MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isbn Structured version   Visualization version   GIF version

Theorem isbn 25259
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 3961 . . 3 (π‘Š ∈ (NrmVec ∩ CMetSp) ↔ (π‘Š ∈ NrmVec ∧ π‘Š ∈ CMetSp))
21anbi1i 623 . 2 ((π‘Š ∈ (NrmVec ∩ CMetSp) ∧ 𝐹 ∈ CMetSp) ↔ ((π‘Š ∈ NrmVec ∧ π‘Š ∈ CMetSp) ∧ 𝐹 ∈ CMetSp))
3 fveq2 6891 . . . . 5 (𝑀 = π‘Š β†’ (Scalarβ€˜π‘€) = (Scalarβ€˜π‘Š))
4 isbn.1 . . . . 5 𝐹 = (Scalarβ€˜π‘Š)
53, 4eqtr4di 2786 . . . 4 (𝑀 = π‘Š β†’ (Scalarβ€˜π‘€) = 𝐹)
65eleq1d 2814 . . 3 (𝑀 = π‘Š β†’ ((Scalarβ€˜π‘€) ∈ CMetSp ↔ 𝐹 ∈ CMetSp))
7 df-bn 25257 . . 3 Ban = {𝑀 ∈ (NrmVec ∩ CMetSp) ∣ (Scalarβ€˜π‘€) ∈ CMetSp}
86, 7elrab2 3684 . 2 (π‘Š ∈ Ban ↔ (π‘Š ∈ (NrmVec ∩ CMetSp) ∧ 𝐹 ∈ CMetSp))
9 df-3an 1087 . 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 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ∩ cin 3944  β€˜cfv 6542  Scalarcsca 17229  NrmVeccnvc 24483  CMetSpccms 25253  Bancbn 25254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-iota 6494  df-fv 6550  df-bn 25257
This theorem is referenced by:  bnsca  25260  bnnvc  25261  bncms  25265  lssbn  25273  srabn  25281  ishl2  25291  cmslssbn  25293
  Copyright terms: Public domain W3C validator