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Theorem nvvop 27773
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 27724 . . 3 Rel CVecOLD
2 nvss 27757 . . . . 5 NrmCVec ⊆ (CVecOLD × V)
3 nvvop.1 . . . . . . . 8 𝑊 = (1st𝑈)
4 eqid 2760 . . . . . . . 8 (normCV𝑈) = (normCV𝑈)
53, 4nvop2 27772 . . . . . . 7 (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, (normCV𝑈)⟩)
65eleq1d 2824 . . . . . 6 (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ↔ ⟨𝑊, (normCV𝑈)⟩ ∈ NrmCVec))
76ibi 256 . . . . 5 (𝑈 ∈ NrmCVec → ⟨𝑊, (normCV𝑈)⟩ ∈ NrmCVec)
82, 7sseldi 3742 . . . 4 (𝑈 ∈ NrmCVec → ⟨𝑊, (normCV𝑈)⟩ ∈ (CVecOLD × V))
9 opelxp1 5307 . . . 4 (⟨𝑊, (normCV𝑈)⟩ ∈ (CVecOLD × V) → 𝑊 ∈ CVecOLD)
108, 9syl 17 . . 3 (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD)
11 1st2nd 7381 . . 3 ((Rel CVecOLD𝑊 ∈ CVecOLD) → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
121, 10, 11sylancr 698 . 2 (𝑈 ∈ NrmCVec → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
13 nvvop.2 . . . . 5 𝐺 = ( +𝑣𝑈)
1413vafval 27767 . . . 4 𝐺 = (1st ‘(1st𝑈))
153fveq2i 6355 . . . 4 (1st𝑊) = (1st ‘(1st𝑈))
1614, 15eqtr4i 2785 . . 3 𝐺 = (1st𝑊)
17 nvvop.4 . . . . 5 𝑆 = ( ·𝑠OLD𝑈)
1817smfval 27769 . . . 4 𝑆 = (2nd ‘(1st𝑈))
193fveq2i 6355 . . . 4 (2nd𝑊) = (2nd ‘(1st𝑈))
2018, 19eqtr4i 2785 . . 3 𝑆 = (2nd𝑊)
2116, 20opeq12i 4558 . 2 𝐺, 𝑆⟩ = ⟨(1st𝑊), (2nd𝑊)⟩
2212, 21syl6eqr 2812 1 (𝑈 ∈ NrmCVec → 𝑊 = ⟨𝐺, 𝑆⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  cop 4327   × cxp 5264  Rel wrel 5271  cfv 6049  1st c1st 7331  2nd c2nd 7332  CVecOLDcvc 27722  NrmCVeccnv 27748   +𝑣 cpv 27749   ·𝑠OLD cns 27751  normCVcnmcv 27754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fo 6055  df-fv 6057  df-oprab 6817  df-1st 7333  df-2nd 7334  df-vc 27723  df-nv 27756  df-va 27759  df-sm 27761  df-nmcv 27764
This theorem is referenced by:  nvi  27778  nvvc  27779  nvop  27840
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