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Theorem phop 28587
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 28584 . . 3 Rel CPreHilOLD
2 1st2nd 7730 . . 3 ((Rel CPreHilOLD𝑈 ∈ CPreHilOLD) → 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
31, 2mpan 688 . 2 (𝑈 ∈ CPreHilOLD𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
4 phop.6 . . . . 5 𝑁 = (normCV𝑈)
54nmcvfval 28376 . . . 4 𝑁 = (2nd𝑈)
65opeq2i 4799 . . 3 ⟨(1st𝑈), 𝑁⟩ = ⟨(1st𝑈), (2nd𝑈)⟩
7 phnv 28583 . . . . 5 (𝑈 ∈ CPreHilOLD𝑈 ∈ NrmCVec)
8 eqid 2819 . . . . . 6 (1st𝑈) = (1st𝑈)
98nvvc 28384 . . . . 5 (𝑈 ∈ NrmCVec → (1st𝑈) ∈ CVecOLD)
10 vcrel 28329 . . . . . . 7 Rel CVecOLD
11 1st2nd 7730 . . . . . . 7 ((Rel CVecOLD ∧ (1st𝑈) ∈ CVecOLD) → (1st𝑈) = ⟨(1st ‘(1st𝑈)), (2nd ‘(1st𝑈))⟩)
1210, 11mpan 688 . . . . . 6 ((1st𝑈) ∈ CVecOLD → (1st𝑈) = ⟨(1st ‘(1st𝑈)), (2nd ‘(1st𝑈))⟩)
13 phop.2 . . . . . . . 8 𝐺 = ( +𝑣𝑈)
1413vafval 28372 . . . . . . 7 𝐺 = (1st ‘(1st𝑈))
15 phop.4 . . . . . . . 8 𝑆 = ( ·𝑠OLD𝑈)
1615smfval 28374 . . . . . . 7 𝑆 = (2nd ‘(1st𝑈))
1714, 16opeq12i 4800 . . . . . 6 𝐺, 𝑆⟩ = ⟨(1st ‘(1st𝑈)), (2nd ‘(1st𝑈))⟩
1812, 17syl6eqr 2872 . . . . 5 ((1st𝑈) ∈ CVecOLD → (1st𝑈) = ⟨𝐺, 𝑆⟩)
197, 9, 183syl 18 . . . 4 (𝑈 ∈ CPreHilOLD → (1st𝑈) = ⟨𝐺, 𝑆⟩)
2019opeq1d 4801 . . 3 (𝑈 ∈ CPreHilOLD → ⟨(1st𝑈), 𝑁⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
216, 20syl5eqr 2868 . 2 (𝑈 ∈ CPreHilOLD → ⟨(1st𝑈), (2nd𝑈)⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
223, 21eqtrd 2854 1 (𝑈 ∈ CPreHilOLD𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1531  wcel 2108  cop 4565  Rel wrel 5553  cfv 6348  1st c1st 7679  2nd c2nd 7680  CVecOLDcvc 28327  NrmCVeccnv 28353   +𝑣 cpv 28354   ·𝑠OLD cns 28356  normCVcnmcv 28359  CPreHilOLDccphlo 28581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  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-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-1st 7681  df-2nd 7682  df-vc 28328  df-nv 28361  df-va 28364  df-ba 28365  df-sm 28366  df-0v 28367  df-nmcv 28369  df-ph 28582
This theorem is referenced by:  phpar  28593
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