Theorem hlcmet 31529

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 31523	. 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan)
2		hlcmet.x	. . 3 ⊢ 𝑋 = (BaseSet‘𝑈)
3		hlcmet.8	. . 3 ⊢ 𝐷 = (IndMet‘𝑈)
4	2, 3	cbncms 31500	. 2 ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))
5	1, 4	syl 18	1 ⊢ (𝑈 ∈ CHilOLD → 𝐷 ∈ (CMet‘𝑋))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ‘cfv 6561  CMetccmet 25608  BaseSetcba 31221  IndMetcims 31226  CBanccbn 31497  CHil_OLDchlo 31520

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6515  df-fv 6569  df-cbn 31498  df-hlo 31521

This theorem is used by:  hlmet  31530  hlcompl  31550