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Theorem lebnumlem3 25027

Description: Lemma for lebnum 25028. By the previous lemmas, 𝐹 is continuous and positive on a compact set, so it has a positive minimum 𝑟. Then setting 𝑑 = 𝑟 / ♯(𝑈), since for each 𝑢 ∈ 𝑈 we have ball(𝑥, 𝑑) ⊆ 𝑢 iff 𝑑 ≤ 𝑑(𝑥, 𝑋 ∖ 𝑢), if ¬ ball(𝑥, 𝑑) ⊆ 𝑢 for all 𝑢 then summing over 𝑢 yields Σ𝑢 ∈ 𝑈𝑑(𝑥, 𝑋 ∖ 𝑢) = 𝐹(𝑥) < Σ𝑢 ∈ 𝑈𝑑 = 𝑟, in contradiction to the assumption that 𝑟 is the minimum of 𝐹. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.) (Revised by AV, 30-Sep-2020.)

**Hypotheses**
Ref	Expression
lebnum.j	⊢ 𝐽 = (MetOpen‘𝐷)
lebnum.d	⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
lebnum.c	⊢ (𝜑 → 𝐽 ∈ Comp)
lebnum.s	⊢ (𝜑 → 𝑈 ⊆ 𝐽)
lebnum.u	⊢ (𝜑 → 𝑋 = ∪ 𝑈)
lebnumlem1.u	⊢ (𝜑 → 𝑈 ∈ Fin)
lebnumlem1.n	⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)
lebnumlem1.f	⊢ 𝐹 = (𝑦 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < ))
lebnumlem2.k	⊢ 𝐾 = (topGen‘ran (,))

**Assertion**
Ref	Expression
lebnumlem3	⊢ (𝜑 → ∃𝑑 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)

Distinct variable groups: 𝑘,𝑑,𝑢,𝑥,𝑦,𝑧,𝐷 𝐽,𝑑,𝑘,𝑥,𝑦,𝑧 𝑈,𝑑,𝑘,𝑢,𝑥,𝑦,𝑧 𝑥,𝐹 𝜑,𝑑,𝑘,𝑥,𝑦,𝑧 𝑋,𝑑,𝑘,𝑢,𝑥,𝑦,𝑧 𝑥,𝐾 Allowed substitution hints: 𝜑(𝑢) 𝐹(𝑦,𝑧,𝑢,𝑘,𝑑) 𝐽(𝑢) 𝐾(𝑦,𝑧,𝑢,𝑘,𝑑)

