Theorem cbncms 29255

Description: The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) Use bncmet 24539 (or preferably bncms 24536) 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 29254	. 2 ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋)))
4	3	simprbi 496	1 ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2101  ‘cfv 6447  CMetccmet 24446  NrmCVeccnv 28974  BaseSetcba 28976  IndMetcims 28981  CBanccbn 29252

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3224  df-v 3436  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-br 5078  df-iota 6399  df-fv 6455  df-cbn 29253

This theorem is referenced by:  bnsscmcl  29258  ubthlem1  29260  ubthlem2  29261  minvecolem4a  29267  hlcmet  29284