Theorem hlcms 8594

Description: The induced metric on a complex Hilbert space is complete.

Hypothesis

Ref	Expression
hlcms.8	|- D = (IndMet` U)

Assertion

Ref	Expression
hlcms	|- (U e. CHil -> D e. CMet)

Proof of Theorem hlcms

Step	Hyp	Ref	Expression
1		hlbn 8588	. 2 |- (U e. CHil -> U e. CBan)
2		hlcms.8	. . 3 |- D = (IndMet` U)
3	2	bncms 8521	. 2 |- (U e. CBan -> D e. CMet)
4	1, 3	syl 10	1 |- (U e. CHil -> D e. CMet)

Syntax hints:   -> wi 3   = wceq 958   e. wcel 960  ` cfv 3188  CMetcms 7918  IndMetcims 8206  CBancbn 8518  CHilchl 8585

This theorem is referenced by:  hlcompl 8613

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204  df-bn 8519  df-hl 8586

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