Proof of Theorem lebnumlem3 Dummy variables 𝑟 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1rp 12999	. . . 4 ⊢ 1 ∈ ℝ⁺
2	1	ne0ii 4298	. . 3 ⊢ ℝ⁺ ≠ ∅
3		ral0 4454	. . . . 5 ⊢ ∀𝑥 ∈ ∅ ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢
4		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑋 = ∅)
5	4	raleqdv 3322	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢 ↔ ∀𝑥 ∈ ∅ ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢))
6	3, 5	mpbiri 260	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
7	6	ralrimivw 3160	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∀𝑑 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
8		r19.2z 4455	. . 3 ⊢ ((ℝ⁺ ≠ ∅ ∧ ∀𝑑 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢) → ∃𝑑 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
9	2, 7, 8	sylancr 596	. 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∃𝑑 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
10		lebnum.j	. . . . . . 7 ⊢ 𝐽 = (MetOpen‘𝐷)
11		lebnum.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
12		lebnum.c	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Comp)
13		lebnum.s	. . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ 𝐽)
14		lebnum.u	. . . . . . 7 ⊢ (𝜑 → 𝑋 = ∪ 𝑈)
15		lebnumlem1.u	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ Fin)
16		lebnumlem1.n	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)
17		lebnumlem1.f	. . . . . . 7 ⊢ 𝐹 = (𝑦 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < ))
18	10, 11, 12, 13, 14, 15, 16, 17	lebnumlem1 25025	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ⁺)
19	18	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐹:𝑋⟶ℝ⁺)
20	19	frnd 6702	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ran 𝐹 ⊆ ℝ⁺)
21		eqid 2764	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
22		lebnumlem2.k	. . . . . . 7 ⊢ 𝐾 = (topGen‘ran (,))
23	12	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐽 ∈ Comp)
24	10, 11, 12, 13, 14, 15, 16, 17, 22	lebnumlem2 25026	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
25	24	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐹 ∈ (𝐽 Cn 𝐾))
26		metxmet 24396	. . . . . . . . . 10 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
27	10	mopnuni 24503	. . . . . . . . . 10 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
28	11, 26, 27	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
29	28	neeq1d 3018	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ≠ ∅ ↔ ∪ 𝐽 ≠ ∅))
30	29	biimpa 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∪ 𝐽 ≠ ∅)
31	21, 22, 23, 25, 30	evth2 25024	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∃𝑤 ∈ ∪ 𝐽∀𝑥 ∈ ∪ 𝐽(𝐹‘𝑤) ≤ (𝐹‘𝑥))
32	28	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 = ∪ 𝐽)
33		raleq 3319	. . . . . . . 8 ⊢ (𝑋 = ∪ 𝐽 → (∀𝑥 ∈ 𝑋 (𝐹‘𝑤) ≤ (𝐹‘𝑥) ↔ ∀𝑥 ∈ ∪ 𝐽(𝐹‘𝑤) ≤ (𝐹‘𝑥)))
34	33	rexeqbi1dv 3333	. . . . . . 7 ⊢ (𝑋 = ∪ 𝐽 → (∃𝑤 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝐹‘𝑤) ≤ (𝐹‘𝑥) ↔ ∃𝑤 ∈ ∪ 𝐽∀𝑥 ∈ ∪ 𝐽(𝐹‘𝑤) ≤ (𝐹‘𝑥)))
35	32, 34	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (∃𝑤 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝐹‘𝑤) ≤ (𝐹‘𝑥) ↔ ∃𝑤 ∈ ∪ 𝐽∀𝑥 ∈ ∪ 𝐽(𝐹‘𝑤) ≤ (𝐹‘𝑥)))
36	31, 35	mpbird 259	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∃𝑤 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝐹‘𝑤) ≤ (𝐹‘𝑥))
37		ffn 6693	. . . . . 6 ⊢ (𝐹:𝑋⟶ℝ⁺ → 𝐹 Fn 𝑋)
38		breq1 5105	. . . . . . . 8 ⊢ (𝑟 = (𝐹‘𝑤) → (𝑟 ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑤) ≤ (𝐹‘𝑥)))
39	38	ralbidv 3187	. . . . . . 7 ⊢ (𝑟 = (𝐹‘𝑤) → (∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥) ↔ ∀𝑥 ∈ 𝑋 (𝐹‘𝑤) ≤ (𝐹‘𝑥)))
40	39	rexrn 7070	. . . . . 6 ⊢ (𝐹 Fn 𝑋 → (∃𝑟 ∈ ran 𝐹∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥) ↔ ∃𝑤 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝐹‘𝑤) ≤ (𝐹‘𝑥)))
41	19, 37, 40	3syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (∃𝑟 ∈ ran 𝐹∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥) ↔ ∃𝑤 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝐹‘𝑤) ≤ (𝐹‘𝑥)))
42	36, 41	mpbird 259	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∃𝑟 ∈ ran 𝐹∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥))
43		ssrexv 4008	. . . 4 ⊢ (ran 𝐹 ⊆ ℝ⁺ → (∃𝑟 ∈ ran 𝐹∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥) → ∃𝑟 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥)))
44	20, 42, 43	sylc 65	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∃𝑟 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥))
45		simpr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ⁺)
46	14	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑋 = ∪ 𝑈)
47		simplr 778	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑋 ≠ ∅)
48	46, 47	eqnetrrd 3027	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → ∪ 𝑈 ≠ ∅)
49		unieq 4878	. . . . . . . . . . 11 ⊢ (𝑈 = ∅ → ∪ 𝑈 = ∪ ∅)
50		uni0 4896	. . . . . . . . . . 11 ⊢ ∪ ∅ = ∅
51	49, 50	eqtrdi 2815	. . . . . . . . . 10 ⊢ (𝑈 = ∅ → ∪ 𝑈 = ∅)
52	51	necon3i 2991	. . . . . . . . 9 ⊢ (∪ 𝑈 ≠ ∅ → 𝑈 ≠ ∅)
53	48, 52	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑈 ≠ ∅)
54	15	ad2antrr 736	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑈 ∈ Fin)
55		hashnncl 14381	. . . . . . . . 9 ⊢ (𝑈 ∈ Fin → ((♯‘𝑈) ∈ ℕ ↔ 𝑈 ≠ ∅))
56	54, 55	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → ((♯‘𝑈) ∈ ℕ ↔ 𝑈 ≠ ∅))
57	53, 56	mpbird 259	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → (♯‘𝑈) ∈ ℕ)
58	57	nnrpd 13037	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → (♯‘𝑈) ∈ ℝ⁺)
59	45, 58	rpdivcld 13056	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → (𝑟 / (♯‘𝑈)) ∈ ℝ⁺)
60		ralnex 3090	. . . . . . . 8 ⊢ (∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢 ↔ ¬ ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)
