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Theorem pwsval 14737
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 3122 . . 3  |-  ( R  e.  V  ->  R  e.  _V )
3 elex 3122 . . 3  |-  ( I  e.  W  ->  I  e.  _V )
4 simpl 457 . . . . . . 7  |-  ( ( r  =  R  /\  i  =  I )  ->  r  =  R )
54fveq2d 5868 . . . . . 6  |-  ( ( r  =  R  /\  i  =  I )  ->  (Scalar `  r )  =  (Scalar `  R )
)
6 pwsval.f . . . . . 6  |-  F  =  (Scalar `  R )
75, 6syl6eqr 2526 . . . . 5  |-  ( ( r  =  R  /\  i  =  I )  ->  (Scalar `  r )  =  F )
8 id 22 . . . . . 6  |-  ( i  =  I  ->  i  =  I )
9 sneq 4037 . . . . . 6  |-  ( r  =  R  ->  { r }  =  { R } )
10 xpeq12 5018 . . . . . 6  |-  ( ( i  =  I  /\  { r }  =  { R } )  ->  (
i  X.  { r } )  =  ( I  X.  { R } ) )
118, 9, 10syl2anr 478 . . . . 5  |-  ( ( r  =  R  /\  i  =  I )  ->  ( i  X.  {
r } )  =  ( I  X.  { R } ) )
127, 11oveq12d 6300 . . . 4  |-  ( ( r  =  R  /\  i  =  I )  ->  ( (Scalar `  r
) X_s ( i  X.  {
r } ) )  =  ( F X_s (
I  X.  { R } ) ) )
13 df-pws 14701 . . . 4  |-  ^s  =  ( r  e.  _V , 
i  e.  _V  |->  ( (Scalar `  r ) X_s ( i  X.  { r } ) ) )
14 ovex 6307 . . . 4  |-  ( F
X_s ( I  X.  { R } ) )  e. 
_V
1512, 13, 14ovmpt2a 6415 . . 3  |-  ( ( R  e.  _V  /\  I  e.  _V )  ->  ( R  ^s  I )  =  ( F X_s (
I  X.  { R } ) ) )
162, 3, 15syl2an 477 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( R  ^s  I )  =  ( F X_s (
I  X.  { R } ) ) )
171, 16syl5eq 2520 1  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Y  =  ( F
X_s ( I  X.  { R } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   {csn 4027    X. cxp 4997   ` cfv 5586  (class class class)co 6282  Scalarcsca 14554   X_scprds 14697    ^s cpws 14698
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-pws 14701
This theorem is referenced by:  pwsbas  14738  pwsplusgval  14741  pwsmulrval  14742  pwsle  14743  pwsvscafval  14745  pwssca  14747  pwsmnd  15769  pws0g  15770  pwspjmhm  15809  pwsgrp  15981  pwsinvg  15982  pwscmn  16662  pwsabl  16663  pwsgsum  16800  pwsgsumOLD  16801  pwsrng  17048  pws1  17049  pwscrng  17050  pwsmgp  17051  pwslmod  17399  frlmpws  18548  frlmlss  18549  frlmpwsfi  18550  frlmbas  18553  frlmbasOLD  18554  frlmip  18576  pwstps  19866  resspwsds  20610  pwsxms  20770  pwsms  20771  rrxprds  21556  cnpwstotbnd  29896  repwsmet  29933  rrnequiv  29934
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