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Theorem phop 27657
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 27654 . . 3 Rel CPreHilOLD
2 1st2nd 7211 . . 3 ((Rel CPreHilOLD𝑈 ∈ CPreHilOLD) → 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
31, 2mpan 706 . 2 (𝑈 ∈ CPreHilOLD𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
4 phop.6 . . . . 5 𝑁 = (normCV𝑈)
54nmcvfval 27446 . . . 4 𝑁 = (2nd𝑈)
65opeq2i 4404 . . 3 ⟨(1st𝑈), 𝑁⟩ = ⟨(1st𝑈), (2nd𝑈)⟩
7 phnv 27653 . . . . 5 (𝑈 ∈ CPreHilOLD𝑈 ∈ NrmCVec)
8 eqid 2621 . . . . . 6 (1st𝑈) = (1st𝑈)
98nvvc 27454 . . . . 5 (𝑈 ∈ NrmCVec → (1st𝑈) ∈ CVecOLD)
10 vcrel 27399 . . . . . . 7 Rel CVecOLD
11 1st2nd 7211 . . . . . . 7 ((Rel CVecOLD ∧ (1st𝑈) ∈ CVecOLD) → (1st𝑈) = ⟨(1st ‘(1st𝑈)), (2nd ‘(1st𝑈))⟩)
1210, 11mpan 706 . . . . . 6 ((1st𝑈) ∈ CVecOLD → (1st𝑈) = ⟨(1st ‘(1st𝑈)), (2nd ‘(1st𝑈))⟩)
13 phop.2 . . . . . . . 8 𝐺 = ( +𝑣𝑈)
1413vafval 27442 . . . . . . 7 𝐺 = (1st ‘(1st𝑈))
15 phop.4 . . . . . . . 8 𝑆 = ( ·𝑠OLD𝑈)
1615smfval 27444 . . . . . . 7 𝑆 = (2nd ‘(1st𝑈))
1714, 16opeq12i 4405 . . . . . 6 𝐺, 𝑆⟩ = ⟨(1st ‘(1st𝑈)), (2nd ‘(1st𝑈))⟩
1812, 17syl6eqr 2673 . . . . 5 ((1st𝑈) ∈ CVecOLD → (1st𝑈) = ⟨𝐺, 𝑆⟩)
197, 9, 183syl 18 . . . 4 (𝑈 ∈ CPreHilOLD → (1st𝑈) = ⟨𝐺, 𝑆⟩)
2019opeq1d 4406 . . 3 (𝑈 ∈ CPreHilOLD → ⟨(1st𝑈), 𝑁⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
216, 20syl5eqr 2669 . 2 (𝑈 ∈ CPreHilOLD → ⟨(1st𝑈), (2nd𝑈)⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
223, 21eqtrd 2655 1 (𝑈 ∈ CPreHilOLD𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  wcel 1989  cop 4181  Rel wrel 5117  cfv 5886  1st c1st 7163  2nd c2nd 7164  CVecOLDcvc 27397  NrmCVeccnv 27423   +𝑣 cpv 27424   ·𝑠OLD cns 27426  normCVcnmcv 27429  CPreHilOLDccphlo 27651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-1st 7165  df-2nd 7166  df-vc 27398  df-nv 27431  df-va 27434  df-ba 27435  df-sm 27436  df-0v 27437  df-nmcv 27439  df-ph 27652
This theorem is referenced by:  phpar  27663
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