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Theorem nvop 28757
Description: A complex inner product space in terms of ordered pair components. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvop.2 𝐺 = ( +𝑣𝑈)
nvop.4 𝑆 = ( ·𝑠OLD𝑈)
nvop.6 𝑁 = (normCV𝑈)
Assertion
Ref Expression
nvop (𝑈 ∈ NrmCVec → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)

Proof of Theorem nvop
StepHypRef Expression
1 nvrel 28683 . . 3 Rel NrmCVec
2 1st2nd 7810 . . 3 ((Rel NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
31, 2mpan 690 . 2 (𝑈 ∈ NrmCVec → 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
4 nvop.6 . . . . 5 𝑁 = (normCV𝑈)
54nmcvfval 28688 . . . 4 𝑁 = (2nd𝑈)
65opeq2i 4788 . . 3 ⟨(1st𝑈), 𝑁⟩ = ⟨(1st𝑈), (2nd𝑈)⟩
7 eqid 2737 . . . . 5 (1st𝑈) = (1st𝑈)
8 nvop.2 . . . . 5 𝐺 = ( +𝑣𝑈)
9 nvop.4 . . . . 5 𝑆 = ( ·𝑠OLD𝑈)
107, 8, 9nvvop 28690 . . . 4 (𝑈 ∈ NrmCVec → (1st𝑈) = ⟨𝐺, 𝑆⟩)
1110opeq1d 4790 . . 3 (𝑈 ∈ NrmCVec → ⟨(1st𝑈), 𝑁⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
126, 11eqtr3id 2792 . 2 (𝑈 ∈ NrmCVec → ⟨(1st𝑈), (2nd𝑈)⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
133, 12eqtrd 2777 1 (𝑈 ∈ NrmCVec → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  cop 4547  Rel wrel 5556  cfv 6380  1st c1st 7759  2nd c2nd 7760  NrmCVeccnv 28665   +𝑣 cpv 28666   ·𝑠OLD cns 28668  normCVcnmcv 28671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fo 6386  df-fv 6388  df-oprab 7217  df-1st 7761  df-2nd 7762  df-vc 28640  df-nv 28673  df-va 28676  df-sm 28678  df-nmcv 28681
This theorem is referenced by:  sspval  28804  isph  28903  hilhhi  29245
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