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

Theorem nvop2 27743
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 27737 . . 3 Rel NrmCVec
2 1st2nd 7369 . . 3 ((Rel NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
31, 2mpan 708 . 2 (𝑈 ∈ NrmCVec → 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
4 nvop2.1 . . 3 𝑊 = (1st𝑈)
5 nvop2.6 . . . 4 𝑁 = (normCV𝑈)
65nmcvfval 27742 . . 3 𝑁 = (2nd𝑈)
74, 6opeq12i 4546 . 2 𝑊, 𝑁⟩ = ⟨(1st𝑈), (2nd𝑈)⟩
83, 7syl6eqr 2800 1 (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, 𝑁⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1620  wcel 2127  cop 4315  Rel wrel 5259  cfv 6037  1st c1st 7319  2nd c2nd 7320  NrmCVeccnv 27719  normCVcnmcv 27725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-iota 6000  df-fun 6039  df-fv 6045  df-oprab 6805  df-1st 7321  df-2nd 7322  df-nv 27727  df-nmcv 27735
This theorem is referenced by:  nvvop  27744  nvi  27749
  Copyright terms: Public domain W3C validator