MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hlpar Unicode version

Theorem hlpar 22249
Description: The parallelogram law satified by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
hlpar.1  |-  X  =  ( BaseSet `  U )
hlpar.2  |-  G  =  ( +v `  U
)
hlpar.4  |-  S  =  ( .s OLD `  U
)
hlpar.6  |-  N  =  ( normCV `  U )
Assertion
Ref Expression
hlpar  |-  ( ( U  e.  CHil OLD  /\  A  e.  X  /\  B  e.  X )  ->  ( ( ( N `
 ( A G B ) ) ^
2 )  +  ( ( N `  ( A G ( -u 1 S B ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) ) ) )

Proof of Theorem hlpar
StepHypRef Expression
1 hlph 22241 . 2  |-  ( U  e.  CHil OLD  ->  U  e.  CPreHil
OLD )
2 hlpar.1 . . 3  |-  X  =  ( BaseSet `  U )
3 hlpar.2 . . 3  |-  G  =  ( +v `  U
)
4 hlpar.4 . . 3  |-  S  =  ( .s OLD `  U
)
5 hlpar.6 . . 3  |-  N  =  ( normCV `  U )
62, 3, 4, 5phpar 22175 . 2  |-  ( ( U  e.  CPreHil OLD  /\  A  e.  X  /\  B  e.  X )  ->  ( ( ( N `
 ( A G B ) ) ^
2 )  +  ( ( N `  ( A G ( -u 1 S B ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) ) ) )
71, 6syl3an1 1217 1  |-  ( ( U  e.  CHil OLD  /\  A  e.  X  /\  B  e.  X )  ->  ( ( ( N `
 ( A G B ) ) ^
2 )  +  ( ( N `  ( A G ( -u 1 S B ) ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717   ` cfv 5396  (class class class)co 6022   1c1 8926    + caddc 8928    x. cmul 8930   -ucneg 9226   2c2 9983   ^cexp 11311   +vcpv 21914   BaseSetcba 21915   .s
OLDcns 21916   normCVcnmcv 21919   CPreHil OLDccphlo 22163   CHil
OLDchlo 22237
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-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-1st 6290  df-2nd 6291  df-vc 21875  df-nv 21921  df-va 21924  df-ba 21925  df-sm 21926  df-0v 21927  df-nmcv 21929  df-ph 22164  df-hlo 22238
  Copyright terms: Public domain W3C validator