Theorem cuspcvg 24244

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 2823	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) ↔ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊))))
2		cuspcvg.2	. . . . . . . . 9 ⊢ 𝐽 = (TopOpen‘𝑊)
3	2	eqcomi 2745	. . . . . . . 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 7428	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((TopOpen‘𝑊) fLim 𝑐) = (𝐽 fLim 𝐶))
7	6	neeq1d 2992	. . . . 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 24242	. . . . . 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 6884	. . . . 5 ⊢ (Fil‘𝐵) = (Fil‘(Base‘𝑊))
15	12, 14	eleqtrdi 2845	. . . 4 ⊢ ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → 𝐶 ∈ (Fil‘(Base‘𝑊)))
16	8, 11, 15	rspcdva 3607	. . 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 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  ∅c0 4313  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  TopOpenctopn 17440  Filcfil 23788   fLim cflim 23877  UnifStcuss 24197  UnifSpcusp 24198  CauFil_uccfilu 24229  CUnifSpccusp 24240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544  df-ov 7413  df-cusp 24241

This theorem is referenced by:  cnextucn  24246