61	54	adantr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑈 ∈ Fin)
62	53	adantr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑈 ≠ ∅)
63		simprl 780	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑥 ∈ 𝑋)
64	63	adantr 484	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝑥 ∈ 𝑋)
65		eqid 2764	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < )) = (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ))
66	65	metdsval 24910	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 → ((𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ))‘𝑥) = inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)), ℝ^, < ))
67	64, 66	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → ((𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ))‘𝑥) = inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)), ℝ^, < ))
68	11	ad2antrr 736	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝐷 ∈ (Met‘𝑋))
69	68	ad2antrr 736	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝐷 ∈ (Met‘𝑋))
70		difssd 4092	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ⊆ 𝑋)
71		elssuni 4899	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝑈 → 𝑘 ⊆ ∪ 𝑈)
72	71	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝑘 ⊆ ∪ 𝑈)
73	46	ad2antrr 736	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝑋 = ∪ 𝑈)
74	72, 73	sseqtrrd 3975	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝑘 ⊆ 𝑋)
75		eleq1 2852	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑋 → (𝑘 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
76	75	notbid 320	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑋 → (¬ 𝑘 ∈ 𝑈 ↔ ¬ 𝑋 ∈ 𝑈))
77	16, 76	syl5ibrcom 249	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑘 = 𝑋 → ¬ 𝑘 ∈ 𝑈))
78	77	necon2ad 2974	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋))
79	78	ad3antrrr 740	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋))
80	79	imp 410	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝑘 ≠ 𝑋)
81		pssdifn0 4323	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ⊆ 𝑋 ∧ 𝑘 ≠ 𝑋) → (𝑋 ∖ 𝑘) ≠ ∅)
82	74, 80, 81	syl2anc 593	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ≠ ∅)
83	65	metdsre 24916	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝑋 ∖ 𝑘) ⊆ 𝑋 ∧ (𝑋 ∖ 𝑘) ≠ ∅) → (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < )):𝑋⟶ℝ)
84	69, 70, 82, 83	syl3anc 1392	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < )):𝑋⟶ℝ)
85	84, 64	ffvelcdmd 7068	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → ((𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < ))‘𝑥) ∈ ℝ)
86	67, 85	eqeltrrd 2865	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)), ℝ^*, < ) ∈ ℝ)
87	59	ad2antrr 736	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑟 / (♯‘𝑈)) ∈ ℝ⁺)
88	87	rpred 13039	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑟 / (♯‘𝑈)) ∈ ℝ)
89		simprr 782	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)
90		sseq2 3964	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = 𝑘 → ((𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢 ↔ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑘))
91	90	notbid 320	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝑘 → (¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢 ↔ ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑘))
92	91	rspccva 3582	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢 ∧ 𝑘 ∈ 𝑈) → ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑘)
93	89, 92	sylan 589	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑘)
94	69, 26	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝐷 ∈ (∞Met‘𝑋))
95	87	rpxrd 13040	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑟 / (♯‘𝑈)) ∈ ℝ^*)
96	65	metdsge 24912	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑋 ∖ 𝑘) ⊆ 𝑋 ∧ 𝑥 ∈ 𝑋) ∧ (𝑟 / (♯‘𝑈)) ∈ ℝ^) → ((𝑟 / (♯‘𝑈)) ≤ ((𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ))‘𝑥) ↔ ((𝑋 ∖ 𝑘) ∩ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈)))) = ∅))
97	94, 70, 64, 95, 96	syl31anc 1394	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → ((𝑟 / (♯‘𝑈)) ≤ ((𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < ))‘𝑥) ↔ ((𝑋 ∖ 𝑘) ∩ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈)))) = ∅))
98		blssm 24480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (𝑟 / (♯‘𝑈)) ∈ ℝ^*) → (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑋)
99	94, 64, 95, 98	syl3anc 1392	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑋)
100		difin0ss 4328	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∖ 𝑘) ∩ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈)))) = ∅ → ((𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑋 → (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑘))
101	99, 100	syl5com 31	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (((𝑋 ∖ 𝑘) ∩ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈)))) = ∅ → (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑘))
102	97, 101	sylbid 242	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → ((𝑟 / (♯‘𝑈)) ≤ ((𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < ))‘𝑥) → (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑘))
103	93, 102	mtod 200	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → ¬ (𝑟 / (♯‘𝑈)) ≤ ((𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < ))‘𝑥))
104	85, 88	ltnled 11332	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (((𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ))‘𝑥) < (𝑟 / (♯‘𝑈)) ↔ ¬ (𝑟 / (♯‘𝑈)) ≤ ((𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ))‘𝑥)))
