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Theorem pwsvscafval 13747
Description: Scalar multiplication in a structure power is pointwise. (Contributed by Mario Carneiro, 11-Jan-2015.)
Hypotheses
Ref Expression
pwsvscaval.y  |-  Y  =  ( R  ^s  I )
pwsvscaval.b  |-  B  =  ( Base `  Y
)
pwsvscaval.s  |-  .x.  =  ( .s `  R )
pwsvscaval.t  |-  .xb  =  ( .s `  Y )
pwsvscaval.f  |-  F  =  (Scalar `  R )
pwsvscaval.k  |-  K  =  ( Base `  F
)
pwsvscaval.r  |-  ( ph  ->  R  e.  V )
pwsvscaval.i  |-  ( ph  ->  I  e.  W )
pwsvscaval.a  |-  ( ph  ->  A  e.  K )
pwsvscaval.x  |-  ( ph  ->  X  e.  B )
Assertion
Ref Expression
pwsvscafval  |-  ( ph  ->  ( A  .xb  X
)  =  ( ( I  X.  { A } )  o F 
.x.  X ) )

Proof of Theorem pwsvscafval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pwsvscaval.t . . . 4  |-  .xb  =  ( .s `  Y )
2 pwsvscaval.r . . . . . 6  |-  ( ph  ->  R  e.  V )
3 pwsvscaval.i . . . . . 6  |-  ( ph  ->  I  e.  W )
4 pwsvscaval.y . . . . . . 7  |-  Y  =  ( R  ^s  I )
5 pwsvscaval.f . . . . . . 7  |-  F  =  (Scalar `  R )
64, 5pwsval 13739 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Y  =  ( F
X_s ( I  X.  { R } ) ) )
72, 3, 6syl2anc 644 . . . . 5  |-  ( ph  ->  Y  =  ( F
X_s ( I  X.  { R } ) ) )
87fveq2d 5761 . . . 4  |-  ( ph  ->  ( .s `  Y
)  =  ( .s
`  ( F X_s (
I  X.  { R } ) ) ) )
91, 8syl5eq 2486 . . 3  |-  ( ph  -> 
.xb  =  ( .s
`  ( F X_s (
I  X.  { R } ) ) ) )
109oveqd 6127 . 2  |-  ( ph  ->  ( A  .xb  X
)  =  ( A ( .s `  ( F X_s ( I  X.  { R } ) ) ) X ) )
11 eqid 2442 . . 3  |-  ( F
X_s ( I  X.  { R } ) )  =  ( F X_s ( I  X.  { R } ) )
12 eqid 2442 . . 3  |-  ( Base `  ( F X_s ( I  X.  { R } ) ) )  =  ( Base `  ( F X_s ( I  X.  { R } ) ) )
13 eqid 2442 . . 3  |-  ( .s
`  ( F X_s (
I  X.  { R } ) ) )  =  ( .s `  ( F X_s ( I  X.  { R } ) ) )
14 pwsvscaval.k . . 3  |-  K  =  ( Base `  F
)
15 fvex 5771 . . . . 5  |-  (Scalar `  R )  e.  _V
165, 15eqeltri 2512 . . . 4  |-  F  e. 
