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Theorem nvop 27867
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 27793 . . 3 Rel NrmCVec
2 1st2nd 7363 . . 3 ((Rel NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
31, 2mpan 670 . 2 (𝑈 ∈ NrmCVec → 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
4 nvop.6 . . . . 5 𝑁 = (normCV𝑈)
54nmcvfval 27798 . . . 4 𝑁 = (2nd𝑈)
65opeq2i 4543 . . 3 ⟨(1st𝑈), 𝑁⟩ = ⟨(1st𝑈), (2nd𝑈)⟩
7 eqid 2771 . . . . 5 (1st𝑈) = (1st𝑈)
8 nvop.2 . . . . 5 𝐺 = ( +𝑣𝑈)
9 nvop.4 . . . . 5 𝑆 = ( ·𝑠OLD𝑈)
107, 8, 9nvvop 27800 . . . 4 (𝑈 ∈ NrmCVec → (1st𝑈) = ⟨𝐺, 𝑆⟩)
1110opeq1d 4545 . . 3 (𝑈 ∈ NrmCVec → ⟨(1st𝑈), 𝑁⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
126, 11syl5eqr 2819 . 2 (𝑈 ∈ NrmCVec → ⟨(1st𝑈), (2nd𝑈)⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
133, 12eqtrd 2805 1 (𝑈 ∈ NrmCVec → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  cop 4322  Rel wrel 5254  cfv 6029  1st c1st 7313  2nd c2nd 7314  NrmCVeccnv 27775   +𝑣 cpv 27776   ·𝑠OLD cns 27778  normCVcnmcv 27781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-fo 6035  df-fv 6037  df-oprab 6796  df-1st 7315  df-2nd 7316  df-vc 27750  df-nv 27783  df-va 27786  df-sm 27788  df-nmcv 27791
This theorem is referenced by:  sspval  27914  isph  28013  hilhhi  28357
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