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Theorem pwsvscafval 14998
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 14990 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Y  =  ( F
X_s ( I  X.  { R } ) ) )
72, 3, 6syl2anc 659 . . . . 5  |-  ( ph  ->  Y  =  ( F
X_s ( I  X.  { R } ) ) )
87fveq2d 5807 . . . 4  |-  ( ph  ->  ( .s `  Y
)  =  ( .s
`  ( F X_s (
I  X.  { R } ) ) ) )
91, 8syl5eq 2453 . . 3  |-  ( ph  -> 
.xb  =  ( .s
`  ( F X_s (
I  X.  { R } ) ) ) )
109oveqd 6249 . 2  |-  ( ph  ->  ( A  .xb  X
)  =  ( A ( .s `  ( F X_s ( I  X.  { R } ) ) ) X ) )
11 eqid 2400 . . 3  |-  ( F
X_s ( I  X.  { R } ) )  =  ( F X_s ( I  X.  { R } ) )
12 eqid 2400 . . 3  |-  ( Base `  ( F X_s ( I  X.  { R } ) ) )  =  ( Base `  ( F X_s ( I  X.  { R } ) ) )
13 eqid 2400 . . 3  |-  ( .s
`  ( F X_s (
I  X.  { R } ) ) )  =  ( .s `  ( F X_s ( I  X.  { R } ) ) )
14 pwsvscaval.k . . 3  |-  K  =  ( Base `  F
)
15 fvex 5813 . . . . 5  |-  (Scalar `  R )  e.  _V
165, 15eqeltri 2484 . . . 4  |-  F  e. 
_V
1716a1i 11 . . 3  |-  ( ph  ->  F  e.  _V )
18 fnconstg 5710 . . . 4  |-  ( R  e.  V  ->  (
I  X.  { R } )  Fn  I
)
192, 18syl 17 . . 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 5807 . . . . 5  |-  ( ph  ->  ( Base `  Y
)  =  ( Base `  ( F X_s ( I  X.  { R } ) ) ) )
2422, 23syl5eq 2453 . . . 4  |-  ( ph  ->  B  =  ( Base `  ( F X_s ( I  X.  { R } ) ) ) )
2521, 24eleqtrd 2490 . . 3  |-  ( ph  ->  X  e.  ( Base `  ( F X_s ( I  X.  { R } ) ) ) )
2611, 12, 13, 14, 17, 3, 19, 20, 25prdsvscaval 14983 . 2  |-  ( ph  ->  ( A ( .s
`  ( F X_s (
I  X.  { R } ) ) ) X )  =  ( x  e.  I  |->  ( A ( .s `  ( ( I  X.  { R } ) `  x ) ) ( X `  x ) ) ) )
27 fvconst2g 6059 . . . . . . . 8  |-  ( ( R  e.  V  /\  x  e.  I )  ->  ( ( I  X.  { R } ) `  x )  =  R )
282, 27sylan 469 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  (
( I  X.  { R } ) `  x
)  =  R )
2928fveq2d 5807 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( .s `  ( ( I  X.  { R }
) `  x )
)  =  ( .s
`  R ) )
30 pwsvscaval.s . . . . . 6  |-  .x.  =  ( .s `  R )
3129, 30syl6eqr 2459 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( .s `  ( ( I  X.  { R }
) `  x )
)  =  .x.  )
3231oveqd 6249 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( A ( .s `  ( ( I  X.  { R } ) `  x ) ) ( X `  x ) )  =  ( A 
.x.  ( X `  x ) ) )
3332mpteq2dva 4478 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( A ( .s
`  ( ( I  X.  { R }
) `  x )
) ( X `  x ) ) )  =  ( x  e.  I  |->  ( A  .x.  ( X `  x ) ) ) )
3420adantr 463 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  A  e.  K )
35 fvex 5813 . . . . 5  |-  ( X `
 x )  e. 
_V
3635a1i 11 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( X `  x )  e.  _V )
37 fconstmpt 4984 . . . . 5  |-  ( I  X.  { A }
)  =  ( x  e.  I  |->  A )
3837a1i 11 . . . 4  |-  ( ph  ->  ( I  X.  { A } )  =  ( x  e.  I  |->  A ) )
39 eqid 2400 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
404, 39, 22, 2, 3, 21pwselbas 14993 . . . . 5  |-  ( ph  ->  X : I --> ( Base `  R ) )
4140feqmptd 5856 . . . 4  |-  ( ph  ->  X  =  ( x  e.  I  |->  ( X `
 x ) ) )
423, 34, 36, 38, 41offval2 6492 . . 3  |-  ( ph  ->  ( ( I  X.  { A } )  oF  .x.  X )  =  ( x  e.  I  |->  ( A  .x.  ( X `  x ) ) ) )
4333, 42eqtr4d 2444 . 2  |-  ( ph  ->  ( x  e.  I  |->  ( A ( .s
`  ( ( I  X.  { R }
) `  x )
) ( X `  x ) ) )  =  ( ( I  X.  { A }
)  oF  .x.  X ) )
4410, 26, 433eqtrd 2445 1  |-  ( ph  ->  ( A  .xb  X
)  =  ( ( I  X.  { A } )  oF  .x.  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840   _Vcvv 3056   {csn 3969    |-> cmpt 4450    X. cxp 4938    Fn wfn 5518   ` cfv 5523  (class class class)co 6232    oFcof 6473   Basecbs 14731  Scalarcsca 14802   .scvsca 14803   X_scprds 14950    ^s cpws 14951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-of 6475  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-1o 7085  df-oadd 7089  df-er 7266  df-map 7377  df-ixp 7426  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-sup 7853  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-nn 10495  df-2 10553  df-3 10554  df-4 10555  df-5 10556  df-6 10557  df-7 10558  df-8 10559  df-9 10560  df-10 10561  df-n0 10755  df-z 10824  df-dec 10938  df-uz 11044  df-fz 11642  df-struct 14733  df-ndx 14734  df-slot 14735  df-base 14736  df-plusg 14812  df-mulr 14813  df-sca 14815  df-vsca 14816  df-ip 14817  df-tset 14818  df-ple 14819  df-ds 14821  df-hom 14823  df-cco 14824  df-prds 14952  df-pws 14954
This theorem is referenced by:  pwsvscaval  14999  pwsdiaglmhm  17913  pwssplit3  17917  frlmvscafval  18985
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