Theorem hlnv 30827

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 30824	. 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan)
2		bnnv 30802	. 2 ⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec)
3	1, 2	syl 17	1 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec)

Syntax hints:   → wi 4   ∈ wcel 2109  NrmCVeccnv 30520  CBanccbn 30798  CHil_OLDchlo 30821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-cbn 30799  df-hlo 30822

This theorem is referenced by:  hlnvi  30828  hlvc  30829  hladdf  30835  hlcom  30836  hlass  30837  hl0cl  30838  hladdid  30839  hlmulf  30840  hlmulid  30841  hlmulass  30842  hldi  30843  hldir  30844  hlmul0  30845  hlipf  30846  hlipcj  30847  hlipgt0  30850