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Theorem nvop 28380
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 28306 . . 3 Rel NrmCVec
2 1st2nd 7727 . . 3 ((Rel NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
31, 2mpan 686 . 2 (𝑈 ∈ NrmCVec → 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
4 nvop.6 . . . . 5 𝑁 = (normCV𝑈)
54nmcvfval 28311 . . . 4 𝑁 = (2nd𝑈)
65opeq2i 4799 . . 3 ⟨(1st𝑈), 𝑁⟩ = ⟨(1st𝑈), (2nd𝑈)⟩
7 eqid 2818 . . . . 5 (1st𝑈) = (1st𝑈)
8 nvop.2 . . . . 5 𝐺 = ( +𝑣𝑈)
9 nvop.4 . . . . 5 𝑆 = ( ·𝑠OLD𝑈)
107, 8, 9nvvop 28313 . . . 4 (𝑈 ∈ NrmCVec → (1st𝑈) = ⟨𝐺, 𝑆⟩)
1110opeq1d 4801 . . 3 (𝑈 ∈ NrmCVec → ⟨(1st𝑈), 𝑁⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
126, 11syl5eqr 2867 . 2 (𝑈 ∈ NrmCVec → ⟨(1st𝑈), (2nd𝑈)⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
133, 12eqtrd 2853 1 (𝑈 ∈ NrmCVec → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  cop 4563  Rel wrel 5553  cfv 6348  1st c1st 7676  2nd c2nd 7677  NrmCVeccnv 28288   +𝑣 cpv 28289   ·𝑠OLD cns 28291  normCVcnmcv 28294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-oprab 7149  df-1st 7678  df-2nd 7679  df-vc 28263  df-nv 28296  df-va 28299  df-sm 28301  df-nmcv 28304
This theorem is referenced by:  sspval  28427  isph  28526  hilhhi  28868
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