Theorem hlcmet 30838

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 30832	. 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan)
2		hlcmet.x	. . 3 ⊢ 𝑋 = (BaseSet‘𝑈)
3		hlcmet.8	. . 3 ⊢ 𝐷 = (IndMet‘𝑈)
4	2, 3	cbncms 30809	. 2 ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))
5	1, 4	syl 17	1 ⊢ (𝑈 ∈ CHilOLD → 𝐷 ∈ (CMet‘𝑋))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ‘cfv 6482  CMetccmet 25152  BaseSetcba 30530  IndMetcims 30535  CBanccbn 30806  CHil_OLDchlo 30829

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 2701

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 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-iota 6438  df-fv 6490  df-cbn 30807  df-hlo 30830

This theorem is referenced by:  hlmet  30839  hlcompl  30859