Theorem hlmet 30718

Description: The induced metric on a complex Hilbert space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
hlmet	⊢ (𝑈 ∈ CHilOLD → 𝐷 ∈ (Met‘𝑋))

Proof of Theorem hlmet

Step	Hyp	Ref	Expression
1		hlcmet.x	. . 3 ⊢ 𝑋 = (BaseSet‘𝑈)
2		hlcmet.8	. . 3 ⊢ 𝐷 = (IndMet‘𝑈)
3	1, 2	hlcmet 30717	. 2 ⊢ (𝑈 ∈ CHilOLD → 𝐷 ∈ (CMet‘𝑋))
4		cmetmet 25227	. 2 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
5	3, 4	syl 17	1 ⊢ (𝑈 ∈ CHilOLD → 𝐷 ∈ (Met‘𝑋))

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  ‘cfv 6548  Metcmet 21265  CMetccmet 25195  BaseSetcba 30409  IndMetcims 30414  CHil_OLDchlo 30708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-cmet 25198  df-cbn 30686  df-hlo 30709

This theorem is referenced by: (None)