MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nvop2 Structured version   Visualization version   GIF version

Theorem nvop2 30637
Description: A normed complex vector space is an ordered pair of a vector space and a norm operation. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvop2.1 𝑊 = (1st𝑈)
nvop2.6 𝑁 = (normCV𝑈)
Assertion
Ref Expression
nvop2 (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, 𝑁⟩)

Proof of Theorem nvop2
StepHypRef Expression
1 nvrel 30631 . . 3 Rel NrmCVec
2 1st2nd 8063 . . 3 ((Rel NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
31, 2mpan 690 . 2 (𝑈 ∈ NrmCVec → 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
4 nvop2.1 . . 3 𝑊 = (1st𝑈)
5 nvop2.6 . . . 4 𝑁 = (normCV𝑈)
65nmcvfval 30636 . . 3 𝑁 = (2nd𝑈)
74, 6opeq12i 4883 . 2 𝑊, 𝑁⟩ = ⟨(1st𝑈), (2nd𝑈)⟩
83, 7eqtr4di 2793 1 (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, 𝑁⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cop 4637  Rel wrel 5694  cfv 6563  1st c1st 8011  2nd c2nd 8012  NrmCVeccnv 30613  normCVcnmcv 30619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-oprab 7435  df-1st 8013  df-2nd 8014  df-nv 30621  df-nmcv 30629
This theorem is referenced by:  nvvop  30638  nvi  30643
  Copyright terms: Public domain W3C validator