Theorem hlcmet 30923

Description: The induced metric on a complex Hilbert space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hlcmet.x	⊢ 𝑋 = (BaseSet‘𝑈)
hlcmet.8	⊢ 𝐷 = (IndMet‘𝑈)

Assertion

Ref	Expression
hlcmet	⊢ (𝑈 ∈ CHilOLD → 𝐷 ∈ (CMet‘𝑋))

Proof of Theorem hlcmet

Step	Hyp	Ref	Expression
1		hlobn 30917	. 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan)
2		hlcmet.x	. . 3 ⊢ 𝑋 = (BaseSet‘𝑈)
3		hlcmet.8	. . 3 ⊢ 𝐷 = (IndMet‘𝑈)
4	2, 3	cbncms 30894	. 2 ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))
5	1, 4	syl 17	1 ⊢ (𝑈 ∈ CHilOLD → 𝐷 ∈ (CMet‘𝑋))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  ‘cfv 6563  CMetccmet 25302  BaseSetcba 30615  IndMetcims 30620  CBanccbn 30891  CHil_OLDchlo 30914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-cbn 30892  df-hlo 30915

This theorem is referenced by:  hlmet  30924  hlcompl  30944