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Theorem hlpar 21476
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 21468 . 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 21402 . 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 1215 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 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   1c1 8738    + caddc 8740    x. cmul 8742   -ucneg 9038   2c2 9795   ^cexp 11104   +vcpv 21141   BaseSetcba 21142   .s
OLDcns 21143   normCVcnmcv 21146   CPreHil OLDccphlo 21390   CHil
OLDchlo 21464
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-1st 6122  df-2nd 6123  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-nmcv 21156  df-ph 21391  df-hlo 21465
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