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Theorem cfilfcls 25181

Description: Similar to ultrafilters (uffclsflim 23925), the cluster points and limit points of a Cauchy filter coincide. (Contributed by Mario Carneiro, 15-Oct-2015.)

**Hypotheses**
Ref	Expression
cfilfcls.1	⊢ 𝐽 = (MetOpen‘𝐷)
cfilfcls.2	⊢ 𝑋 = dom dom 𝐷

**Assertion**
Ref	Expression
cfilfcls	⊢ (𝐹 ∈ (CauFil‘𝐷) → (𝐽 fClus 𝐹) = (𝐽 fLim 𝐹))

Proof of Theorem cfilfcls Dummy variables 𝑥 𝑦 𝑧 𝑓 𝑟 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2730	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	fclselbas 23910	. . . . . . 7 ⊢ (𝑥 ∈ (𝐽 fClus 𝐹) → 𝑥 ∈ ∪ 𝐽)
3	2	adantl 481	. . . . . 6 ⊢ ((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → 𝑥 ∈ ∪ 𝐽)
4		df-cfil 25162	. . . . . . . . . . . 12 ⊢ CauFil = (𝑑 ∈ ∪ ran ∞Met ↦ {𝑓 ∈ (Fil‘dom dom 𝑑) ∣ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
5	4	mptrcl 6980	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (CauFil‘𝐷) → 𝐷 ∈ ∪ ran ∞Met)
6		xmetunirn 24232	. . . . . . . . . . 11 ⊢ (𝐷 ∈ ∪ ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷))
7	5, 6	sylib 218	. . . . . . . . . 10 ⊢ (𝐹 ∈ (CauFil‘𝐷) → 𝐷 ∈ (∞Met‘dom dom 𝐷))
8		cfilfcls.2	. . . . . . . . . . 11 ⊢ 𝑋 = dom dom 𝐷
9	8	fveq2i 6864	. . . . . . . . . 10 ⊢ (∞Met‘𝑋) = (∞Met‘dom dom 𝐷)
10	7, 9	eleqtrrdi 2840	. . . . . . . . 9 ⊢ (𝐹 ∈ (CauFil‘𝐷) → 𝐷 ∈ (∞Met‘𝑋))
11	10	adantr 480	. . . . . . . 8 ⊢ ((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → 𝐷 ∈ (∞Met‘𝑋))
12		cfilfcls.1	. . . . . . . . 9 ⊢ 𝐽 = (MetOpen‘𝐷)
13	12	mopntopon 24334	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
14	11, 13	syl 17	. . . . . . 7 ⊢ ((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → 𝐽 ∈ (TopOn‘𝑋))
15		toponuni 22808	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
16	14, 15	syl 17	. . . . . 6 ⊢ ((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → 𝑋 = ∪ 𝐽)
17	3, 16	eleqtrrd 2832	. . . . 5 ⊢ ((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → 𝑥 ∈ 𝑋)
18	12	mopni2 24388	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦) → ∃𝑟 ∈ ℝ⁺ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑦)
19	18	3expb 1120	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → ∃𝑟 ∈ ℝ⁺ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑦)
20	11, 19	sylan 580	. . . . . . . 8 ⊢ (((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → ∃𝑟 ∈ ℝ⁺ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑦)
21		cfilfil 25174	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → 𝐹 ∈ (Fil‘𝑋))
22	10, 21	mpancom 688	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (CauFil‘𝐷) → 𝐹 ∈ (Fil‘𝑋))
23	22	adantr 480	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → 𝐹 ∈ (Fil‘𝑋))
24	23	ad2antrr 726	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑦)) → 𝐹 ∈ (Fil‘𝑋))
25	11	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) → 𝐷 ∈ (∞Met‘𝑋))
26		simpll 766	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) → 𝐹 ∈ (CauFil‘𝐷))
27		rphalfcl 12987	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ ℝ⁺ → (𝑟 / 2) ∈ ℝ⁺)
28	27	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) → (𝑟 / 2) ∈ ℝ⁺)
29		rphalfcl 12987	. . . . . . . . . . . . 13 ⊢ ((𝑟 / 2) ∈ ℝ⁺ → ((𝑟 / 2) / 2) ∈ ℝ⁺)
30	28, 29	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) → ((𝑟 / 2) / 2) ∈ ℝ⁺)
31		cfil3i 25176	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷) ∧ ((𝑟 / 2) / 2) ∈ ℝ⁺) → ∃𝑦 ∈ 𝑋 (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)
32	25, 26, 30, 31	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) → ∃𝑦 ∈ 𝑋 (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)
33	23	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → 𝐹 ∈ (Fil‘𝑋))
