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Theorem isbn 24705
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 = (Scalarβ€˜π‘Š)
Assertion
Ref Expression
isbn (π‘Š ∈ Ban ↔ (π‘Š ∈ NrmVec ∧ π‘Š ∈ CMetSp ∧ 𝐹 ∈ CMetSp))

Proof of Theorem isbn
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 elin 3927 . . 3 (π‘Š ∈ (NrmVec ∩ CMetSp) ↔ (π‘Š ∈ NrmVec ∧ π‘Š ∈ CMetSp))
21anbi1i 625 . 2 ((π‘Š ∈ (NrmVec ∩ CMetSp) ∧ 𝐹 ∈ CMetSp) ↔ ((π‘Š ∈ NrmVec ∧ π‘Š ∈ CMetSp) ∧ 𝐹 ∈ CMetSp))
3 fveq2 6843 . . . . 5 (𝑀 = π‘Š β†’ (Scalarβ€˜π‘€) = (Scalarβ€˜π‘Š))
4 isbn.1 . . . . 5 𝐹 = (Scalarβ€˜π‘Š)
53, 4eqtr4di 2795 . . . 4 (𝑀 = π‘Š β†’ (Scalarβ€˜π‘€) = 𝐹)
65eleq1d 2823 . . 3 (𝑀 = π‘Š β†’ ((Scalarβ€˜π‘€) ∈ CMetSp ↔ 𝐹 ∈ CMetSp))
7 df-bn 24703 . . 3 Ban = {𝑀 ∈ (NrmVec ∩ CMetSp) ∣ (Scalarβ€˜π‘€) ∈ CMetSp}
86, 7elrab2 3649 . 2 (π‘Š ∈ Ban ↔ (π‘Š ∈ (NrmVec ∩ CMetSp) ∧ 𝐹 ∈ CMetSp))
9 df-3an 1090 . 2 ((π‘Š ∈ NrmVec ∧ π‘Š ∈ CMetSp ∧ 𝐹 ∈ CMetSp) ↔ ((π‘Š ∈ NrmVec ∧ π‘Š ∈ CMetSp) ∧ 𝐹 ∈ CMetSp))
102, 8, 93bitr4i 303 1 (π‘Š ∈ Ban ↔ (π‘Š ∈ NrmVec ∧ π‘Š ∈ CMetSp ∧ 𝐹 ∈ CMetSp))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ∩ cin 3910  β€˜cfv 6497  Scalarcsca 17137  NrmVeccnvc 23940  CMetSpccms 24699  Bancbn 24700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-bn 24703
This theorem is referenced by:  bnsca  24706  bnnvc  24707  bncms  24711  lssbn  24719  srabn  24727  ishl2  24737  cmslssbn  24739
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