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Theorem nvop2 28312
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 28306 . . 3 Rel NrmCVec
2 1st2nd 7727 . . 3 ((Rel NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
31, 2mpan 686 . 2 (𝑈 ∈ NrmCVec → 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
4 nvop2.1 . . 3 𝑊 = (1st𝑈)
5 nvop2.6 . . . 4 𝑁 = (normCV𝑈)
65nmcvfval 28311 . . 3 𝑁 = (2nd𝑈)
74, 6opeq12i 4800 . 2 𝑊, 𝑁⟩ = ⟨(1st𝑈), (2nd𝑈)⟩
83, 7syl6eqr 2871 1 (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, 𝑁⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  cop 4563  Rel wrel 5553  cfv 6348  1st c1st 7676  2nd c2nd 7677  NrmCVeccnv 28288  normCVcnmcv 28294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fv 6356  df-oprab 7149  df-1st 7678  df-2nd 7679  df-nv 28296  df-nmcv 28304
This theorem is referenced by:  nvvop  28313  nvi  28318
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