Theorem cuspcvg 24418

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.

Step	Hyp	Ref	Expression
1		eleq1 2853	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) ↔ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊))))
2		cuspcvg.2	. . . . . . . . 9 ⊢ 𝐽 = (TopOpen‘𝑊)
3	2	eqcomi 2774	. . . . . . . 8 ⊢ (TopOpen‘𝑊) = 𝐽
4	3	a1i 11	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (TopOpen‘𝑊) = 𝐽)
5		id 23	. . . . . . 7 ⊢ (𝑐 = 𝐶 → 𝑐 = 𝐶)
6	4, 5	oveq12d 7418	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((TopOpen‘𝑊) fLim 𝑐) = (𝐽 fLim 𝐶))
7	6	neeq1d 3019	. . . . 5 ⊢ (𝑐 = 𝐶 → (((TopOpen‘𝑊) fLim 𝑐) ≠ ∅ ↔ (𝐽 fLim 𝐶) ≠ ∅))
8	1, 7	imbi12d 347	. . . 4 ⊢ (𝑐 = 𝐶 → ((𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅) ↔ (𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) → (𝐽 fLim 𝐶) ≠ ∅)))
9		iscusp 24416	. . . . . 6 ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
10	9	simprbi 502	. . . . 5 ⊢ (𝑊 ∈ CUnifSp → ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
11	10	adantr 485	. . . 4 ⊢ ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
12		simpr 489	. . . . 5 ⊢ ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → 𝐶 ∈ (Fil‘𝐵))
13		cuspcvg.1	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
14	13	fveq2i 6874	. . . . 5 ⊢ (Fil‘𝐵) = (Fil‘(Base‘𝑊))
15	12, 14	eleqtrdi 2875	. . . 4 ⊢ ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → 𝐶 ∈ (Fil‘(Base‘𝑊)))
16	8, 11, 15	rspcdva 3585	. . 3 ⊢ ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → (𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) → (𝐽 fLim 𝐶) ≠ ∅))
17	16	3impia 1133	. 2 ⊢ ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵) ∧ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊))) → (𝐽 fLim 𝐶) ≠ ∅)
18	17	3com23 1142	1 ⊢ ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) ∧ 𝐶 ∈ (Fil‘𝐵)) → (𝐽 fLim 𝐶) ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∀wral 3079  ∅c0 4288  ‘cfv 6525  (class class class)co 7400  Basecbs 17259  TopOpenctopn 17464  Filcfil 23963   fLim cflim 24052  UnifStcuss 24371  UnifSpcusp 24372  CauFil_uccfilu 24403  CUnifSpccusp 24414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-iota 6481  df-fv 6533  df-ov 7403  df-cusp 24415

This theorem is referenced by:  cnextucn  24420