Theorem iscbn 30865

Description: A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) Use isbn 25285 instead. (New usage is discouraged.)

Hypotheses

Ref	Expression
iscbn.x	⊢ 𝑋 = (BaseSet‘𝑈)
iscbn.8	⊢ 𝐷 = (IndMet‘𝑈)

Assertion

Ref	Expression
iscbn	⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋)))

Proof of Theorem iscbn

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6831	. . . 4 ⊢ (𝑢 = 𝑈 → (IndMet‘𝑢) = (IndMet‘𝑈))
2		iscbn.8	. . . 4 ⊢ 𝐷 = (IndMet‘𝑈)
3	1, 2	eqtr4di 2786	. . 3 ⊢ (𝑢 = 𝑈 → (IndMet‘𝑢) = 𝐷)
4		fveq2 6831	. . . . 5 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
5		iscbn.x	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
6	4, 5	eqtr4di 2786	. . . 4 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
7	6	fveq2d 6835	. . 3 ⊢ (𝑢 = 𝑈 → (CMet‘(BaseSet‘𝑢)) = (CMet‘𝑋))
8	3, 7	eleq12d 2827	. 2 ⊢ (𝑢 = 𝑈 → ((IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢)) ↔ 𝐷 ∈ (CMet‘𝑋)))
9		df-cbn 30864	. 2 ⊢ CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))}
10	8, 9	elrab2 3646	1 ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ‘cfv 6489  CMetccmet 25201  NrmCVeccnv 30585  BaseSetcba 30587  IndMetcims 30592  CBanccbn 30863

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 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-iota 6445  df-fv 6497  df-cbn 30864

This theorem is referenced by:  cbncms  30866  bnnv  30867  bnsscmcl  30869  cnbn  30870  hhhl  31205  hhssbnOLD  31280