Theorem nvvcop 28956

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 28955	. . 3 ⊢ NrmCVec ⊆ (CVecOLD × V)
2	1	sseli 3917	. 2 ⊢ (⟨𝑊, 𝑁⟩ ∈ NrmCVec → ⟨𝑊, 𝑁⟩ ∈ (CVecOLD × V))
3		opelxp1 5630	. 2 ⊢ (⟨𝑊, 𝑁⟩ ∈ (CVecOLD × V) → 𝑊 ∈ CVecOLD)
4	2, 3	syl 17	1 ⊢ (⟨𝑊, 𝑁⟩ ∈ NrmCVec → 𝑊 ∈ CVecOLD)

Syntax hints:   → wi 4   ∈ wcel 2106  Vcvv 3432  ⟨cop 4567   × cxp 5587  CVec_OLDcvc 28920  NrmCVeccnv 28946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5137  df-xp 5595  df-oprab 7279  df-nv 28954

This theorem is referenced by:  nvex  28973