Theorem isbn 23944

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 4172	. . 3 ⊢ (𝑊 ∈ (NrmVec ∩ CMetSp) ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp))
2	1	anbi1i 625	. 2 ⊢ ((𝑊 ∈ (NrmVec ∩ CMetSp) ∧ 𝐹 ∈ CMetSp) ↔ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp) ∧ 𝐹 ∈ CMetSp))
3		fveq2 6673	. . . . 5 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
4		isbn.1	. . . . 5 ⊢ 𝐹 = (Scalar‘𝑊)
5	3, 4	syl6eqr 2877	. . . 4 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
6	5	eleq1d 2900	. . 3 ⊢ (𝑤 = 𝑊 → ((Scalar‘𝑤) ∈ CMetSp ↔ 𝐹 ∈ CMetSp))
7		df-bn 23942	. . 3 ⊢ Ban = {𝑤 ∈ (NrmVec ∩ CMetSp) ∣ (Scalar‘𝑤) ∈ CMetSp}
8	6, 7	elrab2 3686	. 2 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ (NrmVec ∩ CMetSp) ∧ 𝐹 ∈ CMetSp))
9		df-3an 1085	. 2 ⊢ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp) ↔ ((𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp) ∧ 𝐹 ∈ CMetSp))
10	2, 8, 9	3bitr4i 305	1 ⊢ (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113   ∩ cin 3938  ‘cfv 6358  Scalarcsca 16571  NrmVeccnvc 23194  CMetSpccms 23938  Bancbn 23939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-iota 6317  df-fv 6366  df-bn 23942

This theorem is referenced by:  bnsca  23945  bnnvc  23946  bncms  23950  lssbn  23958  srabn  23966  ishl2  23976  cmslssbn  23978