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Theorem pwsvscafval 13393
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 13385 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Y  =  ( F
X_s ( I  X.  { R } ) ) )
72, 3, 6syl2anc 642 . . . . 5  |-  ( ph  ->  Y  =  ( F
X_s ( I  X.  { R } ) ) )
87fveq2d 5529 . . . 4  |-  ( ph  ->  ( .s `  Y
)  =  ( .s
`  ( F X_s (
I  X.  { R } ) ) ) )
91, 8syl5eq 2327 . . 3  |-  ( ph  -> 
.xb  =  ( .s
`  ( F X_s (
I  X.  { R } ) ) ) )
109oveqd 5875 . 2  |-  ( ph  ->  ( A  .xb  X
)  =  ( A ( .s `  ( F X_s ( I  X.  { R } ) ) ) X ) )
11 eqid 2283 . . 3  |-  ( F
X_s ( I  X.  { R } ) )  =  ( F X_s ( I  X.  { R } ) )
12 eqid 2283 . . 3  |-  ( Base `  ( F X_s ( I  X.  { R } ) ) )  =  ( Base `  ( F X_s ( I  X.  { R } ) ) )
13 eqid 2283 . . 3  |-  ( .s
`  ( F X_s (
I  X.  { R } ) ) )  =  ( .s `  ( F X_s ( I  X.  { R } ) ) )
14 pwsvscaval.k . . 3  |-  K  =  ( Base `  F
)
15 fvex 5539 . . . . 5  |-  (Scalar `  R )  e.  _V
165, 15eqeltri 2353 . . . 4  |-  F  e. 
_V
1716a1i 10 . . 3  |-  ( ph  ->  F  e.  _V )
18 fnconstg 5429 . . . 4  |-  ( R  e.  V  ->  (
I  X.  { R } )  Fn  I
)
192, 18syl 15 . . 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 5529 . . . . 5  |-  ( ph  ->  ( Base `  Y
)  =  ( Base `  ( F X_s ( I  X.  { R } ) ) ) )
2422, 23syl5eq 2327 . . . 4  |-  ( ph  ->  B  =  ( Base `  ( F X_s ( I  X.  { R } ) ) ) )
2521, 24eleqtrd 2359 . . 3  |-  ( ph  ->  X  e.  ( Base `  ( F X_s ( I  X.  { R } ) ) ) )
2611, 12, 13, 14, 17, 3, 19, 20, 25prdsvscaval 13378 . 2  |-  ( ph  ->  ( A ( .s
`  ( F X_s (
I  X.  { R } ) ) ) X )  =  ( x  e.  I  |->  ( A ( .s `  ( ( I  X.  { R } ) `  x ) ) ( X `  x ) ) ) )
27 fvconst2g 5727 . . . . . . . 8  |-  ( ( R  e.  V  /\  x  e.  I )  ->  ( ( I  X.  { R } ) `  x )  =  R )
282, 27sylan 457 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  (
( I  X.  { R } ) `  x
)  =  R )
2928fveq2d 5529 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( .s `  ( ( I  X.  { R }
) `  x )
)  =  ( .s
`  R ) )
30 pwsvscaval.s . . . . . 6  |-  .x.  =  ( .s `  R )
3129, 30syl6eqr 2333 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( .s `  ( ( I  X.  { R }
) `  x )
)  =  .x.  )
3231oveqd 5875 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( A ( .s `  ( ( I  X.  { R } ) `  x ) ) ( X `  x ) )  =  ( A 
.x.  ( X `  x ) ) )
3332mpteq2dva 4106 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( A ( .s
`  ( ( I  X.  { R }
) `  x )
) ( X `  x ) ) )  =  ( x  e.  I  |->  ( A  .x.  ( X `  x ) ) ) )
3420adantr 451 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  A  e.  K )
35 fvex 5539 . . . . 5  |-  ( X `
 x )  e. 
_V
3635a1i 10 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( X `  x )  e.  _V )
37 fconstmpt 4732 . . . . 5  |-  ( I  X.  { A }
)  =  ( x  e.  I  |->  A )
3837a1i 10 . . . 4  |-  ( ph  ->  ( I  X.  { A } )  =  ( x  e.  I  |->  A ) )
39 eqid 2283 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
404, 39, 22, 2, 3, 21pwselbas 13388 . . . . 5  |-  ( ph  ->  X : I --> ( Base `  R ) )
4140feqmptd 5575 . . . 4  |-  ( ph  ->  X  =  ( x  e.  I  |->  ( X `
 x ) ) )
423, 34, 36, 38, 41offval2 6095 . . 3  |-  ( ph  ->  ( ( I  X.  { A } )  o F  .x.  X )  =  ( x  e.  I  |->  ( A  .x.  ( X `  x ) ) ) )
4333, 42eqtr4d 2318 . 2  |-  ( ph  ->  ( x  e.  I  |->  ( A ( .s
`  ( ( I  X.  { R }
) `  x )
) ( X `  x ) ) )  =  ( ( I  X.  { A }
)  o F  .x.  X ) )
4410, 26, 433eqtrd 2319 1  |-  ( ph  ->  ( A  .xb  X
)  =  ( ( I  X.  { A } )  o F 
.x.  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640    e. cmpt 4077    X. cxp 4687    Fn wfn 5250   ` cfv 5255  (class class class)co 5858    o Fcof 6076   Basecbs 13148  Scalarcsca 13211   .scvsca 13212   X_scprds 13346    ^s cpws 13347
This theorem is referenced by:  pwsvscaval  13394  pwsdiaglmhm  15814  pwssplit3  27190  frlmvscafval  27230
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-pws 13350
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