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

Step	Hyp	Ref	Expression
1		nvss 27753	. . 3 ⊢ NrmCVec ⊆ (CVecOLD × V)
2	1	sseli 3736	. 2 ⊢ (⟨𝑊, 𝑁⟩ ∈ NrmCVec → ⟨𝑊, 𝑁⟩ ∈ (CVecOLD × V))
3		opelxp1 5303	. 2 ⊢ (⟨𝑊, 𝑁⟩ ∈ (CVecOLD × V) → 𝑊 ∈ CVecOLD)
4	2, 3	syl 17	1 ⊢ (⟨𝑊, 𝑁⟩ ∈ NrmCVec → 𝑊 ∈ CVecOLD)

Syntax hints:   → wi 4   ∈ wcel 2135  Vcvv 3336  ⟨cop 4323   × cxp 5260  CVec_OLDcvc 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