105	103, 104	mpbird 259	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → ((𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < ))‘𝑥) < (𝑟 / (♯‘𝑈)))
106	67, 105	eqbrtrrd 5126	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)), ℝ^*, < ) < (𝑟 / (♯‘𝑈)))
107	61, 62, 86, 88, 106	fsumlt 15830	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)), ℝ^*, < ) < Σ𝑘 ∈ 𝑈 (𝑟 / (♯‘𝑈)))
108		oveq1 7405	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → (𝑦𝐷𝑧) = (𝑥𝐷𝑧))
109	108	mpteq2dv 5196	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)) = (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)))
110	109	rneqd 5916	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)) = ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)))
111	110	infeq1d 9426	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ) = inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)), ℝ^, < ))
112	111	sumeq2sdv 15732	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ) = Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)), ℝ^, < ))
113		sumex 15717	. . . . . . . . . . . . 13 ⊢ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)), ℝ^*, < ) ∈ V
114	112, 17, 113	fvmpt 6977	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 → (𝐹‘𝑥) = Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)), ℝ^*, < ))
115	63, 114	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝐹‘𝑥) = Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)), ℝ^*, < ))
116	59	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝑟 / (♯‘𝑈)) ∈ ℝ⁺)
117	116	rpcnd 13041	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝑟 / (♯‘𝑈)) ∈ ℂ)
118		fsumconst 15819	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ Fin ∧ (𝑟 / (♯‘𝑈)) ∈ ℂ) → Σ𝑘 ∈ 𝑈 (𝑟 / (♯‘𝑈)) = ((♯‘𝑈) · (𝑟 / (♯‘𝑈))))
119	61, 117, 118	syl2anc 593	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → Σ𝑘 ∈ 𝑈 (𝑟 / (♯‘𝑈)) = ((♯‘𝑈) · (𝑟 / (♯‘𝑈))))
120		simplr 778	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑟 ∈ ℝ⁺)
121	120	rpcnd 13041	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑟 ∈ ℂ)
122	57	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (♯‘𝑈) ∈ ℕ)
123	122	nncnd 12228	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (♯‘𝑈) ∈ ℂ)
124	122	nnne0d 12265	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (♯‘𝑈) ≠ 0)
125	121, 123, 124	divcan2d 11971	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → ((♯‘𝑈) · (𝑟 / (♯‘𝑈))) = 𝑟)
126	119, 125	eqtr2d 2800	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑟 = Σ𝑘 ∈ 𝑈 (𝑟 / (♯‘𝑈)))
127	107, 115, 126	3brtr4d 5134	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝐹‘𝑥) < 𝑟)
128	19	ad2antrr 736	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝐹:𝑋⟶ℝ⁺)
129	128, 63	ffvelcdmd 7068	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝐹‘𝑥) ∈ ℝ⁺)
130	129	rpred 13039	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝐹‘𝑥) ∈ ℝ)
131	120	rpred 13039	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑟 ∈ ℝ)
132	130, 131	ltnled 11332	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → ((𝐹‘𝑥) < 𝑟 ↔ ¬ 𝑟 ≤ (𝐹‘𝑥)))
133	127, 132	mpbid 234	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → ¬ 𝑟 ≤ (𝐹‘𝑥))
134	133	expr 460	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝑋) → (∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢 → ¬ 𝑟 ≤ (𝐹‘𝑥)))
135	60, 134	biimtrrid 245	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝑋) → (¬ ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢 → ¬ 𝑟 ≤ (𝐹‘𝑥)))
136	135	con4d 115	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝑋) → (𝑟 ≤ (𝐹‘𝑥) → ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢))
137	136	ralimdva 3176	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → (∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥) → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢))
138		oveq2 7406	. . . . . . . . 9 ⊢ (𝑑 = (𝑟 / (♯‘𝑈)) → (𝑥(ball‘𝐷)𝑑) = (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))))
139	138	sseq1d 3969	. . . . . . . 8 ⊢ (𝑑 = (𝑟 / (♯‘𝑈)) → ((𝑥(ball‘𝐷)𝑑) ⊆ 𝑢 ↔ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢))
140	139	rexbidv 3188	. . . . . . 7 ⊢ (𝑑 = (𝑟 / (♯‘𝑈)) → (∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢 ↔ ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢))
141	140	ralbidv 3187	. . . . . 6 ⊢ (𝑑 = (𝑟 / (♯‘𝑈)) → (∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢 ↔ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢))
142	141	rspcev 3583	. . . . 5 ⊢ (((𝑟 / (♯‘𝑈)) ∈ ℝ⁺ ∧ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢) → ∃𝑑 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
143	59, 137, 142	syl6an 694	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → (∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥) → ∃𝑑 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢))
144	143	rexlimdva 3165	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (∃𝑟 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥) → ∃𝑑 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢))
145	44, 144	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∃𝑑 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
146	9, 145	pm2.61dane 3046	1 ⊢ (𝜑 → ∃𝑑 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1562 ∈ wcel 2144 ≠ wne 2959 ∀wral 3078 ∃wrex 3088 ∖ cdif 3903 ∩ cin 3905 ⊆ wss 3906 ∅c0 4287 ∪ cuni 4867 class class class wbr 5102 ↦ cmpt 5183 ran crn 5650 Fn wfn 6518 ⟶wf 6519 ‘cfv 6523 (class class class)co 7398 Fincfn 8929 infcinf 9389 ℂcc 11073 ℝcr 11074 1c1 11076 · cmul 11080 ℝ^*cxr 11217 < clt 11218 ≤ cle 11219 / cdiv 11846 ℕcn 12212 ℝ⁺crp 12995 (,)cioo 13351 ♯chash 14345 Σcsu 15715 topGenctg 17468 ∞Metcxmet 21411 Metcmet 21412 ballcbl 21413 MetOpencmopn 21416 Cn ccn 23286 Compccmp 23448

