Theorem hlcmet 30134

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 30128	. 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan)
2		hlcmet.x	. . 3 ⊢ 𝑋 = (BaseSet‘𝑈)
3		hlcmet.8	. . 3 ⊢ 𝐷 = (IndMet‘𝑈)
4	2, 3	cbncms 30105	. 2 ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))
5	1, 4	syl 17	1 ⊢ (𝑈 ∈ CHilOLD → 𝐷 ∈ (CMet‘𝑋))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  ‘cfv 6540  CMetccmet 24762  BaseSetcba 29826  IndMetcims 29831  CBanccbn 30102  CHil_OLDchlo 30125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-cbn 30103  df-hlo 30126

This theorem is referenced by:  hlmet  30135  hlcompl  30155