34		simprr 772	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)
35	25	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → 𝐷 ∈ (∞Met‘𝑋))
36	17	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → 𝑥 ∈ 𝑋)
37		rpxr 12968	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
38	37	ad2antlr 727	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → 𝑟 ∈ ℝ^*)
39		blssm 24313	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) → (𝑥(ball‘𝐷)𝑟) ⊆ 𝑋)
40	35, 36, 38, 39	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → (𝑥(ball‘𝐷)𝑟) ⊆ 𝑋)
41		simpllr 775	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → 𝑥 ∈ (𝐽 fClus 𝐹))
42	28	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → (𝑟 / 2) ∈ ℝ⁺)
43	42	rpxrd 13003	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → (𝑟 / 2) ∈ ℝ^*)
44	12	blopn 24395	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (𝑟 / 2) ∈ ℝ^*) → (𝑥(ball‘𝐷)(𝑟 / 2)) ∈ 𝐽)
45	35, 36, 43, 44	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → (𝑥(ball‘𝐷)(𝑟 / 2)) ∈ 𝐽)
46		blcntr 24308	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (𝑟 / 2) ∈ ℝ⁺) → 𝑥 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)))
47	35, 36, 42, 46	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → 𝑥 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)))
48		fclsopni 23909	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (𝐽 fClus 𝐹) ∧ ((𝑥(ball‘𝐷)(𝑟 / 2)) ∈ 𝐽 ∧ 𝑥 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)) ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → ((𝑥(ball‘𝐷)(𝑟 / 2)) ∩ (𝑦(ball‘𝐷)((𝑟 / 2) / 2))) ≠ ∅)
49	41, 45, 47, 34, 48	syl13anc 1374	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → ((𝑥(ball‘𝐷)(𝑟 / 2)) ∩ (𝑦(ball‘𝐷)((𝑟 / 2) / 2))) ≠ ∅)
50		n0 4319	. . . . . . . . . . . . . 14 ⊢ (((𝑥(ball‘𝐷)(𝑟 / 2)) ∩ (𝑦(ball‘𝐷)((𝑟 / 2) / 2))) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ ((𝑥(ball‘𝐷)(𝑟 / 2)) ∩ (𝑦(ball‘𝐷)((𝑟 / 2) / 2))))
51	49, 50	sylib 218	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → ∃𝑧 𝑧 ∈ ((𝑥(ball‘𝐷)(𝑟 / 2)) ∩ (𝑦(ball‘𝐷)((𝑟 / 2) / 2))))
52		elin 3933	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ((𝑥(ball‘𝐷)(𝑟 / 2)) ∩ (𝑦(ball‘𝐷)((𝑟 / 2) / 2))) ↔ (𝑧 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)) ∧ 𝑧 ∈ (𝑦(ball‘𝐷)((𝑟 / 2) / 2))))
53	35	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) ∧ (𝑧 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)) ∧ 𝑧 ∈ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)))) → 𝐷 ∈ (∞Met‘𝑋))
54		simplrl 776	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) ∧ (𝑧 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)) ∧ 𝑧 ∈ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)))) → 𝑦 ∈ 𝑋)
55	42	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) ∧ (𝑧 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)) ∧ 𝑧 ∈ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)))) → (𝑟 / 2) ∈ ℝ⁺)
56	55	rpred 13002	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) ∧ (𝑧 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)) ∧ 𝑧 ∈ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)))) → (𝑟 / 2) ∈ ℝ)
57		simprr 772	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) ∧ (𝑧 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)) ∧ 𝑧 ∈ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)))) → 𝑧 ∈ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)))
58		blhalf 24300	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ ((𝑟 / 2) ∈ ℝ ∧ 𝑧 ∈ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)))) → (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ⊆ (𝑧(ball‘𝐷)(𝑟 / 2)))
