Theorem cuspcvg 24599

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 2849	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) ↔ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊))))
2		cuspcvg.2	. . . . . . . . 9 ⊢ 𝐽 = (TopOpen‘𝑊)
3	2	eqcomi 2770	. . . . . . . 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 7430	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((TopOpen‘𝑊) fLim 𝑐) = (𝐽 fLim 𝐶))
7	6	neeq1d 3015	. . . . 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 24597	. . . . . 6 ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
10	9	simprbi 503	. . . . 5 ⊢ (𝑊 ∈ CUnifSp → ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
11	10	adantr 486	. . . 4 ⊢ ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
12		simpr 490	. . . . 5 ⊢ ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → 𝐶 ∈ (Fil‘𝐵))
13		cuspcvg.1	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
14	13	fveq2i 6880	. . . . 5 ⊢ (Fil‘𝐵) = (Fil‘(Base‘𝑊))
15	12, 14	eleqtrdi 2871	. . . 4 ⊢ ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → 𝐶 ∈ (Fil‘(Base‘𝑊)))
16	8, 11, 15	rspcdva 3578	. . 3 ⊢ ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → (𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) → (𝐽 fLim 𝐶) ≠ ∅))
17	16	3impia 1135	. 2 ⊢ ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵) ∧ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊))) → (𝐽 fLim 𝐶) ≠ ∅)
18	17	3com23 1144	1 ⊢ ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) ∧ 𝐶 ∈ (Fil‘𝐵)) → (𝐽 fLim 𝐶) ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∅c0 4279  ‘cfv 6531  (class class class)co 7412  Basecbs 17367  TopOpenctopn 17572  Filcfil 24144   fLim cflim 24233  UnifStcuss 24552  UnifSpcusp 24553  CauFil_uccfilu 24584  CUnifSpccusp 24595

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6487  df-fv 6539  df-ov 7415  df-cusp 24596

This theorem is used by:  cnextucn  24601