Theorem bcthlem4 23929

Description: Lemma for bcth 23931. Given any open ball (𝐶(ball‘𝐷)𝑅) as starting point (and in particular, a ball in int(∪ ran 𝑀)), the limit point 𝑥 of the centers of the induced sequence of balls 𝑔 is outside ∪ ran 𝑀. Note that a set 𝐴 has empty interior iff every nonempty open set 𝑈 contains points outside 𝐴, i.e. (𝑈 ∖ 𝐴) ≠ ∅. (Contributed by Mario Carneiro, 7-Jan-2014.)

Hypotheses

Ref	Expression
bcth.2	⊢ 𝐽 = (MetOpen‘𝐷)
bcthlem.4	⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
bcthlem.5	⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))})
bcthlem.6	⊢ (𝜑 → 𝑀:ℕ⟶(Clsd‘𝐽))
bcthlem.7	⊢ (𝜑 → 𝑅 ∈ ℝ+)
bcthlem.8	⊢ (𝜑 → 𝐶 ∈ 𝑋)
bcthlem.9	⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 × ℝ+))
bcthlem.10	⊢ (𝜑 → (𝑔‘1) = ⟨𝐶, 𝑅⟩)
bcthlem.11	⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))

Assertion

Ref	Expression
bcthlem4	⊢ (𝜑 → ((𝐶(ball‘𝐷)𝑅) ∖ ∪ ran 𝑀) ≠ ∅)

Distinct variable groups:   𝑘,𝑟,𝑥,𝑧   𝐶,𝑟,𝑥   𝑔,𝑘,𝑟,𝑥,𝑧,𝐷   𝑔,𝐹,𝑘,𝑟,𝑥,𝑧   𝑔,𝐽,𝑘,𝑟,𝑥,𝑧   𝑔,𝑀,𝑘,𝑟,𝑥,𝑧   𝜑,𝑘,𝑟,𝑥,𝑧   𝑥,𝑅   𝑔,𝑋,𝑘,𝑟,𝑥,𝑧

Allowed substitution hints:   𝜑(𝑔)   𝐶(𝑧,𝑔,𝑘)   𝑅(𝑧,𝑔,𝑘,𝑟)

