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Theorem pwsval 13401
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 2809 . . 3  |-  ( R  e.  V  ->  R  e.  _V )
3 elex 2809 . . 3  |-  ( I  e.  W  ->  I  e.  _V )
4 simpl 443 . . . . . . 7  |-  ( ( r  =  R  /\  i  =  I )  ->  r  =  R )
54fveq2d 5545 . . . . . 6  |-  ( ( r  =  R  /\  i  =  I )  ->  (Scalar `  r )  =  (Scalar `  R )
)
6 pwsval.f . . . . . 6  |-  F  =  (Scalar `  R )
75, 6syl6eqr 2346 . . . . 5  |-  ( ( r  =  R  /\  i  =  I )  ->  (Scalar `  r )  =  F )
8 id 19 . . . . . 6  |-  ( i  =  I  ->  i  =  I )
9 sneq 3664 . . . . . 6  |-  ( r  =  R  ->  { r }  =  { R } )
10 xpeq12 4724 . . . . . 6  |-  ( ( i  =  I  /\  { r }  =  { R } )  ->  (
i  X.  { r } )  =  ( I  X.  { R } ) )
118, 9, 10syl2anr 464 . . . . 5  |-  ( ( r  =  R  /\  i  =  I )  ->  ( i  X.  {
r } )  =  ( I  X.  { R } ) )
127, 11oveq12d 5892 . . . 4  |-  ( ( r  =  R  /\  i  =  I )  ->  ( (Scalar `  r
) X_s ( i  X.  {
r } ) )  =  ( F X_s (
I  X.  { R } ) ) )
13 df-pws 13366 . . . 4  |-  ^s  =  ( r  e.  _V , 
i  e.  _V  |->  ( (Scalar `  r ) X_s ( i  X.  { r } ) ) )
14 ovex 5899 . . . 4  |-  ( F
X_s ( I  X.  { R } ) )  e. 
_V
1512, 13, 14ovmpt2a 5994 . . 3  |-  ( ( R  e.  _V  /\  I  e.  _V )  ->  ( R  ^s  I )  =  ( F X_s (
I  X.  { R } ) ) )
162, 3, 15syl2an 463 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( R  ^s  I )  =  ( F X_s (
I  X.  { R } ) ) )
171, 16syl5eq 2340 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 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653    X. cxp 4703   ` cfv 5271  (class class class)co 5874  Scalarcsca 13227   X_scprds 13362    ^s cpws 13363
This theorem is referenced by:  pwsbas  13402  pwsplusgval  13405  pwsmulrval  13406  pwsle  13407  pwsvscafval  13409  pwssca  13411  pwsmnd  14423  pws0g  14424  pwspjmhm  14460  pwsgrp  14622  pwsinvg  14623  pwscmn  15171  pwsabl  15172  pwsgsum  15246  pwsrng  15414  pws1  15415  pwscrng  15416  pwsmgp  15417  pwslmod  15743  pwstps  17340  resspwsds  17952  pwsxms  18094  pwsms  18095  cnpwstotbnd  26624  repwsmet  26661  rrnequiv  26662  frlmpws  27321  frlmlss  27322  frlmpwsfi  27323  frlmbas  27326
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-pws 13366
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