Theorem hlmet 30830

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 30829	. 2 ⊢ (𝑈 ∈ CHilOLD → 𝐷 ∈ (CMet‘𝑋))
4		cmetmet 25192	. 2 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
5	3, 4	syl 17	1 ⊢ (𝑈 ∈ CHilOLD → 𝐷 ∈ (Met‘𝑋))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ‘cfv 6513  Metcmet 21256  CMetccmet 25160  BaseSetcba 30521  IndMetcims 30526  CHil_OLDchlo 30820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-iota 6466  df-fun 6515  df-fv 6521  df-ov 7392  df-cmet 25163  df-cbn 30798  df-hlo 30821

This theorem is referenced by: (None)