ) )
2723, 26syl3an1 1251 . 2  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y  GrpHom  Z ) )
28 snex 4533 . . . . . . . 8  |-  { a }  e.  _V
29 xpexg 6507 . . . . . . . 8  |-  ( ( U  e.  X  /\  { a }  e.  _V )  ->  ( U  X.  { a } )  e.  _V )
308, 28, 29sylancl 662 . . . . . . 7  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  ( U  X.  { a } )  e.  _V )
31 vex 2975 . . . . . . 7  |-  b  e. 
_V
32 offres 6572 . . . . . . 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 662 . . . . . 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 465 . . . . 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 5144 . . . . . . . 8  |-  ( V 
C_  U  ->  (
( U  X.  {
a } )  |`  V )  =  ( V  X.  { a } ) )
36353ad2ant3 1011 . . . . . . 7  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  (
( U  X.  {
a } )  |`  V )  =  ( V  X.  { a } ) )
3736adantr 465 . . . . . 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 6106 . . . . 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 2475 . . . 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 2443 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
41 eqid 2443 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
42 simpl1 991 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  ->  W  e.  LMod )
43 simpl2 992 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  ->  U  e.  X )
4421fveq2d 5695 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  ( Base `  (Scalar `  W
) )  =  (
Base `  (Scalar `  Y
) ) )
4544eleq2d 2510 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  (
a  e.  ( Base `  (Scalar `  W )
)  <->  a  e.  (
Base `  (Scalar `  Y
) ) ) )
4645biimpar 485 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  ( Base `  (Scalar `  Y )
) )  ->  a  e.  ( Base `  (Scalar `  W ) ) )
4746adantrr 716 . . . . . 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 756 . . . . . 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 14432 . . . . 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 5109 . . . 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 5775 . . . . . 6  |-  ( b  e.  B  ->  ( F `  b )  =  ( b  |`  V ) )
5251ad2antll 728 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  (
Base `  (Scalar `  Y
) )  /\  b  e.  B ) )  -> 
( F `  b
)  =  ( b  |`  V ) )
5352oveq2d 6107 . . . 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 2485 . . 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 16965 . . . . . 6  |-  ( ( Y  e.  LMod  /\  a  e.  ( Base `  (Scalar `  Y ) )  /\  b  e.  B )  ->  ( a ( .s
`  Y ) b )  e.  B )
56553expb 1188 . . . . 5  |-  ( ( Y  e.  LMod  /\  (
a  e.  ( Base `  (Scalar `  Y )
)  /\  b  e.  B ) )  -> 
( a ( .s
`  Y ) b )  e.  B )
5711, 56sylan 471 . . . 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 5775 . . . 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 465 . . . 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 17139 . . . . . 6  |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  F : B --> C )
6261ffvelrnda 5843 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  /\  b  e.  B )  ->  ( F `  b
)  e.  C )
6362adantrl 715 . . . 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 14432 . . 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 2485 . 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 17120 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2972    C_ wss 3328   {csn 3877    e. cmpt 4350    X. cxp 4838    |` cres 4842   ` cfv 5418  (class class class)co 6091    oFcof 6318   Basecbs 14174  Scalarcsca 14241   .scvsca 14242    ^s cpws 14385   Grpcgrp 15410    GrpHom cghm 15744   LModclmod 16948   LMHom clmhm 17100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-fz 11438  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-plusg 14251  df-mulr 14252  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-hom 14262  df-cco 14263  df-0g 14380  df-prds 14386  df-pws 14388  df-mnd 15415  df-grp 15545  df-minusg 15546  df-ghm 15745  df-mgp 16592  df-ur 16604  df-rng 16647  df-lmod 16950  df-lmhm 17103
This theorem is referenced by:  frlmsplit2  18197  pwssplit4  29442  pwslnmlem2  29446
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