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Theorem nvop2 28369
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 28363 . . 3 Rel NrmCVec
2 1st2nd 7713 . . 3 ((Rel NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
31, 2mpan 689 . 2 (𝑈 ∈ NrmCVec → 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
4 nvop2.1 . . 3 𝑊 = (1st𝑈)
5 nvop2.6 . . . 4 𝑁 = (normCV𝑈)
65nmcvfval 28368 . . 3 𝑁 = (2nd𝑈)
74, 6opeq12i 4781 . 2 𝑊, 𝑁⟩ = ⟨(1st𝑈), (2nd𝑈)⟩
83, 7syl6eqr 2874 1 (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, 𝑁⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  cop 4546  Rel wrel 5533  cfv 6328  1st c1st 7662  2nd c2nd 7663  NrmCVeccnv 28345  normCVcnmcv 28351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-iota 6287  df-fun 6330  df-fv 6336  df-oprab 7134  df-1st 7664  df-2nd 7665  df-nv 28353  df-nmcv 28361
This theorem is referenced by:  nvvop  28370  nvi  28375
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