Proof of Theorem bcthlem4

Dummy variables 𝑛 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bcthlem.4	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
2		cmetmet 23888	. . . . . . 7 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
4		metxmet 22943	. . . . . 6 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
5	3, 4	syl 17	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
6		bcthlem.9	. . . . 5 ⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 × ℝ+))
7		bcth.2	. . . . . 6 ⊢ 𝐽 = (MetOpen‘𝐷)
8		bcthlem.5	. . . . . 6 ⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))})
9		bcthlem.6	. . . . . 6 ⊢ (𝜑 → 𝑀:ℕ⟶(Clsd‘𝐽))
10		bcthlem.7	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℝ+)
11		bcthlem.8	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
12		bcthlem.10	. . . . . 6 ⊢ (𝜑 → (𝑔‘1) = ⟨𝐶, 𝑅⟩)
13		bcthlem.11	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))
14	7, 1, 8, 9, 10, 11, 6, 12, 13	bcthlem2 23927	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))
15		elrp 12390	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ+ ↔ (𝑟 ∈ ℝ ∧ 0 < 𝑟))
16		nnrecl 11894	. . . . . . . . 9 ⊢ ((𝑟 ∈ ℝ ∧ 0 < 𝑟) → ∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟)
17	15, 16	sylbi 219	. . . . . . . 8 ⊢ (𝑟 ∈ ℝ+ → ∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟)
18	17	adantl 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → ∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟)
19		peano2nn 11649	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
20	19	adantl 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℕ)
21		fvoveq1 7178	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → (𝑔‘(𝑘 + 1)) = (𝑔‘(𝑚 + 1)))
22		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑚 → 𝑘 = 𝑚)
23		fveq2 6669	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑚 → (𝑔‘𝑘) = (𝑔‘𝑚))
24	22, 23	oveq12d 7173	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → (𝑘𝐹(𝑔‘𝑘)) = (𝑚𝐹(𝑔‘𝑚)))
25	21, 24	eleq12d 2907	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → ((𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ↔ (𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔‘𝑚))))
26	25	rspccva 3621	. . . . . . . . . . . . . 14 ⊢ ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ∧ 𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔‘𝑚)))
27	13, 26	sylan 582	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔‘𝑚)))
28	6	ffvelrnda 6850	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑔‘𝑚) ∈ (𝑋 × ℝ+))
29	7, 1, 8	bcthlem1 23926	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔‘𝑚) ∈ (𝑋 × ℝ+))) → ((𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔‘𝑚)) ↔ ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) ∧ (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑚)) ∖ (𝑀‘𝑚)))))
30	29	expr 459	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑔‘𝑚) ∈ (𝑋 × ℝ+) → ((𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔‘𝑚)) ↔ ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) ∧ (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑚)) ∖ (𝑀‘𝑚))))))
31	28, 30	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔‘𝑚)) ↔ ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) ∧ (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑚)) ∖ (𝑀‘𝑚)))))
32	27, 31	mpbid 234	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) ∧ (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑚)) ∖ (𝑀‘𝑚))))
33	32	simp2d 1139	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚))
34	33	adantlr 713	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚))
35	32	simp1d 1138	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+))
36		xp2nd 7721	. . . . . . . . . . . . . 14 ⊢ ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ+)
37	35, 36	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ+)
38	37	rpred 12430	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ)
39	38	adantlr 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ)
40		nnrecre 11678	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (1 / 𝑚) ∈ ℝ)
41	40	adantl 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (1 / 𝑚) ∈ ℝ)
42		rpre 12396	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ)
43	42	ad2antlr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → 𝑟 ∈ ℝ)
44		lttr 10716	. . . . . . . . . . 11 ⊢ (((2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ ∧ (1 / 𝑚) ∈ ℝ ∧ 𝑟 ∈ ℝ) → (((2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚) ∧ (1 / 𝑚) < 𝑟) → (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
