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Theorem cfilfcls 24782

Description: Similar to ultrafilters (uffclsflim 23526), 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 2732	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	fclselbas 23511	. . . . . . 7 ⊢ (𝑥 ∈ (𝐽 fClus 𝐹) → 𝑥 ∈ ∪ 𝐽)
3	2	adantl 482	. . . . . 6 ⊢ ((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → 𝑥 ∈ ∪ 𝐽)
4		df-cfil 24763	. . . . . . . . . . . 12 ⊢ CauFil = (𝑑 ∈ ∪ ran ∞Met ↦ {𝑓 ∈ (Fil‘dom dom 𝑑) ∣ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
5	4	mptrcl 7004	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (CauFil‘𝐷) → 𝐷 ∈ ∪ ran ∞Met)
6		xmetunirn 23834	. . . . . . . . . . 11 ⊢ (𝐷 ∈ ∪ ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷))
7	5, 6	sylib 217	. . . . . . . . . 10 ⊢ (𝐹 ∈ (CauFil‘𝐷) → 𝐷 ∈ (∞Met‘dom dom 𝐷))
8		cfilfcls.2	. . . . . . . . . . 11 ⊢ 𝑋 = dom dom 𝐷
9	8	fveq2i 6891	. . . . . . . . . 10 ⊢ (∞Met‘𝑋) = (∞Met‘dom dom 𝐷)
10	7, 9	eleqtrrdi 2844	. . . . . . . . 9 ⊢ (𝐹 ∈ (CauFil‘𝐷) → 𝐷 ∈ (∞Met‘𝑋))
11	10	adantr 481	. . . . . . . 8 ⊢ ((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → 𝐷 ∈ (∞Met‘𝑋))
12		cfilfcls.1	. . . . . . . . 9 ⊢ 𝐽 = (MetOpen‘𝐷)
13	12	mopntopon 23936	. . . . . . . 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 22407	. . . . . . 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 2836	. . . . 5 ⊢ ((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → 𝑥 ∈ 𝑋)
18	12	mopni2 23993	. . . . . . . . . 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 24775	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → 𝐹 ∈ (Fil‘𝑋))
22	10, 21	mpancom 686	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (CauFil‘𝐷) → 𝐹 ∈ (Fil‘𝑋))
23	22	adantr 481	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → 𝐹 ∈ (Fil‘𝑋))
24	23	ad2antrr 724	. . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) → 𝐷 ∈ (∞Met‘𝑋))
26		simpll 765	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) → 𝐹 ∈ (CauFil‘𝐷))
27		rphalfcl 12997	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ ℝ⁺ → (𝑟 / 2) ∈ ℝ⁺)
28	27	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) → (𝑟 / 2) ∈ ℝ⁺)
29		rphalfcl 12997	. . . . . . . . . . . . 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 24777	. . . . . . . . . . . 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 1371	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) → ∃𝑦 ∈ 𝑋 (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)
33	23	ad2antrr 724	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → 𝐹 ∈ (Fil‘𝑋))
34		simprr 771	. . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 724	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → 𝑥 ∈ 𝑋)
37		rpxr 12979	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
38	37	ad2antlr 725	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝑋 ∧ (𝑦(ball‘𝐷)((𝑟 / 2) / 2)) ∈ 𝐹)) → 𝑟 ∈ ℝ^*)
39		blssm 23915	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*) → (𝑥(ball‘𝐷)𝑟) ⊆ 𝑋)
40	35, 36, 38, 39	syl3anc 1371	. . . . . . . . . . . 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 774	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 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 13013	. . . . . . . . . . . . . . . 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 24000	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (𝑟 / 2) ∈ ℝ^*) → (𝑥(ball‘𝐷)(𝑟 / 2)) ∈ 𝐽)
45	35, 36, 43, 44	syl3anc 1371	. . . . . . . . . . . . . . 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 23910	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (𝑟 / 2) ∈ ℝ⁺) → 𝑥 ∈ (𝑥(ball‘𝐷)(𝑟 / 2)))
47	35, 36, 42, 46	syl3anc 1371	. . . . . . . . . . . . . . 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 23510	. . . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . . 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 4345	. . . . . . . . . . . . . 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 217	. . . . . . . . . . . . 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 3963	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . 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 13012	. . . . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . . 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 23902	. . . . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . . . 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 23915	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (𝑟 / 2) ∈ ℝ^*) → (𝑥(ball‘𝐷)(𝑟 / 2)) ⊆ 𝑋)
61	35, 36, 43, 60	syl3anc 1371	. . . . . . . . . . . . . . . . . . . 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 3981	. . . . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . . . . . 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 13012	. . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . 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 13013	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . 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 23891	. . . . . . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . . 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 23902	. . . . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . . . 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 3991	. . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . 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 241	. . . . . . . . . . . . . 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 1936	. . . . . . . . . . . . 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 23348	. . . . . . . . . . . 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 1372	. . . . . . . . . . 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 3161	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑟 ∈ ℝ⁺) → (𝑥(ball‘𝐷)𝑟) ∈ 𝐹)
82	81	ad2ant2r 745	. . . . . . . . 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 22420	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → 𝑦 ⊆ 𝑋)
84	83	adantrr 715	. . . . . . . . . . 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 481	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) ∧ (𝑟 ∈ ℝ⁺ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑦)) → 𝑦 ⊆ 𝑋)
87		simprr 771	. . . . . . . . 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 23348	. . . . . . . . 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 1372	. . . . . . . 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 3161	. . . . . . 7 ⊢ (((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → 𝑦 ∈ 𝐹)
91	90	expr 457	. . . . . 6 ⊢ (((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) ∧ 𝑦 ∈ 𝐽) → (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹))
92	91	ralrimiva 3146	. . . . 5 ⊢ ((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → ∀𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹))
93		flimopn 23470	. . . . . 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 711	. . . 4 ⊢ ((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ (𝐽 fClus 𝐹)) → 𝑥 ∈ (𝐽 fLim 𝐹))
96	95	ex 413	. . 3 ⊢ (𝐹 ∈ (CauFil‘𝐷) → (𝑥 ∈ (𝐽 fClus 𝐹) → 𝑥 ∈ (𝐽 fLim 𝐹)))
97	96	ssrdv 3987	. 2 ⊢ (𝐹 ∈ (CauFil‘𝐷) → (𝐽 fClus 𝐹) ⊆ (𝐽 fLim 𝐹))
98		flimfcls 23521	. . 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 3998	1 ⊢ (𝐹 ∈ (CauFil‘𝐷) → (𝐽 fClus 𝐹) = (𝐽 fLim 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3432 ∩ cin 3946 ⊆ wss 3947 ∅c0 4321 ∪ cuni 4907 × cxp 5673 dom cdm 5675 ran crn 5676 “ cima 5678 ‘cfv 6540 (class class class)co 7405 ℝcr 11105 0cc0 11106 ℝ^*cxr 11243 / cdiv 11867 2c2 12263 ℝ⁺crp 12970 [,)cico 13322 ∞Metcxmet 20921 ballcbl 20923 MetOpencmopn 20926 TopOnctopon 22403 Filcfil 23340 fLim cflim 23429 fClus cfcls 23431 CauFilccfil 24760

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ico 13326 df-topgen 17385 df-psmet 20928 df-xmet 20929 df-bl 20931 df-mopn 20932 df-fbas 20933 df-top 22387 df-topon 22404 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-fil 23341 df-flim 23434 df-fcls 23436 df-cfil 24763

This theorem is referenced by: relcmpcmet 24826

W3C validator