Theorem hlmet 30135

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 30134	. 2 ⊢ (𝑈 ∈ CHilOLD → 𝐷 ∈ (CMet‘𝑋))
4		cmetmet 24794	. 2 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
5	3, 4	syl 17	1 ⊢ (𝑈 ∈ CHilOLD → 𝐷 ∈ (Met‘𝑋))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  ‘cfv 6540  Metcmet 20922  CMetccmet 24762  BaseSetcba 29826  IndMetcims 29831  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-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

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-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  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-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-cmet 24765  df-cbn 30103  df-hlo 30126

This theorem is referenced by: (None)