Theorem iscbn 30952

Description: A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) Use isbn 25306 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 6842	. . . 4 ⊢ (𝑢 = 𝑈 → (IndMet‘𝑢) = (IndMet‘𝑈))
2		iscbn.8	. . . 4 ⊢ 𝐷 = (IndMet‘𝑈)
3	1, 2	eqtr4di 2790	. . 3 ⊢ (𝑢 = 𝑈 → (IndMet‘𝑢) = 𝐷)
4		fveq2 6842	. . . . 5 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
5		iscbn.x	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
6	4, 5	eqtr4di 2790	. . . 4 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
7	6	fveq2d 6846	. . 3 ⊢ (𝑢 = 𝑈 → (CMet‘(BaseSet‘𝑢)) = (CMet‘𝑋))
8	3, 7	eleq12d 2831	. 2 ⊢ (𝑢 = 𝑈 → ((IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢)) ↔ 𝐷 ∈ (CMet‘𝑋)))
9		df-cbn 30951	. 2 ⊢ CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))}
10	8, 9	elrab2 3651	1 ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ‘cfv 6500  CMetccmet 25222  NrmCVeccnv 30672  BaseSetcba 30674  IndMetcims 30679  CBanccbn 30950

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 2709

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 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-cbn 30951

This theorem is referenced by:  cbncms  30953  bnnv  30954  bnsscmcl  30956  cnbn  30957  hhhl  31292  hhssbnOLD  31367