Theorem cbncms 30851

Description: The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) Use bncmet 25304 (or preferably bncms 25301) 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

Step	Hyp	Ref	Expression
1		iscbn.x	. . 3 ⊢ 𝑋 = (BaseSet‘𝑈)
2		iscbn.8	. . 3 ⊢ 𝐷 = (IndMet‘𝑈)
3	1, 2	iscbn 30850	. 2 ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋)))
4	3	simprbi 496	1 ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ‘cfv 6536  CMetccmet 25211  NrmCVeccnv 30570  BaseSetcba 30572  IndMetcims 30577  CBanccbn 30848

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544  df-cbn 30849

This theorem is referenced by:  bnsscmcl  30854  ubthlem1  30856  ubthlem2  30857  minvecolem4a  30863  hlcmet  30880