Theorem hlnv 30962

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 30959	. 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan)
2		bnnv 30937	. 2 ⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec)
3	1, 2	syl 17	1 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec)

Syntax hints:   → wi 4   ∈ wcel 2114  NrmCVeccnv 30655  CBanccbn 30933  CHil_OLDchlo 30956

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 2708

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 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-cbn 30934  df-hlo 30957

This theorem is referenced by:  hlnvi  30963  hlvc  30964  hladdf  30970  hlcom  30971  hlass  30972  hl0cl  30973  hladdid  30974  hlmulf  30975  hlmulid  30976  hlmulass  30977  hldi  30978  hldir  30979  hlmul0  30980  hlipf  30981  hlipcj  30982  hlipgt0  30985