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Theorem pwsvscafval 14766
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 14758 . . . . . 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 5876 . . . 4  |-  ( ph  ->  ( .s `  Y
)  =  ( .s
`  ( F X_s (
I  X.  { R } ) ) ) )
91, 8syl5eq 2520 . . 3  |-  ( ph  -> 
.xb  =  ( .s
`  ( F X_s (
I  X.  { R } ) ) ) )
109oveqd 6312 . 2  |-  ( ph  ->  ( A  .xb  X
)  =  ( A ( .s `  ( F X_s ( I  X.  { R } ) ) ) X ) )
11 eqid 2467 . . 3  |-  ( F
X_s ( I  X.  { R } ) )  =  ( F X_s ( I  X.  { R } ) )
12 eqid 2467 . . 3  |-  ( Base `  ( F X_s ( I  X.  { R } ) ) )  =  ( Base `  ( F X_s ( I  X.  { R } ) ) )
13 eqid 2467 . . 3  |-  ( .s
`  ( F X_s (
I  X.  { R } ) ) )  =  ( .s `  ( F X_s ( I  X.  { R } ) ) )
14 pwsvscaval.k . . 3  |-  K  =  ( Base `  F
)
15 fvex 5882 . . . . 5  |-  (Scalar `  R )  e.  _V
165, 15eqeltri 2551 . . . 4  |-  F  e. 
_V
1716a1i 11 . . 3  |-  ( ph  ->  F  e.  _V )
18 fnconstg 5779 . . . 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 5876 . . . . 5  |-  ( ph  ->  ( Base `  Y
)  =  ( Base `  ( F X_s ( I  X.  { R } ) ) ) )
2422, 23syl5eq 2520 . . . 4  |-  ( ph  ->  B  =  ( Base `  ( F X_s ( I  X.  { R } ) ) ) )
2521, 24eleqtrd 2557 . . 3  |-  ( ph  ->  X  e.  ( Base `  ( F X_s ( I  X.  { R } ) ) ) )
2611, 12, 13, 14, 17, 3, 19, 20, 25prdsvscaval 14751 . 2  |-  ( ph  ->  ( A ( .s
`  ( F X_s (
I  X.  { R } ) ) ) X )  =  ( x  e.  I  |->  ( A ( .s `  ( ( I  X.  { R } ) `  x ) ) ( X `  x ) ) ) )
27 fvconst2g 6125 . . . . . . . 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 5876 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( .s `  ( ( I  X.  { R }
) `  x )
)  =  ( .s
`  R ) )
30 pwsvscaval.s . . . . . 6  |-  .x.  =  ( .s `  R )
3129, 30syl6eqr 2526 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( .s `  ( ( I  X.  { R }
) `  x )
)  =  .x.  )
3231oveqd 6312 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( A ( .s `  ( ( I  X.  { R } ) `  x ) ) ( X `  x ) )  =  ( A 
.x.  ( X `  x ) ) )
3332mpteq2dva 4539 . . 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 5882 . . . . 5  |-  ( X `
 x )  e. 
_V
3635a1i 11 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( X `  x )  e.  _V )
37 fconstmpt 5049 . . . . 5  |-  ( I  X.  { A }
)  =  ( x  e.  I  |->  A )
3837a1i 11 . . . 4  |-  ( ph  ->  ( I  X.  { A } )  =  ( x  e.  I  |->  A ) )
39 eqid 2467 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
404, 39, 22, 2, 3, 21pwselbas 14761 . . . . 5  |-  ( ph  ->  X : I --> ( Base `  R ) )
4140feqmptd 5927 . . . 4  |-  ( ph  ->  X  =  ( x  e.  I  |->  ( X `
 x ) ) )
423, 34, 36, 38, 41offval2 6551 . . 3  |-  ( ph  ->  ( ( I  X.  { A } )  oF  .x.  X )  =  ( x  e.  I  |->  ( A  .x.  ( X `  x ) ) ) )
4333, 42eqtr4d 2511 . 2  |-  ( ph  ->  ( x  e.  I  |->  ( A ( .s
`  ( ( I  X.  { R }
) `  x )
) ( X `  x ) ) )  =  ( ( I  X.  { A }
)  oF  .x.  X ) )
4410, 26, 433eqtrd 2512 1  |-  ( ph  ->  ( A  .xb  X
)  =  ( ( I  X.  { A } )  oF  .x.  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118   {csn 4033    |-> cmpt 4511    X. cxp 5003    Fn wfn 5589   ` cfv 5594  (class class class)co 6295    oFcof 6533   Basecbs 14507  Scalarcsca 14575   .scvsca 14576   X_scprds 14718    ^s cpws 14719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-hom 14596  df-cco 14597  df-prds 14720  df-pws 14722
This theorem is referenced by:  pwsvscaval  14767  pwsdiaglmhm  17574  pwssplit3  17578  frlmvscafval  18668
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