Theorem hlmet 30842

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 30841	. 2 ⊢ (𝑈 ∈ CHilOLD → 𝐷 ∈ (CMet‘𝑋))
4		cmetmet 25256	. 2 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
5	3, 4	syl 17	1 ⊢ (𝑈 ∈ CHilOLD → 𝐷 ∈ (Met‘𝑋))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  ‘cfv 6541  Metcmet 21312  CMetccmet 25224  BaseSetcba 30533  IndMetcims 30538  CHil_OLDchlo 30832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-iota 6494  df-fun 6543  df-fv 6549  df-ov 7416  df-cmet 25227  df-cbn 30810  df-hlo 30833

This theorem is referenced by: (None)