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Theorem cbncms 30885
Description: The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) Use bncmet 25382 (or preferably bncms 25379) instead. (New usage is discouraged.)
Hypotheses
Ref Expression
iscbn.x 𝑋 = (BaseSet‘𝑈)
iscbn.8 𝐷 = (IndMet‘𝑈)
Assertion
Ref Expression
cbncms (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))

Proof of Theorem cbncms
StepHypRef Expression
1 iscbn.x . . 3 𝑋 = (BaseSet‘𝑈)
2 iscbn.8 . . 3 𝐷 = (IndMet‘𝑈)
31, 2iscbn 30884 . 2 (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋)))
43simprbi 496 1 (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  cfv 6560  CMetccmet 25289  NrmCVeccnv 30604  BaseSetcba 30606  IndMetcims 30611  CBanccbn 30882
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-ext 2707
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-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-cbn 30883
This theorem is referenced by:  bnsscmcl  30888  ubthlem1  30890  ubthlem2  30891  minvecolem4a  30897  hlcmet  30914
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