Theorem hlmet 28666

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 28665	. 2 ⊢ (𝑈 ∈ CHilOLD → 𝐷 ∈ (CMet‘𝑋))
4		cmetmet 23883	. 2 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
5	3, 4	syl 17	1 ⊢ (𝑈 ∈ CHilOLD → 𝐷 ∈ (Met‘𝑋))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110  ‘cfv 6350  Metcmet 20525  CMetccmet 23851  BaseSetcba 28357  IndMetcims 28362  CHil_OLDchlo 28656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-iota 6309  df-fun 6352  df-fv 6358  df-ov 7153  df-cmet 23854  df-cbn 28634  df-hlo 28657

This theorem is referenced by: (None)