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Theorem nvvcop 27754
Description: A normed complex vector space is a vector space. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 1-May-2015.) (New usage is discouraged.)
Assertion
Ref Expression
nvvcop (⟨𝑊, 𝑁⟩ ∈ NrmCVec → 𝑊 ∈ CVecOLD)

Proof of Theorem nvvcop
StepHypRef Expression
1 nvss 27753 . . 3 NrmCVec ⊆ (CVecOLD × V)
21sseli 3736 . 2 (⟨𝑊, 𝑁⟩ ∈ NrmCVec → ⟨𝑊, 𝑁⟩ ∈ (CVecOLD × V))
3 opelxp1 5303 . 2 (⟨𝑊, 𝑁⟩ ∈ (CVecOLD × V) → 𝑊 ∈ CVecOLD)
42, 3syl 17 1 (⟨𝑊, 𝑁⟩ ∈ NrmCVec → 𝑊 ∈ CVecOLD)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2135  Vcvv 3336  cop 4323   × cxp 5260  CVecOLDcvc 27718  NrmCVeccnv 27744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pr 5051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-opab 4861  df-xp 5268  df-oprab 6813  df-nv 27752
This theorem is referenced by:  nvex  27771
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