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Theorem nvvop 30537
Description: The vector space component of a normed complex vector space is an ordered pair of the underlying group and a scalar product. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvvop.1 𝑊 = (1st𝑈)
nvvop.2 𝐺 = ( +𝑣𝑈)
nvvop.4 𝑆 = ( ·𝑠OLD𝑈)
Assertion
Ref Expression
nvvop (𝑈 ∈ NrmCVec → 𝑊 = ⟨𝐺, 𝑆⟩)

Proof of Theorem nvvop
StepHypRef Expression
1 vcrel 30488 . . 3 Rel CVecOLD
2 nvss 30521 . . . . 5 NrmCVec ⊆ (CVecOLD × V)
3 nvvop.1 . . . . . . . 8 𝑊 = (1st𝑈)
4 eqid 2726 . . . . . . . 8 (normCV𝑈) = (normCV𝑈)
53, 4nvop2 30536 . . . . . . 7 (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, (normCV𝑈)⟩)
65eleq1d 2811 . . . . . 6 (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ↔ ⟨𝑊, (normCV𝑈)⟩ ∈ NrmCVec))
76ibi 266 . . . . 5 (𝑈 ∈ NrmCVec → ⟨𝑊, (normCV𝑈)⟩ ∈ NrmCVec)
82, 7sselid 3977 . . . 4 (𝑈 ∈ NrmCVec → ⟨𝑊, (normCV𝑈)⟩ ∈ (CVecOLD × V))
9 opelxp1 5715 . . . 4 (⟨𝑊, (normCV𝑈)⟩ ∈ (CVecOLD × V) → 𝑊 ∈ CVecOLD)
108, 9syl 17 . . 3 (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD)
11 1st2nd 8043 . . 3 ((Rel CVecOLD𝑊 ∈ CVecOLD) → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
121, 10, 11sylancr 585 . 2 (𝑈 ∈ NrmCVec → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
13 nvvop.2 . . . . 5 𝐺 = ( +𝑣𝑈)
1413vafval 30531 . . . 4 𝐺 = (1st ‘(1st𝑈))
153fveq2i 6894 . . . 4 (1st𝑊) = (1st ‘(1st𝑈))
1614, 15eqtr4i 2757 . . 3 𝐺 = (1st𝑊)
17 nvvop.4 . . . . 5 𝑆 = ( ·𝑠OLD𝑈)
1817smfval 30533 . . . 4 𝑆 = (2nd ‘(1st𝑈))
193fveq2i 6894 . . . 4 (2nd𝑊) = (2nd ‘(1st𝑈))
2018, 19eqtr4i 2757 . . 3 𝑆 = (2nd𝑊)
2116, 20opeq12i 4877 . 2 𝐺, 𝑆⟩ = ⟨(1st𝑊), (2nd𝑊)⟩
2212, 21eqtr4di 2784 1 (𝑈 ∈ NrmCVec → 𝑊 = ⟨𝐺, 𝑆⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  Vcvv 3463  cop 4630   × cxp 5671  Rel wrel 5678  cfv 6544  1st c1st 7991  2nd c2nd 7992  CVecOLDcvc 30486  NrmCVeccnv 30512   +𝑣 cpv 30513   ·𝑠OLD cns 30515  normCVcnmcv 30518
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 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5295  ax-nul 5302  ax-pr 5424  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3421  df-v 3465  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-oprab 7418  df-1st 7993  df-2nd 7994  df-vc 30487  df-nv 30520  df-va 30523  df-sm 30525  df-nmcv 30528
This theorem is referenced by:  nvi  30542  nvvc  30543  nvop  30604
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