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Theorem cbncms 29904
Description: The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) Use bncmet 24763 (or preferably bncms 24760) instead. (New usage is discouraged.)
Hypotheses
Ref Expression
iscbn.x 𝑋 = (BaseSetβ€˜π‘ˆ)
iscbn.8 𝐷 = (IndMetβ€˜π‘ˆ)
Assertion
Ref Expression
cbncms (π‘ˆ ∈ CBan β†’ 𝐷 ∈ (CMetβ€˜π‘‹))

Proof of Theorem cbncms
StepHypRef Expression
1 iscbn.x . . 3 𝑋 = (BaseSetβ€˜π‘ˆ)
2 iscbn.8 . . 3 𝐷 = (IndMetβ€˜π‘ˆ)
31, 2iscbn 29903 . 2 (π‘ˆ ∈ CBan ↔ (π‘ˆ ∈ NrmCVec ∧ 𝐷 ∈ (CMetβ€˜π‘‹)))
43simprbi 497 1 (π‘ˆ ∈ CBan β†’ 𝐷 ∈ (CMetβ€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  β€˜cfv 6516  CMetccmet 24670  NrmCVeccnv 29623  BaseSetcba 29625  IndMetcims 29630  CBanccbn 29901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3419  df-v 3461  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-br 5126  df-iota 6468  df-fv 6524  df-cbn 29902
This theorem is referenced by:  bnsscmcl  29907  ubthlem1  29909  ubthlem2  29910  minvecolem4a  29916  hlcmet  29933
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