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Theorem nvvop 28402
 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 28353 . . 3 Rel CVecOLD
2 nvss 28386 . . . . 5 NrmCVec ⊆ (CVecOLD × V)
3 nvvop.1 . . . . . . . 8 𝑊 = (1st𝑈)
4 eqid 2798 . . . . . . . 8 (normCV𝑈) = (normCV𝑈)
53, 4nvop2 28401 . . . . . . 7 (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, (normCV𝑈)⟩)
65eleq1d 2874 . . . . . 6 (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ↔ ⟨𝑊, (normCV𝑈)⟩ ∈ NrmCVec))
76ibi 270 . . . . 5 (𝑈 ∈ NrmCVec → ⟨𝑊, (normCV𝑈)⟩ ∈ NrmCVec)
82, 7sseldi 3913 . . . 4 (𝑈 ∈ NrmCVec → ⟨𝑊, (normCV𝑈)⟩ ∈ (CVecOLD × V))
9 opelxp1 5561 . . . 4 (⟨𝑊, (normCV𝑈)⟩ ∈ (CVecOLD × V) → 𝑊 ∈ CVecOLD)
108, 9syl 17 . . 3 (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD)
11 1st2nd 7723 . . 3 ((Rel CVecOLD𝑊 ∈ CVecOLD) → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
121, 10, 11sylancr 590 . 2 (𝑈 ∈ NrmCVec → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
13 nvvop.2 . . . . 5 𝐺 = ( +𝑣𝑈)
1413vafval 28396 . . . 4 𝐺 = (1st ‘(1st𝑈))
153fveq2i 6649 . . . 4 (1st𝑊) = (1st ‘(1st𝑈))
1614, 15eqtr4i 2824 . . 3 𝐺 = (1st𝑊)
17 nvvop.4 . . . . 5 𝑆 = ( ·𝑠OLD𝑈)
1817smfval 28398 . . . 4 𝑆 = (2nd ‘(1st𝑈))
193fveq2i 6649 . . . 4 (2nd𝑊) = (2nd ‘(1st𝑈))
2018, 19eqtr4i 2824 . . 3 𝑆 = (2nd𝑊)
2116, 20opeq12i 4771 . 2 𝐺, 𝑆⟩ = ⟨(1st𝑊), (2nd𝑊)⟩
2212, 21eqtr4di 2851 1 (𝑈 ∈ NrmCVec → 𝑊 = ⟨𝐺, 𝑆⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ⟨cop 4531   × cxp 5518  Rel wrel 5525  ‘cfv 6325  1st c1st 7672  2nd c2nd 7673  CVecOLDcvc 28351  NrmCVeccnv 28377   +𝑣 cpv 28378   ·𝑠OLD cns 28380  normCVcnmcv 28383 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-fo 6331  df-fv 6333  df-oprab 7140  df-1st 7674  df-2nd 7675  df-vc 28352  df-nv 28385  df-va 28388  df-sm 28390  df-nmcv 28393 This theorem is referenced by:  nvi  28407  nvvc  28408  nvop  28469
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