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Theorem hlpar 27137
 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 = ( +𝑣𝑈)
hlpar.4 𝑆 = ( ·𝑠OLD𝑈)
hlpar.6 𝑁 = (normCV𝑈)
Assertion
Ref Expression
hlpar ((𝑈 ∈ CHilOLD𝐴𝑋𝐵𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2))))

Proof of Theorem hlpar
StepHypRef Expression
1 hlph 27129 . 2 (𝑈 ∈ CHilOLD𝑈 ∈ CPreHilOLD)
2 hlpar.1 . . 3 𝑋 = (BaseSet‘𝑈)
3 hlpar.2 . . 3 𝐺 = ( +𝑣𝑈)
4 hlpar.4 . . 3 𝑆 = ( ·𝑠OLD𝑈)
5 hlpar.6 . . 3 𝑁 = (normCV𝑈)
62, 3, 4, 5phpar 27063 . 2 ((𝑈 ∈ CPreHilOLD𝐴𝑋𝐵𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2))))
71, 6syl3an1 1351 1 ((𝑈 ∈ CHilOLD𝐴𝑋𝐵𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  1c1 9816   + caddc 9818   · cmul 9820  -cneg 10146  2c2 10947  ↑cexp 12722   +𝑣 cpv 26824  BaseSetcba 26825   ·𝑠OLD cns 26826  normCVcnmcv 26829  CPreHilOLDccphlo 27051  CHilOLDchlo 27125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-1st 7059  df-2nd 7060  df-vc 26798  df-nv 26831  df-va 26834  df-ba 26835  df-sm 26836  df-0v 26837  df-nmcv 26839  df-ph 27052  df-hlo 27126 This theorem is referenced by: (None)
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