Theorem bcthlem4 25225

Description: Lemma for bcth 25227. 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 25184	. . . . . . 7 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
4		metxmet 24220	. . . . . 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 25223	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))
15		elrp 12895	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ+ ↔ (𝑟 ∈ ℝ ∧ 0 < 𝑟))
16		nnrecl 12382	. . . . . . . . 9 ⊢ ((𝑟 ∈ ℝ ∧ 0 < 𝑟) → ∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟)
17	15, 16	sylbi 217	. . . . . . . 8 ⊢ (𝑟 ∈ ℝ+ → ∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟)
18	17	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → ∃𝑚 ∈ ℕ (1 / 𝑚) < 𝑟)
19		peano2nn 12140	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
20	19	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℕ)
21		fvoveq1 7372	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → (𝑔‘(𝑘 + 1)) = (𝑔‘(𝑚 + 1)))
22		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑚 → 𝑘 = 𝑚)
23		fveq2 6822	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑚 → (𝑔‘𝑘) = (𝑔‘𝑚))
24	22, 23	oveq12d 7367	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → (𝑘𝐹(𝑔‘𝑘)) = (𝑚𝐹(𝑔‘𝑚)))
25	21, 24	eleq12d 2822	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → ((𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ↔ (𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔‘𝑚))))
26	25	rspccva 3576	. . . . . . . . . . . . . 14 ⊢ ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ∧ 𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔‘𝑚)))
27	13, 26	sylan 580	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ∈ (𝑚𝐹(𝑔‘𝑚)))
28	6	ffvelcdmda 7018	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑔‘𝑚) ∈ (𝑋 × ℝ+))
29	7, 1, 8	bcthlem1 25222	. . . . . . . . . . . . . . 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 456	. . . . . . . . . . . . . 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 232	. . . . . . . . . . . 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 1143	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚))
34	33	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚))
35	32	simp1d 1142	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+))
36		xp2nd 7957	. . . . . . . . . . . . . 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 12937	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ)
39	38	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ)
40		nnrecre 12170	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (1 / 𝑚) ∈ ℝ)
41	40	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (1 / 𝑚) ∈ ℝ)
42		rpre 12902	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ)
43	42	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → 𝑟 ∈ ℝ)
44		lttr 11192	. . . . . . . . . . 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 1373	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (((2nd ‘(𝑔‘(𝑚 + 1))) < (1 / 𝑚) ∧ (1 / 𝑚) < 𝑟) → (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
46	34, 45	mpand 695	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → ((1 / 𝑚) < 𝑟 → (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
47		2fveq3 6827	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑚 + 1) → (2nd ‘(𝑔‘𝑛)) = (2nd ‘(𝑔‘(𝑚 + 1))))
48	47	breq1d 5102	. . . . . . . . . 10 ⊢ (𝑛 = (𝑚 + 1) → ((2nd ‘(𝑔‘𝑛)) < 𝑟 ↔ (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟))
49	48	rspcev 3577	. . . . . . . . 9 ⊢ (((𝑚 + 1) ∈ ℕ ∧ (2nd ‘(𝑔‘(𝑚 + 1))) < 𝑟) → ∃𝑛 ∈ ℕ (2nd ‘(𝑔‘𝑛)) < 𝑟)
50	20, 46, 49	syl6an 684	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → ((1 / 𝑚) < 𝑟 → ∃𝑛 ∈ ℕ (2nd ‘(𝑔‘𝑛)) < 𝑟))
