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Theorem nvop2 30755
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 30749 . . 3 Rel NrmCVec
2 1st2nd 8014 . . 3 ((Rel NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
31, 2mpan 700 . 2 (𝑈 ∈ NrmCVec → 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
4 nvop2.1 . . 3 𝑊 = (1st𝑈)
5 nvop2.6 . . . 4 𝑁 = (normCV𝑈)
65nmcvfval 30754 . . 3 𝑁 = (2nd𝑈)
74, 6opeq12i 4835 . 2 𝑊, 𝑁⟩ = ⟨(1st𝑈), (2nd𝑈)⟩
83, 7eqtr4di 2814 1 (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, 𝑁⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1559  wcel 2141  cop 4587  Rel wrel 5650  cfv 6515  1st c1st 7962  2nd c2nd 7963  NrmCVeccnv 30731  normCVcnmcv 30737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7712
This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-iota 6471  df-fun 6517  df-fv 6523  df-oprab 7394  df-1st 7964  df-2nd 7965  df-nv 30739  df-nmcv 30747
This theorem is referenced by:  nvvop  30756  nvi  30761
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