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Theorem phop 30838
Description: A complex inner product space in terms of ordered pair components. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
phop.2 𝐺 = ( +𝑣𝑈)
phop.4 𝑆 = ( ·𝑠OLD𝑈)
phop.6 𝑁 = (normCV𝑈)
Assertion
Ref Expression
phop (𝑈 ∈ CPreHilOLD𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)

Proof of Theorem phop
StepHypRef Expression
1 phrel 30835 . . 3 Rel CPreHilOLD
2 1st2nd 8065 . . 3 ((Rel CPreHilOLD𝑈 ∈ CPreHilOLD) → 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
31, 2mpan 690 . 2 (𝑈 ∈ CPreHilOLD𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
4 phop.6 . . . . 5 𝑁 = (normCV𝑈)
54nmcvfval 30627 . . . 4 𝑁 = (2nd𝑈)
65opeq2i 4876 . . 3 ⟨(1st𝑈), 𝑁⟩ = ⟨(1st𝑈), (2nd𝑈)⟩
7 phnv 30834 . . . . 5 (𝑈 ∈ CPreHilOLD𝑈 ∈ NrmCVec)
8 eqid 2736 . . . . . 6 (1st𝑈) = (1st𝑈)
98nvvc 30635 . . . . 5 (𝑈 ∈ NrmCVec → (1st𝑈) ∈ CVecOLD)
10 vcrel 30580 . . . . . . 7 Rel CVecOLD
11 1st2nd 8065 . . . . . . 7 ((Rel CVecOLD ∧ (1st𝑈) ∈ CVecOLD) → (1st𝑈) = ⟨(1st ‘(1st𝑈)), (2nd ‘(1st𝑈))⟩)
1210, 11mpan 690 . . . . . 6 ((1st𝑈) ∈ CVecOLD → (1st𝑈) = ⟨(1st ‘(1st𝑈)), (2nd ‘(1st𝑈))⟩)
13 phop.2 . . . . . . . 8 𝐺 = ( +𝑣𝑈)
1413vafval 30623 . . . . . . 7 𝐺 = (1st ‘(1st𝑈))
15 phop.4 . . . . . . . 8 𝑆 = ( ·𝑠OLD𝑈)
1615smfval 30625 . . . . . . 7 𝑆 = (2nd ‘(1st𝑈))
1714, 16opeq12i 4877 . . . . . 6 𝐺, 𝑆⟩ = ⟨(1st ‘(1st𝑈)), (2nd ‘(1st𝑈))⟩
1812, 17eqtr4di 2794 . . . . 5 ((1st𝑈) ∈ CVecOLD → (1st𝑈) = ⟨𝐺, 𝑆⟩)
197, 9, 183syl 18 . . . 4 (𝑈 ∈ CPreHilOLD → (1st𝑈) = ⟨𝐺, 𝑆⟩)
2019opeq1d 4878 . . 3 (𝑈 ∈ CPreHilOLD → ⟨(1st𝑈), 𝑁⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
216, 20eqtr3id 2790 . 2 (𝑈 ∈ CPreHilOLD → ⟨(1st𝑈), (2nd𝑈)⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
223, 21eqtrd 2776 1 (𝑈 ∈ CPreHilOLD𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  cop 4631  Rel wrel 5689  cfv 6560  1st c1st 8013  2nd c2nd 8014  CVecOLDcvc 30578  NrmCVeccnv 30604   +𝑣 cpv 30605   ·𝑠OLD cns 30607  normCVcnmcv 30610  CPreHilOLDccphlo 30832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-1st 8015  df-2nd 8016  df-vc 30579  df-nv 30612  df-va 30615  df-ba 30616  df-sm 30617  df-0v 30618  df-nmcv 30620  df-ph 30833
This theorem is referenced by:  phpar  30844
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