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Theorem nvvop 29551
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 29502 . . 3 Rel CVecOLD
2 nvss 29535 . . . . 5 NrmCVec ⊆ (CVecOLD × V)
3 nvvop.1 . . . . . . . 8 𝑊 = (1st𝑈)
4 eqid 2736 . . . . . . . 8 (normCV𝑈) = (normCV𝑈)
53, 4nvop2 29550 . . . . . . 7 (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, (normCV𝑈)⟩)
65eleq1d 2822 . . . . . 6 (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ↔ ⟨𝑊, (normCV𝑈)⟩ ∈ NrmCVec))
76ibi 266 . . . . 5 (𝑈 ∈ NrmCVec → ⟨𝑊, (normCV𝑈)⟩ ∈ NrmCVec)
82, 7sselid 3942 . . . 4 (𝑈 ∈ NrmCVec → ⟨𝑊, (normCV𝑈)⟩ ∈ (CVecOLD × V))
9 opelxp1 5674 . . . 4 (⟨𝑊, (normCV𝑈)⟩ ∈ (CVecOLD × V) → 𝑊 ∈ CVecOLD)
108, 9syl 17 . . 3 (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD)
11 1st2nd 7971 . . 3 ((Rel CVecOLD𝑊 ∈ CVecOLD) → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
121, 10, 11sylancr 587 . 2 (𝑈 ∈ NrmCVec → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
13 nvvop.2 . . . . 5 𝐺 = ( +𝑣𝑈)
1413vafval 29545 . . . 4 𝐺 = (1st ‘(1st𝑈))
153fveq2i 6845 . . . 4 (1st𝑊) = (1st ‘(1st𝑈))
1614, 15eqtr4i 2767 . . 3 𝐺 = (1st𝑊)
17 nvvop.4 . . . . 5 𝑆 = ( ·𝑠OLD𝑈)
1817smfval 29547 . . . 4 𝑆 = (2nd ‘(1st𝑈))
193fveq2i 6845 . . . 4 (2nd𝑊) = (2nd ‘(1st𝑈))
2018, 19eqtr4i 2767 . . 3 𝑆 = (2nd𝑊)
2116, 20opeq12i 4835 . 2 𝐺, 𝑆⟩ = ⟨(1st𝑊), (2nd𝑊)⟩
2212, 21eqtr4di 2794 1 (𝑈 ∈ NrmCVec → 𝑊 = ⟨𝐺, 𝑆⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  Vcvv 3445  cop 4592   × cxp 5631  Rel wrel 5638  cfv 6496  1st c1st 7919  2nd c2nd 7920  CVecOLDcvc 29500  NrmCVeccnv 29526   +𝑣 cpv 29527   ·𝑠OLD cns 29529  normCVcnmcv 29532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fo 6502  df-fv 6504  df-oprab 7361  df-1st 7921  df-2nd 7922  df-vc 29501  df-nv 29534  df-va 29537  df-sm 29539  df-nmcv 29542
This theorem is referenced by:  nvi  29556  nvvc  29557  nvop  29618
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