Theorem hlnv 30978

Description: Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Assertion

Ref	Expression
hlnv	⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec)

Proof of Theorem hlnv

Step	Hyp	Ref	Expression
1		hlobn 30975	. 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan)
2		bnnv 30953	. 2 ⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec)
3	1, 2	syl 17	1 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec)

Syntax hints:   → wi 4   ∈ wcel 2114  NrmCVeccnv 30671  CBanccbn 30949  CHil_OLDchlo 30972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-cbn 30950  df-hlo 30973

This theorem is referenced by:  hlnvi  30979  hlvc  30980  hladdf  30986  hlcom  30987  hlass  30988  hl0cl  30989  hladdid  30990  hlmulf  30991  hlmulid  30992  hlmulass  30993  hldi  30994  hldir  30995  hlmul0  30996  hlipf  30997  hlipcj  30998  hlipgt0  31001