_V
1716a1i 11 . . 3  |-  ( ph  ->  F  e.  _V )
18 fnconstg 5660 . . . 4  |-  ( R  e.  V  ->  (
I  X.  { R } )  Fn  I
)
192, 18syl 16 . . 3  |-  ( ph  ->  ( I  X.  { R } )  Fn  I
)
20 pwsvscaval.a . . 3  |-  ( ph  ->  A  e.  K )
21 pwsvscaval.x . . . 4  |-  ( ph  ->  X  e.  B )
22 pwsvscaval.b . . . . 5  |-  B  =  ( Base `  Y
)
237fveq2d 5761 . . . . 5  |-  ( ph  ->  ( Base `  Y
)  =  ( Base `  ( F X_s ( I  X.  { R } ) ) ) )
2422, 23syl5eq 2486 . . . 4  |-  ( ph  ->  B  =  ( Base `  ( F X_s ( I  X.  { R } ) ) ) )
2521, 24eleqtrd 2518 . . 3  |-  ( ph  ->  X  e.  ( Base `  ( F X_s ( I  X.  { R } ) ) ) )
2611, 12, 13, 14, 17, 3, 19, 20, 25prdsvscaval 13732 . 2  |-  ( ph  ->  ( A ( .s
`  ( F X_s (
I  X.  { R } ) ) ) X )  =  ( x  e.  I  |->  ( A ( .s `  ( ( I  X.  { R } ) `  x ) ) ( X `  x ) ) ) )
27 fvconst2g 5974 . . . . . . . 8  |-  ( ( R  e.  V  /\  x  e.  I )  ->  ( ( I  X.  { R } ) `  x )  =  R )
282, 27sylan 459 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  (
( I  X.  { R } ) `  x
)  =  R )
2928fveq2d 5761 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( .s `  ( ( I  X.  { R }
) `  x )
)  =  ( .s
`  R ) )
30 pwsvscaval.s . . . . . 6  |-  .x.  =  ( .s `  R )
3129, 30syl6eqr 2492 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( .s `  ( ( I  X.  { R }
) `  x )
)  =  .x.  )
3231oveqd 6127 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( A ( .s `  ( ( I  X.  { R } ) `  x ) ) ( X `  x ) )  =  ( A 
.x.  ( X `  x ) ) )
3332mpteq2dva 4320 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( A ( .s
`  ( ( I  X.  { R }
) `  x )
) ( X `  x ) ) )  =  ( x  e.  I  |->  ( A  .x.  ( X `  x ) ) ) )
3420adantr 453 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  A  e.  K )
35 fvex 5771 . . . . 5  |-  ( X `
 x )  e. 
_V
3635a1i 11 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( X `  x )  e.  _V )
37 fconstmpt 4950 . . . . 5  |-  ( I  X.  { A }
)  =  ( x  e.  I  |->  A )
3837a1i 11 . . . 4  |-  ( ph  ->  ( I  X.  { A } )  =  ( x  e.  I  |->  A ) )
39 eqid 2442 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
404, 39, 22, 2, 3, 21pwselbas 13742 . . . . 5  |-  ( ph  ->  X : I --> ( Base `  R ) )
4140feqmptd 5808 . . . 4  |-  ( ph  ->  X  =  ( x  e.  I  |->  ( X `
 x ) ) )
423, 34, 36, 38, 41offval2 6351 . . 3  |-  ( ph  ->  ( ( I  X.  { A } )  o F  .x.  X )  =  ( x  e.  I  |->  ( A  .x.  ( X `  x ) ) ) )
4333, 42eqtr4d 2477 . 2  |-  ( ph  ->  ( x  e.  I  |->  ( A ( .s
`  ( ( I  X.  { R }
) `  x )
) ( X `  x ) ) )  =  ( ( I  X.  { A }
)  o F  .x.  X ) )
4410, 26, 433eqtrd 2478 1  |-  ( ph  ->  ( A  .xb  X
)  =  ( ( I  X.  { A } )  o F 
.x.  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1727   _Vcvv 2962   {csn 3838    e. cmpt 4291    X. cxp 4905    Fn wfn 5478   ` cfv 5483  (class class class)co 6110    o Fcof 6332   Basecbs 13500  Scalarcsca 13563   .scvsca 13564   X_scprds 13700    ^s cpws 13701
This theorem is referenced by:  pwsvscaval  13748  pwsdiaglmhm  16164  pwssplit3  27205  frlmvscafval  27245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6334  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-oadd 6757  df-er 6934  df-map 7049  df-ixp 7093  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-sup 7475  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-2 10089  df-3 10090  df-4 10091  df-5 10092  df-6 10093  df-7 10094  df-8 10095  df-9 10096  df-10 10097  df-n0 10253  df-z 10314  df-dec 10414  df-uz 10520  df-fz 11075  df-struct 13502  df-ndx 13503  df-slot 13504  df-base 13505  df-plusg 13573  df-mulr 13574  df-sca 13576  df-vsca 13577  df-tset 13579  df-ple 13580  df-ds 13582  df-hom 13584  df-cco 13585  df-prds 13702  df-pws 13704
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