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Theorem phop 26891
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 26888 . . 3 Rel CPreHilOLD
2 1st2nd 7083 . . 3 ((Rel CPreHilOLD𝑈 ∈ CPreHilOLD) → 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
31, 2mpan 701 . 2 (𝑈 ∈ CPreHilOLD𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
4 phop.6 . . . . 5 𝑁 = (normCV𝑈)
54nmcvfval 26658 . . . 4 𝑁 = (2nd𝑈)
65opeq2i 4338 . . 3 ⟨(1st𝑈), 𝑁⟩ = ⟨(1st𝑈), (2nd𝑈)⟩
7 phnv 26887 . . . . 5 (𝑈 ∈ CPreHilOLD𝑈 ∈ NrmCVec)
8 eqid 2609 . . . . . 6 (1st𝑈) = (1st𝑈)
98nvvc 26666 . . . . 5 (𝑈 ∈ NrmCVec → (1st𝑈) ∈ CVecOLD)
10 vcrel 26596 . . . . . . 7 Rel CVecOLD
11 1st2nd 7083 . . . . . . 7 ((Rel CVecOLD ∧ (1st𝑈) ∈ CVecOLD) → (1st𝑈) = ⟨(1st ‘(1st𝑈)), (2nd ‘(1st𝑈))⟩)
1210, 11mpan 701 . . . . . 6 ((1st𝑈) ∈ CVecOLD → (1st𝑈) = ⟨(1st ‘(1st𝑈)), (2nd ‘(1st𝑈))⟩)
13 phop.2 . . . . . . . 8 𝐺 = ( +𝑣𝑈)
1413vafval 26654 . . . . . . 7 𝐺 = (1st ‘(1st𝑈))
15 phop.4 . . . . . . . 8 𝑆 = ( ·𝑠OLD𝑈)
1615smfval 26656 . . . . . . 7 𝑆 = (2nd ‘(1st𝑈))
1714, 16opeq12i 4339 . . . . . 6 𝐺, 𝑆⟩ = ⟨(1st ‘(1st𝑈)), (2nd ‘(1st𝑈))⟩
1812, 17syl6eqr 2661 . . . . 5 ((1st𝑈) ∈ CVecOLD → (1st𝑈) = ⟨𝐺, 𝑆⟩)
197, 9, 183syl 18 . . . 4 (𝑈 ∈ CPreHilOLD → (1st𝑈) = ⟨𝐺, 𝑆⟩)
2019opeq1d 4340 . . 3 (𝑈 ∈ CPreHilOLD → ⟨(1st𝑈), 𝑁⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
216, 20syl5eqr 2657 . 2 (𝑈 ∈ CPreHilOLD → ⟨(1st𝑈), (2nd𝑈)⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
223, 21eqtrd 2643 1 (𝑈 ∈ CPreHilOLD𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  cop 4130  Rel wrel 5033  cfv 5790  1st c1st 7035  2nd c2nd 7036  CVecOLDcvc 26594  NrmCVeccnv 26635   +𝑣 cpv 26636   ·𝑠OLD cns 26638  normCVcnmcv 26641  CPreHilOLDccphlo 26885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-1st 7037  df-2nd 7038  df-vc 26595  df-nv 26643  df-va 26646  df-ba 26647  df-sm 26648  df-0v 26649  df-nmcv 26651  df-ph 26886
This theorem is referenced by:  phpar  26897
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