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Theorem pwsval 13636
Description: Value of a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
Hypotheses
Ref Expression
pwsval.y  |-  Y  =  ( R  ^s  I )
pwsval.f  |-  F  =  (Scalar `  R )
Assertion
Ref Expression
pwsval  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Y  =  ( F
X_s ( I  X.  { R } ) ) )

Proof of Theorem pwsval
Dummy variables  i 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwsval.y . 2  |-  Y  =  ( R  ^s  I )
2 elex 2908 . . 3  |-  ( R  e.  V  ->  R  e.  _V )
3 elex 2908 . . 3  |-  ( I  e.  W  ->  I  e.  _V )
4 simpl 444 . . . . . . 7  |-  ( ( r  =  R  /\  i  =  I )  ->  r  =  R )
54fveq2d 5673 . . . . . 6  |-  ( ( r  =  R  /\  i  =  I )  ->  (Scalar `  r )  =  (Scalar `  R )
)
6 pwsval.f . . . . . 6  |-  F  =  (Scalar `  R )
75, 6syl6eqr 2438 . . . . 5  |-  ( ( r  =  R  /\  i  =  I )  ->  (Scalar `  r )  =  F )
8 id 20 . . . . . 6  |-  ( i  =  I  ->  i  =  I )
9 sneq 3769 . . . . . 6  |-  ( r  =  R  ->  { r }  =  { R } )
10 xpeq12 4838 . . . . . 6  |-  ( ( i  =  I  /\  { r }  =  { R } )  ->  (
i  X.  { r } )  =  ( I  X.  { R } ) )
118, 9, 10syl2anr 465 . . . . 5  |-  ( ( r  =  R  /\  i  =  I )  ->  ( i  X.  {
r } )  =  ( I  X.  { R } ) )
127, 11oveq12d 6039 . . . 4  |-  ( ( r  =  R  /\  i  =  I )  ->  ( (Scalar `  r
) X_s ( i  X.  {
r } ) )  =  ( F X_s (
I  X.  { R } ) ) )
13 df-pws 13601 . . . 4  |-  ^s  =  ( r  e.  _V , 
i  e.  _V  |->  ( (Scalar `  r ) X_s ( i  X.  { r } ) ) )
14 ovex 6046 . . . 4  |-  ( F
X_s ( I  X.  { R } ) )  e. 
_V
1512, 13, 14ovmpt2a 6144 . . 3  |-  ( ( R  e.  _V  /\  I  e.  _V )  ->  ( R  ^s  I )  =  ( F X_s (
I  X.  { R } ) ) )
162, 3, 15syl2an 464 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( R  ^s  I )  =  ( F X_s (
I  X.  { R } ) ) )
171, 16syl5eq 2432 1  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Y  =  ( F
X_s ( I  X.  { R } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2900   {csn 3758    X. cxp 4817   ` cfv 5395  (class class class)co 6021  Scalarcsca 13460   X_scprds 13597    ^s cpws 13598
This theorem is referenced by:  pwsbas  13637  pwsplusgval  13640  pwsmulrval  13641  pwsle  13642  pwsvscafval  13644  pwssca  13646  pwsmnd  14658  pws0g  14659  pwspjmhm  14695  pwsgrp  14857  pwsinvg  14858  pwscmn  15406  pwsabl  15407  pwsgsum  15481  pwsrng  15649  pws1  15650  pwscrng  15651  pwsmgp  15652  pwslmod  15974  pwstps  17584  resspwsds  18311  pwsxms  18453  pwsms  18454  cnpwstotbnd  26198  repwsmet  26235  rrnequiv  26236  frlmpws  26888  frlmlss  26889  frlmpwsfi  26890  frlmbas  26893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-iota 5359  df-fun 5397  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-pws 13601
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