45	39, 41, 43, 44	syl3anc 1367	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (((2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚) ∧ (1 / 𝑚) < 𝑟) → (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
46	34, 45	mpand 693	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → ((1 / 𝑚) < 𝑟 → (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
47		2fveq3 6674	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑚 + 1) → (2nd ‘(𝑔‘𝑛)) = (2nd ‘(𝑔‘(𝑚 + 1))))
48	47	breq1d 5075	. . . . . . . . . 10 ⊢ (𝑛 = (𝑚 + 1) → ((2nd ‘(𝑔‘𝑛)) < 𝑟 ↔ (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
49	48	rspcev 3622	. . . . . . . . 9 ⊢ (((𝑚 + 1) ∈ ℕ ∧ (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟) → ∃𝑛 ∈ ℕ (2nd ‘(𝑔‘𝑛)) < 𝑟)
50	20, 46, 49	syl6an 682	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → ((1 / 𝑚) < 𝑟 → ∃𝑛 ∈ ℕ (2nd ‘(𝑔‘𝑛)) < 𝑟))
51	50	rexlimdva 3284	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → (∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟 → ∃𝑛 ∈ ℕ (2nd ‘(𝑔‘𝑛)) < 𝑟))
52	18, 51	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → ∃𝑛 ∈ ℕ (2nd ‘(𝑔‘𝑛)) < 𝑟)
53	52	ralrimiva 3182	. . . . 5 ⊢ (𝜑 → ∀𝑟 ∈ ℝ+ ∃𝑛 ∈ ℕ (2nd ‘(𝑔‘𝑛)) < 𝑟)
54	5, 6, 14, 53	caubl 23910	. . . 4 ⊢ (𝜑 → (1st ∘ 𝑔) ∈ (Cau‘𝐷))
55	7	cmetcau 23891	. . . 4 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (1st ∘ 𝑔) ∈ (Cau‘𝐷)) → (1st ∘ 𝑔) ∈ dom (⇝𝑡‘𝐽))
56	1, 54, 55	syl2anc 586	. . 3 ⊢ (𝜑 → (1st ∘ 𝑔) ∈ dom (⇝𝑡‘𝐽))
57		fo1st 7708	. . . . . 6 ⊢ 1st :V–onto→V
58		fofun 6590	. . . . . 6 ⊢ (1st :V–onto→V → Fun 1st )
59	57, 58	ax-mp 5	. . . . 5 ⊢ Fun 1st
60		vex 3497	. . . . 5 ⊢ 𝑔 ∈ V
61		cofunexg 7649	. . . . 5 ⊢ ((Fun 1st ∧ 𝑔 ∈ V) → (1st ∘ 𝑔) ∈ V)
62	59, 60, 61	mp2an 690	. . . 4 ⊢ (1st ∘ 𝑔) ∈ V
63	62	eldm 5768	. . 3 ⊢ ((1st ∘ 𝑔) ∈ dom (⇝𝑡‘𝐽) ↔ ∃𝑥(1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥)
64	56, 63	sylib 220	. 2 ⊢ (𝜑 → ∃𝑥(1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥)
65		1nn 11648	. . . . . 6 ⊢ 1 ∈ ℕ
66	7, 1, 8, 9, 10, 11, 6, 12, 13	bcthlem3 23928	. . . . . 6 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ 1 ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘1)))
67	65, 66	mp3an3 1446	. . . . 5 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘1)))
68	12	fveq2d 6673	. . . . . . 7 ⊢ (𝜑 → ((ball‘𝐷)‘(𝑔‘1)) = ((ball‘𝐷)‘⟨𝐶, 𝑅⟩))
69		df-ov 7158	. . . . . . 7 ⊢ (𝐶(ball‘𝐷)𝑅) = ((ball‘𝐷)‘⟨𝐶, 𝑅⟩)
70	68, 69	syl6eqr 2874	. . . . . 6 ⊢ (𝜑 → ((ball‘𝐷)‘(𝑔‘1)) = (𝐶(ball‘𝐷)𝑅))
71	70	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥) → ((ball‘𝐷)‘(𝑔‘1)) = (𝐶(ball‘𝐷)𝑅))
72	67, 71	eleqtrd 2915	. . . 4 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥) → 𝑥 ∈ (𝐶(ball‘𝐷)𝑅))
73	7	mopntop 23049	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
74	5, 73	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ Top)
75	74	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐽 ∈ Top)
76	5	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋))
77		xp1st 7720	. . . . . . . . . . . . . . 15 ⊢ ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) → (1st ‘(𝑔‘(𝑚 + 1))) ∈ 𝑋)
78	35, 77	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1st ‘(𝑔‘(𝑚 + 1))) ∈ 𝑋)
79	37	rpxrd 12431	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ*)
80		blssm 23027	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑔‘(𝑚 + 1))) ∈ 𝑋 ∧ (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ*) → ((1st ‘(𝑔‘(𝑚 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑚 + 1)))) ⊆ 𝑋)
81	76, 78, 79, 80	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝑔‘(𝑚 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑚 + 1)))) ⊆ 𝑋)
82		1st2nd2 7727	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) → (𝑔‘(𝑚 + 1)) = ⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩)
83	35, 82	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) = ⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩)
84	83	fveq2d 6673	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) = ((ball‘𝐷)‘⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩))
85		df-ov 7158	. . . . . . . . . . . . . 14 ⊢ ((1st ‘(𝑔‘(𝑚 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑚 + 1)))) = ((ball‘𝐷)‘⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩)
86	84, 85	syl6reqr 2875	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝑔‘(𝑚 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑚 + 1)))) = ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))))
87	7	mopnuni 23050	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
88	5, 87	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
89	88	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋 = ∪ 𝐽)
90	81, 86, 89	3sstr3d 4012	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ ∪ 𝐽)