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-inf2 9598 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 ax-addf 11154

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-se 5603 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-isom 6532 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-of 7662 df-om 7849 df-1st 7972 df-2nd 7973 df-supp 8143 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-2o 8440 df-er 8680 df-ec 8682 df-map 8812 df-ixp 8882 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-card 9899 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-div 11847 df-nn 12213 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12484 df-z 12571 df-dec 12691 df-uz 12842 df-q 12952 df-rp 12996 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-ioo 13355 df-ico 13357 df-icc 13358 df-fz 13515 df-fzo 13662 df-seq 14017 df-exp 14077 df-hash 14346 df-cj 15128 df-re 15129 df-im 15130 df-sqrt 15264 df-abs 15265 df-clim 15517 df-sum 15716 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17248 df-ress 17269 df-plusg 17301 df-mulr 17302 df-starv 17303 df-sca 17304 df-vsca 17305 df-ip 17306 df-tset 17307 df-ple 17308 df-ds 17310 df-unif 17311 df-hom 17312 df-cco 17313 df-rest 17453 df-topn 17454 df-0g 17472 df-gsum 17473 df-topgen 17474 df-pt 17475 df-prds 17478 df-xrs 17534 df-qtop 17539 df-imas 17540 df-xps 17542 df-mre 17616 df-mrc 17617 df-acs 17619 df-mgm 18676 df-sgrp 18755 df-mnd 18771 df-submnd 18820 df-mulg 19112 df-cntz 19359 df-cmn 19824 df-psmet 21418 df-xmet 21419 df-met 21420 df-bl 21421 df-mopn 21422 df-cnfld 21427 df-top 22956 df-topon 22973 df-topsp 22995 df-bases 23008 df-cld 23081 df-ntr 23082 df-cls 23083 df-cn 23289 df-cnp 23290 df-cmp 23449 df-tx 23624 df-hmeo 23817 df-xms 24382 df-ms 24383 df-tms 24384

This theorem is referenced by: lebnum 25028

W3C validator