Theorem hlcmet 29256

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 29250	. 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan)
2		hlcmet.x	. . 3 ⊢ 𝑋 = (BaseSet‘𝑈)
3		hlcmet.8	. . 3 ⊢ 𝐷 = (IndMet‘𝑈)
4	2, 3	cbncms 29227	. 2 ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))
5	1, 4	syl 17	1 ⊢ (𝑈 ∈ CHilOLD → 𝐷 ∈ (CMet‘𝑋))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ‘cfv 6433  CMetccmet 24418  BaseSetcba 28948  IndMetcims 28953  CBanccbn 29224  CHil_OLDchlo 29247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-cbn 29225  df-hlo 29248

This theorem is referenced by:  hlmet  29257  hlcompl  29277