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Theorem cbncms 30400
Description: The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) Use bncmet 25108 (or preferably bncms 25105) 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 30399 . 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 1540  wcel 2105  cfv 6543  CMetccmet 25015  NrmCVeccnv 30119  BaseSetcba 30121  IndMetcims 30126  CBanccbn 30397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-cbn 30398
This theorem is referenced by:  bnsscmcl  30403  ubthlem1  30405  ubthlem2  30406  minvecolem4a  30412  hlcmet  30429
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