Theorem cbncms 31354

Description: The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) Use bncmet 25581 (or preferably bncms 25578) 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 31353	. 2 ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋)))
4	3	simprbi 503	1 ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ‘cfv 6537  CMetccmet 25488  NrmCVeccnv 31073  BaseSetcba 31075  IndMetcims 31080  CBanccbn 31351

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-cbn 31352

This theorem is used by:  bnsscmcl  31357  ubthlem1  31359  ubthlem2  31360  minvecolem4a  31366  hlcmet  31383