Theorem hlnv 30818

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 30815	. 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan)
2		bnnv 30793	. 2 ⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec)
3	1, 2	syl 17	1 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec)

Syntax hints:   → wi 4   ∈ wcel 2108  NrmCVeccnv 30511  CBanccbn 30789  CHil_OLDchlo 30812

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6483  df-fv 6538  df-cbn 30790  df-hlo 30813

This theorem is referenced by:  hlnvi  30819  hlvc  30820  hladdf  30826  hlcom  30827  hlass  30828  hl0cl  30829  hladdid  30830  hlmulf  30831  hlmulid  30832  hlmulass  30833  hldi  30834  hldir  30835  hlmul0  30836  hlipf  30837  hlipcj  30838  hlipgt0  30841