Theorem cuspcvg 24331

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 2832	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) ↔ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊))))
2		cuspcvg.2	. . . . . . . . 9 ⊢ 𝐽 = (TopOpen‘𝑊)
3	2	eqcomi 2749	. . . . . . . 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 7466	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((TopOpen‘𝑊) fLim 𝑐) = (𝐽 fLim 𝐶))
7	6	neeq1d 3006	. . . . 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 24329	. . . . . 6 ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
10	9	simprbi 496	. . . . 5 ⊢ (𝑊 ∈ CUnifSp → ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
11	10	adantr 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)
14	13	fveq2i 6923	. . . . 5 ⊢ (Fil‘𝐵) = (Fil‘(Base‘𝑊))
15	12, 14	eleqtrdi 2854	. . . 4 ⊢ ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → 𝐶 ∈ (Fil‘(Base‘𝑊)))
16	8, 11, 15	rspcdva 3636	. . 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 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∅c0 4352  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  TopOpenctopn 17481  Filcfil 23874   fLim cflim 23963  UnifStcuss 24283  UnifSpcusp 24284  CauFil_uccfilu 24316  CUnifSpccusp 24327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451  df-cusp 24328

This theorem is referenced by:  cnextucn  24333