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Theorem nvvop 28178
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 28129 . . 3 Rel CVecOLD
2 nvss 28162 . . . . 5 NrmCVec ⊆ (CVecOLD × V)
3 nvvop.1 . . . . . . . 8 𝑊 = (1st𝑈)
4 eqid 2780 . . . . . . . 8 (normCV𝑈) = (normCV𝑈)
53, 4nvop2 28177 . . . . . . 7 (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, (normCV𝑈)⟩)
65eleq1d 2852 . . . . . 6 (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ↔ ⟨𝑊, (normCV𝑈)⟩ ∈ NrmCVec))
76ibi 259 . . . . 5 (𝑈 ∈ NrmCVec → ⟨𝑊, (normCV𝑈)⟩ ∈ NrmCVec)
82, 7sseldi 3858 . . . 4 (𝑈 ∈ NrmCVec → ⟨𝑊, (normCV𝑈)⟩ ∈ (CVecOLD × V))
9 opelxp1 5452 . . . 4 (⟨𝑊, (normCV𝑈)⟩ ∈ (CVecOLD × V) → 𝑊 ∈ CVecOLD)
108, 9syl 17 . . 3 (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD)
11 1st2nd 7556 . . 3 ((Rel CVecOLD𝑊 ∈ CVecOLD) → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
121, 10, 11sylancr 579 . 2 (𝑈 ∈ NrmCVec → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
13 nvvop.2 . . . . 5 𝐺 = ( +𝑣𝑈)
1413vafval 28172 . . . 4 𝐺 = (1st ‘(1st𝑈))
153fveq2i 6507 . . . 4 (1st𝑊) = (1st ‘(1st𝑈))
1614, 15eqtr4i 2807 . . 3 𝐺 = (1st𝑊)
17 nvvop.4 . . . . 5 𝑆 = ( ·𝑠OLD𝑈)
1817smfval 28174 . . . 4 𝑆 = (2nd ‘(1st𝑈))
193fveq2i 6507 . . . 4 (2nd𝑊) = (2nd ‘(1st𝑈))
2018, 19eqtr4i 2807 . . 3 𝑆 = (2nd𝑊)
2116, 20opeq12i 4687 . 2 𝐺, 𝑆⟩ = ⟨(1st𝑊), (2nd𝑊)⟩
2212, 21syl6eqr 2834 1 (𝑈 ∈ NrmCVec → 𝑊 = ⟨𝐺, 𝑆⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1508  wcel 2051  Vcvv 3417  cop 4450   × cxp 5409  Rel wrel 5416  cfv 6193  1st c1st 7505  2nd c2nd 7506  CVecOLDcvc 28127  NrmCVeccnv 28153   +𝑣 cpv 28154   ·𝑠OLD cns 28156  normCVcnmcv 28159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2752  ax-sep 5064  ax-nul 5071  ax-pow 5123  ax-pr 5190  ax-un 7285
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2551  df-eu 2589  df-clab 2761  df-cleq 2773  df-clel 2848  df-nfc 2920  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3419  df-sbc 3684  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4182  df-if 4354  df-sn 4445  df-pr 4447  df-op 4451  df-uni 4718  df-br 4935  df-opab 4997  df-mpt 5014  df-id 5316  df-xp 5417  df-rel 5418  df-cnv 5419  df-co 5420  df-dm 5421  df-rn 5422  df-res 5423  df-ima 5424  df-iota 6157  df-fun 6195  df-fn 6196  df-f 6197  df-fo 6199  df-fv 6201  df-oprab 6986  df-1st 7507  df-2nd 7508  df-vc 28128  df-nv 28161  df-va 28164  df-sm 28166  df-nmcv 28169
This theorem is referenced by:  nvi  28183  nvvc  28184  nvop  28245
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