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Theorem pwsvscafval 14437
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 } )  oF  .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 14429 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Y  =  ( F
X_s ( I  X.  { R } ) ) )
72, 3, 6syl2anc 661 . . . . 5  |-  ( ph  ->  Y  =  ( F
X_s ( I  X.  { R } ) ) )
87fveq2d 5700 . . . 4  |-  ( ph  ->  ( .s `  Y
)  =  ( .s
`  ( F X_s (
I  X.  { R } ) ) ) )
91, 8syl5eq 2487 . . 3  |-  ( ph  -> 
.xb  =  ( .s
`  ( F X_s (
I  X.  { R } ) ) ) )
109oveqd 6113 . 2  |-  ( ph  ->  ( A  .xb  X
)  =  ( A ( .s `  ( F X_s ( I  X.  { R } ) ) ) X ) )
11 eqid 2443 . . 3  |-  ( F
X_s ( I  X.  { R } ) )  =  ( F X_s ( I  X.  { R } ) )
12 eqid 2443 . . 3  |-  ( Base `  ( F X_s ( I  X.  { R } ) ) )  =  ( Base `  ( F X_s ( I  X.  { R } ) ) )
13 eqid 2443 . . 3  |-  ( .s
`  ( F X_s (
I  X.  { R } ) ) )  =  ( .s `  ( F X_s ( I  X.  { R } ) ) )
14 pwsvscaval.k . . 3  |-  K  =  ( Base `  F
)
15 fvex 5706 . . . . 5  |-  (Scalar `  R )  e.  _V
165, 15eqeltri 2513 . . . 4  |-  F  e. 
_V
1716a1i 11 . . 3  |-  ( ph  ->  F  e.  _V )
18 fnconstg 5603 . . . 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 5700 . . . . 5  |-  ( ph  ->  ( Base `  Y
)  =  ( Base `  ( F X_s ( I  X.  { R } ) ) ) )
2422, 23syl5eq 2487 . . . 4  |-  ( ph  ->  B  =  ( Base `  ( F X_s ( I  X.  { R } ) ) ) )
2521, 24eleqtrd 2519 . . 3  |-  ( ph  ->  X  e.  ( Base `  ( F X_s ( I  X.  { R } ) ) ) )
2611, 12, 13, 14, 17, 3, 19, 20, 25prdsvscaval 14422 . 2  |-  ( ph  ->  ( A ( .s
`  ( F X_s (
I  X.  { R } ) ) ) X )  =  ( x  e.  I  |->  ( A ( .s `  ( ( I  X.  { R } ) `  x ) ) ( X `  x ) ) ) )
27 fvconst2g 5936 . . . . . . . 8  |-  ( ( R  e.  V  /\  x  e.  I )  ->  ( ( I  X.  { R } ) `  x )  =  R )
282, 27sylan 471 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  (
( I  X.  { R } ) `  x
)  =  R )
2928fveq2d 5700 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( .s `  ( ( I  X.  { R }
) `  x )
)  =  ( .s
`  R ) )
30 pwsvscaval.s . . . . . 6  |-  .x.  =  ( .s `  R )
3129, 30syl6eqr 2493 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( .s `  ( ( I  X.  { R }
) `  x )
)  =  .x.  )
3231oveqd 6113 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( A ( .s `  ( ( I  X.  { R } ) `  x ) ) ( X `  x ) )  =  ( A 
.x.  ( X `  x ) ) )
3332mpteq2dva 4383 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( A ( .s
`  ( ( I  X.  { R }
) `  x )
) ( X `  x ) ) )  =  ( x  e.  I  |->  ( A  .x.  ( X `  x ) ) ) )
3420adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  A  e.  K )
35 fvex 5706 . . . . 5  |-  ( X `
 x )  e. 
_V
3635a1i 11 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( X `  x )  e.  _V )
37 fconstmpt 4887 . . . . 5  |-  ( I  X.  { A }
)  =  ( x  e.  I  |->  A )
3837a1i 11 . . . 4  |-  ( ph  ->  ( I  X.  { A } )  =  ( x  e.  I  |->  A ) )
39 eqid 2443 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
404, 39, 22, 2, 3, 21pwselbas 14432 . . . . 5  |-  ( ph  ->  X : I --> ( Base `  R ) )
4140feqmptd 5749 . . . 4  |-  ( ph  ->  X  =  ( x  e.  I  |->  ( X `
 x ) ) )
423, 34, 36, 38, 41offval2 6341 . . 3  |-  ( ph  ->  ( ( I  X.  { A } )  oF  .x.  X )  =  ( x  e.  I  |->  ( A  .x.  ( X `  x ) ) ) )
4333, 42eqtr4d 2478 . 2  |-  ( ph  ->  ( x  e.  I  |->  ( A ( .s
`  ( ( I  X.  { R }
) `  x )
) ( X `  x ) ) )  =  ( ( I  X.  { A }
)  oF  .x.  X ) )
4410, 26, 433eqtrd 2479 1  |-  ( ph  ->  ( A  .xb  X
)  =  ( ( I  X.  { A } )  oF  .x.  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2977   {csn 3882    e. cmpt 4355    X. cxp 4843    Fn wfn 5418   ` cfv 5423  (class class class)co 6096    oFcof 6323   Basecbs 14179  Scalarcsca 14246   .scvsca 14247   X_scprds 14389    ^s cpws 14390
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 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
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 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-fz 11443  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-plusg 14256  df-mulr 14257  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-hom 14267  df-cco 14268  df-prds 14391  df-pws 14393
This theorem is referenced by:  pwsvscaval  14438  pwsdiaglmhm  17143  pwssplit3  17147  frlmvscafval  18198
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