Theorem iscbn 30893

Description: A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) Use isbn 25386 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 6907	. . . 4 ⊢ (𝑢 = 𝑈 → (IndMet‘𝑢) = (IndMet‘𝑈))
2		iscbn.8	. . . 4 ⊢ 𝐷 = (IndMet‘𝑈)
3	1, 2	eqtr4di 2793	. . 3 ⊢ (𝑢 = 𝑈 → (IndMet‘𝑢) = 𝐷)
4		fveq2 6907	. . . . 5 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
5		iscbn.x	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
6	4, 5	eqtr4di 2793	. . . 4 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
7	6	fveq2d 6911	. . 3 ⊢ (𝑢 = 𝑈 → (CMet‘(BaseSet‘𝑢)) = (CMet‘𝑋))
8	3, 7	eleq12d 2833	. 2 ⊢ (𝑢 = 𝑈 → ((IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢)) ↔ 𝐷 ∈ (CMet‘𝑋)))
9		df-cbn 30892	. 2 ⊢ CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))}
10	8, 9	elrab2 3698	1 ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ‘cfv 6563  CMetccmet 25302  NrmCVeccnv 30613  BaseSetcba 30615  IndMetcims 30620  CBanccbn 30891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-cbn 30892

This theorem is referenced by:  cbncms  30894  bnnv  30895  bnsscmcl  30897  cnbn  30898  hhhl  31233  hhssbnOLD  31308