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Theorem nvvcop 27840
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 27839 . . 3 NrmCVec ⊆ (CVecOLD × V)
21sseli 3757 . 2 (⟨𝑊, 𝑁⟩ ∈ NrmCVec → ⟨𝑊, 𝑁⟩ ∈ (CVecOLD × V))
3 opelxp1 5318 . 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 2155  Vcvv 3350  cop 4340   × cxp 5275  CVecOLDcvc 27804  NrmCVeccnv 27830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-opab 4872  df-xp 5283  df-oprab 6846  df-nv 27838
This theorem is referenced by:  nvex  27857
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