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Theorem hlpar 22389
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 22381 . 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 22315 . 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 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   1c1 8981    + caddc 8983    x. cmul 8985   -ucneg 9282   2c2 10039   ^cexp 11372   +vcpv 22054   BaseSetcba 22055   .s
OLDcns 22056   normCVcnmcv 22059   CPreHil OLDccphlo 22303   CHil
OLDchlo 22377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-1st 6341  df-2nd 6342  df-vc 22015  df-nv 22061  df-va 22064  df-ba 22065  df-sm 22066  df-0v 22067  df-nmcv 22069  df-ph 22304  df-hlo 22378
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