Theorem nvex 30707

Description: The components of a normed complex vector space are sets. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 1-May-2015.) (New usage is discouraged.)

Assertion

Ref	Expression
nvex	⊢ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec → (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V))

Proof of Theorem nvex

Step	Hyp	Ref	Expression
1		nvvcop 30690	. . 3 ⊢ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec → ⟨𝐺, 𝑆⟩ ∈ CVecOLD)
2		vcex 30674	. . 3 ⊢ (⟨𝐺, 𝑆⟩ ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V))
3	1, 2	syl 17	. 2 ⊢ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec → (𝐺 ∈ V ∧ 𝑆 ∈ V))
4		nvss 30689	. . . 4 ⊢ NrmCVec ⊆ (CVecOLD × V)
5	4	sseli 3918	. . 3 ⊢ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec → ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ (CVecOLD × V))
6		opelxp2 5668	. . 3 ⊢ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ (CVecOLD × V) → 𝑁 ∈ V)
7	5, 6	syl 17	. 2 ⊢ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec → 𝑁 ∈ V)
8		df-3an 1094	. 2 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) ↔ ((𝐺 ∈ V ∧ 𝑆 ∈ V) ∧ 𝑁 ∈ V))
9	3, 7, 8	sylanbrc 589	1 ⊢ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec → (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   ∈ wcel 2119  Vcvv 3432  ⟨cop 4568   × cxp 5623  CVec_OLDcvc 30654  NrmCVeccnv 30680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-11 2168  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-xp 5631  df-rel 5632  df-oprab 7367  df-vc 30655  df-nv 30688

This theorem is referenced by:  isnv  30708  h2hva  31070  h2hsm  31071  h2hnm  31072