Theorem isbn 24407

Description: A Banach space is a normed vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2015.)

Hypothesis

Ref	Expression
isbn.1	⊢ 𝐹 = (Scalar‘𝑊)

Assertion

Ref	Expression
isbn	⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp))

Proof of Theorem isbn

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3899	. . 3 ⊢ (𝑊 ∈ (NrmVec ∩ CMetSp) ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp))
2	1	anbi1i 623	. 2 ⊢ ((𝑊 ∈ (NrmVec ∩ CMetSp) ∧ 𝐹 ∈ CMetSp) ↔ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp) ∧ 𝐹 ∈ CMetSp))
3		fveq2 6756	. . . . 5 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
4		isbn.1	. . . . 5 ⊢ 𝐹 = (Scalar‘𝑊)
5	3, 4	eqtr4di 2797	. . . 4 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
6	5	eleq1d 2823	. . 3 ⊢ (𝑤 = 𝑊 → ((Scalar‘𝑤) ∈ CMetSp ↔ 𝐹 ∈ CMetSp))
7		df-bn 24405	. . 3 ⊢ Ban = {𝑤 ∈ (NrmVec ∩ CMetSp) ∣ (Scalar‘𝑤) ∈ CMetSp}
8	6, 7	elrab2 3620	. 2 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ (NrmVec ∩ CMetSp) ∧ 𝐹 ∈ CMetSp))
9		df-3an 1087	. 2 ⊢ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp) ↔ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp) ∧ 𝐹 ∈ CMetSp))
10	2, 8, 9	3bitr4i 302	1 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ∩ cin 3882  ‘cfv 6418  Scalarcsca 16891  NrmVeccnvc 23643  CMetSpccms 24401  Bancbn 24402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-bn 24405

This theorem is referenced by:  bnsca  24408  bnnvc  24409  bncms  24413  lssbn  24421  srabn  24429  ishl2  24439  cmslssbn  24441