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Theorem pwsval 14420
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 2979 . . 3  |-  ( R  e.  V  ->  R  e.  _V )
3 elex 2979 . . 3  |-  ( I  e.  W  ->  I  e.  _V )
4 simpl 454 . . . . . . 7  |-  ( ( r  =  R  /\  i  =  I )  ->  r  =  R )
54fveq2d 5692 . . . . . 6  |-  ( ( r  =  R  /\  i  =  I )  ->  (Scalar `  r )  =  (Scalar `  R )
)
6 pwsval.f . . . . . 6  |-  F  =  (Scalar `  R )
75, 6syl6eqr 2491 . . . . 5  |-  ( ( r  =  R  /\  i  =  I )  ->  (Scalar `  r )  =  F )
8 id 22 . . . . . 6  |-  ( i  =  I  ->  i  =  I )
9 sneq 3884 . . . . . 6  |-  ( r  =  R  ->  { r }  =  { R } )
10 xpeq12 4855 . . . . . 6  |-  ( ( i  =  I  /\  { r }  =  { R } )  ->  (
i  X.  { r } )  =  ( I  X.  { R } ) )
118, 9, 10syl2anr 475 . . . . 5  |-  ( ( r  =  R  /\  i  =  I )  ->  ( i  X.  {
r } )  =  ( I  X.  { R } ) )
127, 11oveq12d 6108 . . . 4  |-  ( ( r  =  R  /\  i  =  I )  ->  ( (Scalar `  r
) X_s ( i  X.  {
r } ) )  =  ( F X_s (
I  X.  { R } ) ) )
13 df-pws 14384 . . . 4  |-  ^s  =  ( r  e.  _V , 
i  e.  _V  |->  ( (Scalar `  r ) X_s ( i  X.  { r } ) ) )
14 ovex 6115 . . . 4  |-  ( F
X_s ( I  X.  { R } ) )  e. 
_V
1512, 13, 14ovmpt2a 6220 . . 3  |-  ( ( R  e.  _V  /\  I  e.  _V )  ->  ( R  ^s  I )  =  ( F X_s (
I  X.  { R } ) ) )
162, 3, 15syl2an 474 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( R  ^s  I )  =  ( F X_s (
I  X.  { R } ) ) )
171, 16syl5eq 2485 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 1364    e. wcel 1761   _Vcvv 2970   {csn 3874    X. cxp 4834   ` cfv 5415  (class class class)co 6090  Scalarcsca 14237   X_scprds 14380    ^s cpws 14381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-iota 5378  df-fun 5417  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-pws 14384
This theorem is referenced by:  pwsbas  14421  pwsplusgval  14424  pwsmulrval  14425  pwsle  14426  pwsvscafval  14428  pwssca  14430  pwsmnd  15452  pws0g  15453  pwspjmhm  15491  pwsgrp  15659  pwsinvg  15660  pwscmn  16338  pwsabl  16339  pwsgsum  16463  pwsgsumOLD  16464  pwsrng  16697  pws1  16698  pwscrng  16699  pwsmgp  16700  pwslmod  17029  frlmpws  18134  frlmlss  18135  frlmpwsfi  18136  frlmbas  18139  frlmbasOLD  18140  frlmip  18162  pwstps  19162  resspwsds  19906  pwsxms  20066  pwsms  20067  rrxprds  20852  cnpwstotbnd  28621  repwsmet  28658  rrnequiv  28659
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