51	50	rexlimdva 3130	. . . . . . 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 3121	. . . . 5 ⊢ (𝜑 → ∀𝑟 ∈ ℝ+ ∃𝑛 ∈ ℕ (2nd ‘(𝑔‘𝑛)) < 𝑟)
54	5, 6, 14, 53	caubl 25206	. . . 4 ⊢ (𝜑 → (1st ∘ 𝑔) ∈ (Cau‘𝐷))
55	7	cmetcau 25187	. . . 4 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (1st ∘ 𝑔) ∈ (Cau‘𝐷)) → (1st ∘ 𝑔) ∈ dom (⇝𝑡‘𝐽))
56	1, 54, 55	syl2anc 584	. . 3 ⊢ (𝜑 → (1st ∘ 𝑔) ∈ dom (⇝𝑡‘𝐽))
57		fo1st 7944	. . . . . 6 ⊢ 1st :V–onto→V
58		fofun 6737	. . . . . 6 ⊢ (1st :V–onto→V → Fun 1st )
59	57, 58	ax-mp 5	. . . . 5 ⊢ Fun 1st
60		vex 3440	. . . . 5 ⊢ 𝑔 ∈ V
61		cofunexg 7884	. . . . 5 ⊢ ((Fun 1st ∧ 𝑔 ∈ V) → (1st ∘ 𝑔) ∈ V)
62	59, 60, 61	mp2an 692	. . . 4 ⊢ (1st ∘ 𝑔) ∈ V
63	62	eldm 5843	. . 3 ⊢ ((1st ∘ 𝑔) ∈ dom (⇝𝑡‘𝐽) ↔ ∃𝑥(1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥)
64	56, 63	sylib 218	. 2 ⊢ (𝜑 → ∃𝑥(1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥)
65		1nn 12139	. . . . . 6 ⊢ 1 ∈ ℕ
66	7, 1, 8, 9, 10, 11, 6, 12, 13	bcthlem3 25224	. . . . . 6 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ 1 ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘1)))
67	65, 66	mp3an3 1452	. . . . 5 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘1)))
68	12	fveq2d 6826	. . . . . . 7 ⊢ (𝜑 → ((ball‘𝐷)‘(𝑔‘1)) = ((ball‘𝐷)‘⟨𝐶, 𝑅⟩))
69		df-ov 7352	. . . . . . 7 ⊢ (𝐶(ball‘𝐷)𝑅) = ((ball‘𝐷)‘⟨𝐶, 𝑅⟩)
70	68, 69	eqtr4di 2782	. . . . . 6 ⊢ (𝜑 → ((ball‘𝐷)‘(𝑔‘1)) = (𝐶(ball‘𝐷)𝑅))
71	70	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥) → ((ball‘𝐷)‘(𝑔‘1)) = (𝐶(ball‘𝐷)𝑅))
72	67, 71	eleqtrd 2830	. . . 4 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥) → 𝑥 ∈ (𝐶(ball‘𝐷)𝑅))
73	7	mopntop 24326	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
74	5, 73	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ Top)
75	74	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐽 ∈ Top)
76	5	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋))
77		xp1st 7956	. . . . . . . . . . . . . . 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 12938	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd ‘(𝑔‘(𝑚 + 1))) ∈ ℝ*)
80		blssm 24304	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝑔‘(𝑚 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑚 + 1)))) ⊆ 𝑋)
82		df-ov 7352	. . . . . . . . . . . . . 14 ⊢ ((1st ‘(𝑔‘(𝑚 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑚 + 1)))) = ((ball‘𝐷)‘⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩)
83		1st2nd2 7963	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔‘(𝑚 + 1)) ∈ (𝑋 × ℝ+) → (𝑔‘(𝑚 + 1)) = ⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩)
84	35, 83	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑔‘(𝑚 + 1)) = ⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩)
85	84	fveq2d 6826	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) = ((ball‘𝐷)‘⟨(1st ‘(𝑔‘(𝑚 + 1))), (2nd ‘(𝑔‘(𝑚 + 1)))⟩))
86	82, 85	eqtr4id 2783	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((1st ‘(𝑔‘(𝑚 + 1)))(ball‘𝐷)(2nd ‘(𝑔‘(𝑚 + 1)))) = ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))))
87	7	mopnuni 24327	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
88	5, 87	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
89	88	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋 = ∪ 𝐽)
90	81, 86, 89	3sstr3d 3990	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ ∪ 𝐽)
91		eqid 2729	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
92	91	sscls 22941	. . . . . . . . . . . 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 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))))
94	32	simp3d 1144	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑚 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑚)) ∖ (𝑀‘𝑚)))
95	93, 94	sstrd 3946	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑚)) ∖ (𝑀‘𝑚)))