59	53, 54, 56, 57, 58	syl22anc 838	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) ∧ (𝑧 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)) ∧ 𝑧 ∈ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)))) → (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ⊆ (𝑧(ball‘𝐷)(𝑟 / 2)))
60		blssm 24313	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (𝑟 / 2) ∈ ℝ^*) → (𝑥(ball‘𝐷)(𝑟 / 2)) ⊆ 𝑋)
61	35, 36, 43, 60	syl3anc 1373	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → (𝑥(ball‘𝐷)(𝑟 / 2)) ⊆ 𝑋)
62	61	sselda 3949	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) ∧ 𝑧 ∈ (𝑥(ball‘𝐷)(𝑟 / 2))) → 𝑧 ∈ 𝑋)
63	62	adantrr 717	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) ∧ (𝑧 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)) ∧ 𝑧 ∈ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)))) → 𝑧 ∈ 𝑋)
64		simpllr 775	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) ∧ (𝑧 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)) ∧ 𝑧 ∈ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)))) → 𝑟 ∈ ℝ⁺)
65	64	rpred 13002	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) ∧ (𝑧 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)) ∧ 𝑧 ∈ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)))) → 𝑟 ∈ ℝ)
66		simprl 770	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) ∧ (𝑧 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)) ∧ 𝑧 ∈ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)))) → 𝑧 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)))
67	55	rpxrd 13003	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) ∧ (𝑧 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)) ∧ 𝑧 ∈ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)))) → (𝑟 / 2) ∈ ℝ^*)
68	36	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) ∧ (𝑧 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)) ∧ 𝑧 ∈ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)))) → 𝑥 ∈ 𝑋)
69		blcom 24289	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑟 / 2) ∈ ℝ^*) ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)) ↔ 𝑥 ∈ (𝑧(ball‘𝐷)(𝑟 / 2))))
70	53, 67, 68, 63, 69	syl22anc 838	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) ∧ (𝑧 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)) ∧ 𝑧 ∈ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)))) → (𝑧 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)) ↔ 𝑥 ∈ (𝑧(ball‘𝐷)(𝑟 / 2))))
71	66, 70	mpbid 232	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) ∧ (𝑧 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)) ∧ 𝑧 ∈ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)))) → 𝑥 ∈ (𝑧(ball‘𝐷)(𝑟 / 2)))
72		blhalf 24300	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝑋) ∧ (𝑟 ∈ ℝ ∧ 𝑥 ∈ (𝑧(ball‘𝐷)(𝑟 / 2)))) → (𝑧(ball‘𝐷)(𝑟 / 2)) ⊆ (𝑥(ball‘𝐷)𝑟))
73	53, 63, 65, 71, 72	syl22anc 838	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) ∧ (𝑧 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)) ∧ 𝑧 ∈ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)))) → (𝑧(ball‘𝐷)(𝑟 / 2)) ⊆ (𝑥(ball‘𝐷)𝑟))
74	59, 73	sstrd 3960	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) ∧ (𝑧 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)) ∧ 𝑧 ∈ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)))) → (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ⊆ (𝑥(ball‘𝐷)𝑟))
75	74	ex 412	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → ((𝑧 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)) ∧ 𝑧 ∈ (𝑦(ball‘𝐷)((𝑟 / 2) / 2))) → (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ⊆ (𝑥(ball‘𝐷)𝑟)))
76	52, 75	biimtrid 242	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → (𝑧 ∈ ((𝑥(ball‘𝐷)(𝑟 / 2)) ∩ (𝑦(ball‘𝐷)((𝑟 / 2) / 2))) → (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ⊆ (𝑥(ball‘𝐷)𝑟)))