91		eqid 2821	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
92	91	sscls 21663	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ ∪ 𝐽) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))))
93	75, 90, 92	syl2anc 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))))
94	32	simp3d 1140	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑚)) ∖ (𝑀‘𝑚)))
95	93, 94	sstrd 3976	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑚)) ∖ (𝑀‘𝑚)))
96	95	3adant2 1127	. . . . . . . . 9 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ 𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑚)) ∖ (𝑀‘𝑚)))
97	7, 1, 8, 9, 10, 11, 6, 12, 13	bcthlem3 23928	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ (𝑚 + 1) ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))))
98	19, 97	syl3an3 1161	. . . . . . . . 9 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ 𝑚 ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))))
99	96, 98	sseldd 3967	. . . . . . . 8 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ 𝑚 ∈ ℕ) → 𝑥 ∈ (((ball‘𝐷)‘(𝑔‘𝑚)) ∖ (𝑀‘𝑚)))
100	99	eldifbd 3948	. . . . . . 7 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ 𝑚 ∈ ℕ) → ¬ 𝑥 ∈ (𝑀‘𝑚))
101	100	3expa 1114	. . . . . 6 ⊢ (((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥) ∧ 𝑚 ∈ ℕ) → ¬ 𝑥 ∈ (𝑀‘𝑚))
102	101	ralrimiva 3182	. . . . 5 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥) → ∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀‘𝑚))
103		eluni2 4841	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ ran 𝑀 ↔ ∃𝑦 ∈ ran 𝑀 𝑥 ∈ 𝑦)
104	9	ffnd 6514	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 Fn ℕ)
105		eleq2 2901	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑀‘𝑚) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑀‘𝑚)))
106	105	rexrn 6852	. . . . . . . . . 10 ⊢ (𝑀 Fn ℕ → (∃𝑦 ∈ ran 𝑀 𝑥 ∈ 𝑦 ↔ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀‘𝑚)))
107	104, 106	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑦 ∈ ran 𝑀 𝑥 ∈ 𝑦 ↔ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀‘𝑚)))
108	103, 107	syl5bb 285	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ∪ ran 𝑀 ↔ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀‘𝑚)))
109	108	notbid 320	. . . . . . 7 ⊢ (𝜑 → (¬ 𝑥 ∈ ∪ ran 𝑀 ↔ ¬ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀‘𝑚)))
110		ralnex 3236	. . . . . . 7 ⊢ (∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀‘𝑚) ↔ ¬ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀‘𝑚))
111	109, 110	syl6bbr 291	. . . . . 6 ⊢ (𝜑 → (¬ 𝑥 ∈ ∪ ran 𝑀 ↔ ∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀‘𝑚)))
112	111	biimpar 480	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀‘𝑚)) → ¬ 𝑥 ∈ ∪ ran 𝑀)
113	102, 112	syldan 593	. . . 4 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥) → ¬ 𝑥 ∈ ∪ ran 𝑀)
114	72, 113	eldifd 3946	. . 3 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥) → 𝑥 ∈ ((𝐶(ball‘𝐷)𝑅) ∖ ∪ ran 𝑀))
115	114	ne0d 4300	. 2 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥) → ((𝐶(ball‘𝐷)𝑅) ∖ ∪ ran 𝑀) ≠ ∅)
116	64, 115	exlimddv 1932	1 ⊢ (𝜑 → ((𝐶(ball‘𝐷)𝑅) ∖ ∪ ran 𝑀) ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533  ∃wex 1776   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  Vcvv 3494   ∖ cdif 3932   ⊆ wss 3935  ∅c0 4290  ⟨cop 4572  ∪ cuni 4837   class class class wbr 5065  {copab 5127   × cxp 5552  dom cdm 5554  ran crn 5555   ∘ ccom 5558  Fun wfun 6348   Fn wfn 6349  ⟶wf 6350  –onto→wfo 6352  ‘cfv 6354  (class class class)co 7155   ∈ cmpo 7157  1^st c1st 7686  2^nd c2nd 7687  ℝcr 10535  0cc0 10536  1c1 10537   + caddc 10539  ℝ^*cxr 10673   < clt 10674   / cdiv 11296  ℕcn 11637  ℝ⁺crp 12388  ∞Metcxmet 20529  Metcmet 20530  ballcbl 20531  MetOpencmopn 20534  Topctop 21500  Clsdccld 21623  clsccl 21625  ⇝_𝑡clm 21833  Cauccau 23855  CMetccmet 23856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-iin 4921  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-er 8288  df-map 8407  df-pm 8408  df-en 8509  df-dom 8510  df-sdom 8511  df-sup 8905  df-inf 8906  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-nn 11638  df-2 11699  df-n0 11897  df-z 11981  df-uz 12243  df-q 12348  df-rp 12389  df-xneg 12506  df-xadd 12507  df-xmul 12508  df-ico 12743  df-rest 16695  df-topgen 16716  df-psmet 20536  df-xmet 20537  df-met 20538  df-bl 20539  df-mopn 20540  df-fbas 20541  df-fg 20542  df-top 21501  df-topon 21518  df-bases 21553  df-cld 21626  df-ntr 21627  df-cls 21628  df-nei 21705  df-lm 21836  df-fil 22453  df-fm 22545  df-flim 22546  df-flf 22547  df-cfil 23857  df-cau 23858  df-cmet 23859

This theorem is referenced by:  bcthlem5  23930