Theorem iscbn 30935

Description: A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) Use isbn 25305 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 6840	. . . 4 ⊢ (𝑢 = 𝑈 → (IndMet‘𝑢) = (IndMet‘𝑈))
2		iscbn.8	. . . 4 ⊢ 𝐷 = (IndMet‘𝑈)
3	1, 2	eqtr4di 2789	. . 3 ⊢ (𝑢 = 𝑈 → (IndMet‘𝑢) = 𝐷)
4		fveq2 6840	. . . . 5 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
5		iscbn.x	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
6	4, 5	eqtr4di 2789	. . . 4 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
7	6	fveq2d 6844	. . 3 ⊢ (𝑢 = 𝑈 → (CMet‘(BaseSet‘𝑢)) = (CMet‘𝑋))
8	3, 7	eleq12d 2830	. 2 ⊢ (𝑢 = 𝑈 → ((IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢)) ↔ 𝐷 ∈ (CMet‘𝑋)))
9		df-cbn 30934	. 2 ⊢ CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))}
10	8, 9	elrab2 3637	1 ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ‘cfv 6498  CMetccmet 25221  NrmCVeccnv 30655  BaseSetcba 30657  IndMetcims 30662  CBanccbn 30933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-cbn 30934

This theorem is referenced by:  cbncms  30936  bnnv  30937  bnsscmcl  30939  cnbn  30940  hhhl  31275  hhssbnOLD  31350