77	76	exlimdv 1933	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → (∃𝑧 𝑧 ∈ ((𝑥(ball‘𝐷)(𝑟 / 2)) ∩ (𝑦(ball‘𝐷)((𝑟 / 2) / 2))) → (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ⊆ (𝑥(ball‘𝐷)𝑟)))
78	51, 77	mpd 15	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ⊆ (𝑥(ball‘𝐷)𝑟))
79		filss 23747	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ((𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹 ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ⊆ (𝑥(ball‘𝐷)𝑟))) → (𝑥(ball‘𝐷)𝑟) ∈ 𝐹)
80	33, 34, 40, 78, 79	syl13anc 1374	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → (𝑥(ball‘𝐷)𝑟) ∈ 𝐹)
81	32, 80	rexlimddv 3141	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) → (𝑥(ball‘𝐷)𝑟) ∈ 𝐹)
82	81	ad2ant2r 747	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑦)) → (𝑥(ball‘𝐷)𝑟) ∈ 𝐹)
83		toponss 22821	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → 𝑦 ⊆ 𝑋)
84	83	adantrr 717	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → 𝑦 ⊆ 𝑋)
85	14, 84	sylan 580	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → 𝑦 ⊆ 𝑋)
86	85	adantr 480	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑦)) → 𝑦 ⊆ 𝑋)
87		simprr 772	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑦)) → (𝑥(ball‘𝐷)𝑟) ⊆ 𝑦)
88		filss 23747	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ((𝑥(ball‘𝐷)𝑟) ∈ 𝐹 ∧ 𝑦 ⊆ 𝑋 ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑦)) → 𝑦 ∈ 𝐹)
89	24, 82, 86, 87, 88	syl13anc 1374	. . . . . . . 8 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑦)) → 𝑦 ∈ 𝐹)
90	20, 89	rexlimddv 3141	. . . . . . 7 ⊢ (((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → 𝑦 ∈ 𝐹)
91	90	expr 456	. . . . . 6 ⊢ (((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑦 ∈ 𝐽) → (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹))
92	91	ralrimiva 3126	. . . . 5 ⊢ ((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → ∀𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹))
93		flimopn 23869	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹))))
94	14, 23, 93	syl2anc 584	. . . . 5 ⊢ ((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹))))
95	17, 92, 94	mpbir2and 713	. . . 4 ⊢ ((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → 𝑥 ∈ (𝐽 fLim 𝐹))
96	95	ex 412	. . 3 ⊢ (𝐹 ∈ (CauFil‘𝐷) → (𝑥 ∈ (𝐽 fClus 𝐹) → 𝑥 ∈ (𝐽 fLim 𝐹)))
97	96	ssrdv 3955	. 2 ⊢ (𝐹 ∈ (CauFil‘𝐷) → (𝐽 fClus 𝐹) ⊆ (𝐽 fLim 𝐹))
98		flimfcls 23920	. . 3 ⊢ (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹)
99	98	a1i 11	. 2 ⊢ (𝐹 ∈ (CauFil‘𝐷) → (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹))
100	97, 99	eqssd 3967	1 ⊢ (𝐹 ∈ (CauFil‘𝐷) → (𝐽 fClus 𝐹) = (𝐽 fLim 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2109 ≠ wne 2926 ∀wral 3045 ∃wrex 3054 {crab 3408 ∩ cin 3916 ⊆ wss 3917 ∅c0 4299 ∪ cuni 4874 × cxp 5639 dom cdm 5641 ran crn 5642 “ cima 5644 ‘cfv 6514 (class class class)co 7390 ℝcr 11074 0cc0 11075 ℝ^*cxr 11214 / cdiv 11842 2c2 12248 ℝ⁺crp 12958 [,)cico 13315 ∞Metcxmet 21256 ballcbl 21258 MetOpencmopn 21261 TopOnctopon 22804 Filcfil 23739 fLim cflim 23828 fClus cfcls 23830 CauFilccfil 25159

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-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-sup 9400 df-inf 9401 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-n0 12450 df-z 12537 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ico 13319 df-topgen 17413 df-psmet 21263 df-xmet 21264 df-bl 21266 df-mopn 21267 df-fbas 21268 df-top 22788 df-topon 22805 df-bases 22840 df-cld 22913 df-ntr 22914 df-cls 22915 df-nei 22992 df-fil 23740 df-flim 23833 df-fcls 23835 df-cfil 25162

This theorem is referenced by: relcmpcmet 25225

W3C validator