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Theorem cuspcvg 24311
Description: In a complete uniform space, any Cauchy filter 𝐶 has a limit. (Contributed by Thierry Arnoux, 3-Dec-2017.)
Hypotheses
Ref Expression
cuspcvg.1 𝐵 = (Base‘𝑊)
cuspcvg.2 𝐽 = (TopOpen‘𝑊)
Assertion
Ref Expression
cuspcvg ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) ∧ 𝐶 ∈ (Fil‘𝐵)) → (𝐽 fLim 𝐶) ≠ ∅)

Proof of Theorem cuspcvg
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2828 . . . . 5 (𝑐 = 𝐶 → (𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) ↔ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊))))
2 cuspcvg.2 . . . . . . . . 9 𝐽 = (TopOpen‘𝑊)
32eqcomi 2745 . . . . . . . 8 (TopOpen‘𝑊) = 𝐽
43a1i 11 . . . . . . 7 (𝑐 = 𝐶 → (TopOpen‘𝑊) = 𝐽)
5 id 22 . . . . . . 7 (𝑐 = 𝐶𝑐 = 𝐶)
64, 5oveq12d 7450 . . . . . 6 (𝑐 = 𝐶 → ((TopOpen‘𝑊) fLim 𝑐) = (𝐽 fLim 𝐶))
76neeq1d 2999 . . . . 5 (𝑐 = 𝐶 → (((TopOpen‘𝑊) fLim 𝑐) ≠ ∅ ↔ (𝐽 fLim 𝐶) ≠ ∅))
81, 7imbi12d 344 . . . 4 (𝑐 = 𝐶 → ((𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅) ↔ (𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) → (𝐽 fLim 𝐶) ≠ ∅)))
9 iscusp 24309 . . . . . 6 (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
109simprbi 496 . . . . 5 (𝑊 ∈ CUnifSp → ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
1110adantr 480 . . . 4 ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
12 simpr 484 . . . . 5 ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → 𝐶 ∈ (Fil‘𝐵))
13 cuspcvg.1 . . . . . 6 𝐵 = (Base‘𝑊)
1413fveq2i 6908 . . . . 5 (Fil‘𝐵) = (Fil‘(Base‘𝑊))
1512, 14eleqtrdi 2850 . . . 4 ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → 𝐶 ∈ (Fil‘(Base‘𝑊)))
168, 11, 15rspcdva 3622 . . 3 ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → (𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) → (𝐽 fLim 𝐶) ≠ ∅))
17163impia 1117 . 2 ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵) ∧ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊))) → (𝐽 fLim 𝐶) ≠ ∅)
18173com23 1126 1 ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) ∧ 𝐶 ∈ (Fil‘𝐵)) → (𝐽 fLim 𝐶) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wcel 2107  wne 2939  wral 3060  c0 4332  cfv 6560  (class class class)co 7432  Basecbs 17248  TopOpenctopn 17467  Filcfil 23854   fLim cflim 23943  UnifStcuss 24263  UnifSpcusp 24264  CauFiluccfilu 24296  CUnifSpccusp 24307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435  df-cusp 24308
This theorem is referenced by:  cnextucn  24313
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