Theorem cbncms 31349

Description: The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) Use bncmet 25578 (or preferably bncms 25575) 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 31348	. 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 6533  CMetccmet 25485  NrmCVeccnv 31068  BaseSetcba 31070  IndMetcims 31075  CBanccbn 31346

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 2732

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 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-cbn 31347

This theorem is used by:  bnsscmcl  31352  ubthlem1  31354  ubthlem2  31355  minvecolem4a  31361  hlcmet  31378