96	95	3adant2 1131	. . . . . . . . 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 25224	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ (𝑚 + 1) ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))))
98	19, 97	syl3an3 1165	. . . . . . . . 9 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ 𝑚 ∈ ℕ) → 𝑥 ∈ ((ball‘𝐷)‘(𝑔‘(𝑚 + 1))))
99	96, 98	sseldd 3936	. . . . . . . 8 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ 𝑚 ∈ ℕ) → 𝑥 ∈ (((ball‘𝐷)‘(𝑔‘𝑚)) ∖ (𝑀‘𝑚)))
100	99	eldifbd 3916	. . . . . . 7 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥 ∧ 𝑚 ∈ ℕ) → ¬ 𝑥 ∈ (𝑀‘𝑚))
101	100	3expa 1118	. . . . . 6 ⊢ (((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥) ∧ 𝑚 ∈ ℕ) → ¬ 𝑥 ∈ (𝑀‘𝑚))
102	101	ralrimiva 3121	. . . . 5 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥) → ∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀‘𝑚))
103		eluni2 4862	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ ran 𝑀 ↔ ∃𝑦 ∈ ran 𝑀 𝑥 ∈ 𝑦)
104	9	ffnd 6653	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 Fn ℕ)
105		eleq2 2817	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑀‘𝑚) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑀‘𝑚)))
106	105	rexrn 7021	. . . . . . . . . 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	bitrid 283	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ∪ ran 𝑀 ↔ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀‘𝑚)))
109	108	notbid 318	. . . . . . 7 ⊢ (𝜑 → (¬ 𝑥 ∈ ∪ ran 𝑀 ↔ ¬ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀‘𝑚)))
110		ralnex 3055	. . . . . . 7 ⊢ (∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀‘𝑚) ↔ ¬ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑀‘𝑚))
111	109, 110	bitr4di 289	. . . . . 6 ⊢ (𝜑 → (¬ 𝑥 ∈ ∪ ran 𝑀 ↔ ∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀‘𝑚)))
112	111	biimpar 477	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑚 ∈ ℕ ¬ 𝑥 ∈ (𝑀‘𝑚)) → ¬ 𝑥 ∈ ∪ ran 𝑀)
113	102, 112	syldan 591	. . . 4 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥) → ¬ 𝑥 ∈ ∪ ran 𝑀)
114	72, 113	eldifd 3914	. . 3 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥) → 𝑥 ∈ ((𝐶(ball‘𝐷)𝑅) ∖ ∪ ran 𝑀))
115	114	ne0d 4293	. 2 ⊢ ((𝜑 ∧ (1st ∘ 𝑔)(⇝𝑡‘𝐽)𝑥) → ((𝐶(ball‘𝐷)𝑅) ∖ ∪ ran 𝑀) ≠ ∅)
116	64, 115	exlimddv 1935	1 ⊢ (𝜑 → ((𝐶(ball‘𝐷)𝑅) ∖ ∪ ran 𝑀) ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  Vcvv 3436   ∖ cdif 3900   ⊆ wss 3903  ∅c0 4284  ⟨cop 4583  ∪ cuni 4858   class class class wbr 5092  {copab 5154   × cxp 5617  dom cdm 5619  ran crn 5620   ∘ ccom 5623  Fun wfun 6476   Fn wfn 6477  ⟶wf 6478  –onto→wfo 6480  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351  1^st c1st 7922  2^nd c2nd 7923  ℝcr 11008  0cc0 11009  1c1 11010   + caddc 11012  ℝ^*cxr 11148   < clt 11149   / cdiv 11777  ℕcn 12128  ℝ⁺crp 12893  ∞Metcxmet 21246  Metcmet 21247  ballcbl 21248  MetOpencmopn 21251  Topctop 22778  Clsdccld 22901  clsccl 22903  ⇝_𝑡clm 23111  Cauccau 25151  CMetccmet 25152

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 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-er 8625  df-map 8755  df-pm 8756  df-en 8873  df-dom 8874  df-sdom 8875  df-sup 9332  df-inf 9333  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-n0 12385  df-z 12472  df-uz 12736  df-q 12850  df-rp 12894  df-xneg 13014  df-xadd 13015  df-xmul 13016  df-ico 13254  df-rest 17326  df-topgen 17347  df-psmet 21253  df-xmet 21254  df-met 21255  df-bl 21256  df-mopn 21257  df-fbas 21258  df-fg 21259  df-top 22779  df-topon 22796  df-bases 22831  df-cld 22904  df-ntr 22905  df-cls 22906  df-nei 22983  df-lm 23114  df-fil 23731  df-fm 23823  df-flim 23824  df-flf 23825  df-cfil 25153  df-cau 25154  df-cmet 25155

This theorem is referenced by:  bcthlem5  25226