Theorem cuspcvg 23690

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 2820	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) ↔ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊))))
2		cuspcvg.2	. . . . . . . . 9 ⊢ 𝐽 = (TopOpen‘𝑊)
3	2	eqcomi 2740	. . . . . . . 8 ⊢ (TopOpen‘𝑊) = 𝐽
4	3	a1i 11	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (TopOpen‘𝑊) = 𝐽)
5		id 22	. . . . . . 7 ⊢ (𝑐 = 𝐶 → 𝑐 = 𝐶)
6	4, 5	oveq12d 7380	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((TopOpen‘𝑊) fLim 𝑐) = (𝐽 fLim 𝐶))
7	6	neeq1d 2999	. . . . 5 ⊢ (𝑐 = 𝐶 → (((TopOpen‘𝑊) fLim 𝑐) ≠ ∅ ↔ (𝐽 fLim 𝐶) ≠ ∅))
8	1, 7	imbi12d 344	. . . 4 ⊢ (𝑐 = 𝐶 → ((𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅) ↔ (𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) → (𝐽 fLim 𝐶) ≠ ∅)))
9		iscusp 23688	. . . . . 6 ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
10	9	simprbi 497	. . . . 5 ⊢ (𝑊 ∈ CUnifSp → ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
11	10	adantr 481	. . . 4 ⊢ ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
12		simpr 485	. . . . 5 ⊢ ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → 𝐶 ∈ (Fil‘𝐵))
13		cuspcvg.1	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
14	13	fveq2i 6850	. . . . 5 ⊢ (Fil‘𝐵) = (Fil‘(Base‘𝑊))
15	12, 14	eleqtrdi 2842	. . . 4 ⊢ ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → 𝐶 ∈ (Fil‘(Base‘𝑊)))
16	8, 11, 15	rspcdva 3583	. . 3 ⊢ ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → (𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) → (𝐽 fLim 𝐶) ≠ ∅))
17	16	3impia 1117	. 2 ⊢ ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵) ∧ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊))) → (𝐽 fLim 𝐶) ≠ ∅)
18	17	3com23 1126	1 ⊢ ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) ∧ 𝐶 ∈ (Fil‘𝐵)) → (𝐽 fLim 𝐶) ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2939  ∀wral 3060  ∅c0 4287  ‘cfv 6501  (class class class)co 7362  Basecbs 17094  TopOpenctopn 17317  Filcfil 23233   fLim cflim 23322  UnifStcuss 23642  UnifSpcusp 23643  CauFil_uccfilu 23675  CUnifSpccusp 23686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-iota 6453  df-fv 6509  df-ov 7365  df-cusp 23687

This theorem is referenced by:  cnextucn  23692