Theorem cbncms 30847

Description: The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) Use bncmet 25275 (or preferably bncms 25272) 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 30846	. 2 ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋)))
4	3	simprbi 496	1 ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ‘cfv 6486  CMetccmet 25182  NrmCVeccnv 30566  BaseSetcba 30568  IndMetcims 30573  CBanccbn 30844

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-iota 6442  df-fv 6494  df-cbn 30845

This theorem is referenced by:  bnsscmcl  30850  ubthlem1  30852  ubthlem2  30853  minvecolem4a  30859  hlcmet  30876