Theorem hlcmet 27750

Description: The induced metric on a complex Hilbert space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hlcmet.x	|- X = ( BaseSet ` U )
hlcmet.8	|- D = ( IndMet ` U )

Assertion

Ref	Expression
hlcmet	|- ( U e. CHilOLD -> D e. ( CMet ` X ) )

Proof of Theorem hlcmet

Step	Hyp	Ref	Expression
1		hlobn 27744	. 2 |- ( U e. CHilOLD -> U e. CBan )
2		hlcmet.x	. . 3 |- X = ( BaseSet ` U )
3		hlcmet.8	. . 3 |- D = ( IndMet ` U )
4	2, 3	cbncms 27721	. 2 |- ( U e. CBan -> D e. ( CMet ` X ) )
5	1, 4	syl 17	1 |- ( U e. CHilOLD -> D e. ( CMet ` X ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888   CMetcms 23052   BaseSetcba 27441   IndMetcims 27446   CBanccbn 27718   CHilOLDchlo 27741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-cbn 27719  df-hlo 27742

This theorem is referenced by:  hlmet  27751  hlcompl  27771