Theorem nvex 27466

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 27449	. . 3 ⊢ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec → ⟨𝐺, 𝑆⟩ ∈ CVecOLD)
2		vcex 27433	. . 3 ⊢ (⟨𝐺, 𝑆⟩ ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V))
3	1, 2	syl 17	. 2 ⊢ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec → (𝐺 ∈ V ∧ 𝑆 ∈ V))
4		nvss 27448	. . . 4 ⊢ NrmCVec ⊆ (CVecOLD × V)
5	4	sseli 3599	. . 3 ⊢ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec → ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ (CVecOLD × V))
6		opelxp2 5151	. . 3 ⊢ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ (CVecOLD × V) → 𝑁 ∈ V)
7	5, 6	syl 17	. 2 ⊢ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec → 𝑁 ∈ V)
8		df-3an 1039	. 2 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) ↔ ((𝐺 ∈ V ∧ 𝑆 ∈ V) ∧ 𝑁 ∈ V))
9	3, 7, 8	sylanbrc 698	1 ⊢ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec → (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   ∈ wcel 1990  Vcvv 3200  ⟨cop 4183   × cxp 5112  CVec_OLDcvc 27413  NrmCVeccnv 27439

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-oprab 6654  df-vc 27414  df-nv 27447

This theorem is referenced by:  isnv  27467  h2hva  27831  h2hsm  27832  h2hnm  27833