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Theorem cfilfcls 25241

Description: Similar to ultrafilters (uffclsflim 23996), 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 2736	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	fclselbas 23981	. . . . . . 7 ⊢ (𝑥 ∈ (𝐽 fClus 𝐹) → 𝑥 ∈ ∪ 𝐽)
3	2	adantl 481	. . . . . 6 ⊢ ((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → 𝑥 ∈ ∪ 𝐽)
4		df-cfil 25222	. . . . . . . . . . . 12 ⊢ CauFil = (𝑑 ∈ ∪ ran ∞Met ↦ {𝑓 ∈ (Fil‘dom dom 𝑑) ∣ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
5	4	mptrcl 6957	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (CauFil‘𝐷) → 𝐷 ∈ ∪ ran ∞Met)
6		xmetunirn 24302	. . . . . . . . . . 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 6843	. . . . . . . . . 10 ⊢ (∞Met‘𝑋) = (∞Met‘dom dom 𝐷)
10	7, 9	eleqtrrdi 2847	. . . . . . . . 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 24404	. . . . . . . 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 22879	. . . . . . 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 2839	. . . . 5 ⊢ ((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → 𝑥 ∈ 𝑋)
18	12	mopni2 24458	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦) → ∃𝑟 ∈ ℝ⁺ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑦)
19	18	3expb 1121	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → ∃𝑟 ∈ ℝ⁺ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑦)
20	11, 19	sylan 581	. . . . . . . 8 ⊢ (((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → ∃𝑟 ∈ ℝ⁺ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑦)
21		cfilfil 25234	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → 𝐹 ∈ (Fil‘𝑋))
22	10, 21	mpancom 689	. . . . . . . . . . 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 727	. . . . . . . . 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 767	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) → 𝐹 ∈ (CauFil‘𝐷))
27		rphalfcl 12971	. . . . . . . . . . . . . 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 12971	. . . . . . . . . . . . 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 25236	. . . . . . . . . . . 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 1374	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) → ∃𝑦 ∈ 𝑋 (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)
33	23	ad2antrr 727	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → 𝐹 ∈ (Fil‘𝑋))
34		simprr 773	. . . . . . . . . . . 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 727	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → 𝑥 ∈ 𝑋)
37		rpxr 12952	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
38	37	ad2antlr 728	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → 𝑟 ∈ ℝ^*)
39		blssm 24383	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) → (𝑥(ball‘𝐷)𝑟) ⊆ 𝑋)
40	35, 36, 38, 39	syl3anc 1374	. . . . . . . . . . . 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 776	. . . . . . . . . . . . . . 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 12987	. . . . . . . . . . . . . . . 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 24465	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (𝑟 / 2) ∈ ℝ^*) → (𝑥(ball‘𝐷)(𝑟 / 2)) ∈ 𝐽)
45	35, 36, 43, 44	syl3anc 1374	. . . . . . . . . . . . . . 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 24378	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (𝑟 / 2) ∈ ℝ⁺) → 𝑥 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)))
47	35, 36, 42, 46	syl3anc 1374	. . . . . . . . . . . . . . 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 23980	. . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . . 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 4293	. . . . . . . . . . . . . 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 3905	. . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . . . 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 12986	. . . . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . . 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 24370	. . . . . . . . . . . . . . . . . 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 839	. . . . . . . . . . . . . . . . 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 24383	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (𝑟 / 2) ∈ ℝ^*) → (𝑥(ball‘𝐷)(𝑟 / 2)) ⊆ 𝑋)
61	35, 36, 43, 60	syl3anc 1374	. . . . . . . . . . . . . . . . . . . 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 3921	. . . . . . . . . . . . . . . . . . 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 718	. . . . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . . . 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 12986	. . . . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . . . 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 12987	. . . . . . . . . . . . . . . . . . . 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 24359	. . . . . . . . . . . . . . . . . . . 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 839	. . . . . . . . . . . . . . . . . . 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 24370	. . . . . . . . . . . . . . . . . 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 839	. . . . . . . . . . . . . . . . 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 3932	. . . . . . . . . . . . . . . 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 1935	. . . . . . . . . . . . 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 23818	. . . . . . . . . . . 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 1375	. . . . . . . . . . 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 3144	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) → (𝑥(ball‘𝐷)𝑟) ∈ 𝐹)
82	81	ad2ant2r 748	. . . . . . . . 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 22892	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → 𝑦 ⊆ 𝑋)
84	83	adantrr 718	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → 𝑦 ⊆ 𝑋)
85	14, 84	sylan 581	. . . . . . . . . 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 773	. . . . . . . . 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 23818	. . . . . . . . 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 1375	. . . . . . . 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 3144	. . . . . . 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 3129	. . . . 5 ⊢ ((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → ∀𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹))
93		flimopn 23940	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹))))
94	14, 23, 93	syl2anc 585	. . . . 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 714	. . . 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 3927	. 2 ⊢ (𝐹 ∈ (CauFil‘𝐷) → (𝐽 fClus 𝐹) ⊆ (𝐽 fLim 𝐹))
98		flimfcls 23991	. . 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 3939	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 1542 ∃wex 1781 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 ∃wrex 3061 {crab 3389 ∩ cin 3888 ⊆ wss 3889 ∅c0 4273 ∪ cuni 4850 × cxp 5629 dom cdm 5631 ran crn 5632 “ cima 5634 ‘cfv 6498 (class class class)co 7367 ℝcr 11037 0cc0 11038 ℝ^*cxr 11178 / cdiv 11807 2c2 12236 ℝ⁺crp 12942 [,)cico 13300 ∞Metcxmet 21337 ballcbl 21339 MetOpencmopn 21342 TopOnctopon 22875 Filcfil 23810 fLim cflim 23899 fClus cfcls 23901 CauFilccfil 25219

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-sup 9355 df-inf 9356 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-n0 12438 df-z 12525 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ico 13304 df-topgen 17406 df-psmet 21344 df-xmet 21345 df-bl 21347 df-mopn 21348 df-fbas 21349 df-top 22859 df-topon 22876 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-fil 23811 df-flim 23904 df-fcls 23906 df-cfil 25222

This theorem is referenced by: relcmpcmet 25285

W3C validator