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Theorem lebnumlem3 25177

Description: Lemma for lebnum 25178. 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 13040	. . . 4 ⊢ 1 ∈ ℝ⁺
2	1	ne0ii 4297	. . 3 ⊢ ℝ⁺ ≠ ∅
3		ral0 4461	. . . . 5 ⊢ ∀𝑥 ∈ ∅ ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢
4		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑋 = ∅)
5	4	raleqdv 3325	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢 ↔ ∀𝑥 ∈ ∅ ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢))
6	3, 5	mpbiri 261	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
7	6	ralrimivw 3163	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∀𝑑 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
8		r19.2z 4462	. . 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 599	. 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 25175	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ⁺)
19	18	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐹:𝑋⟶ℝ⁺)
20	19	frnd 6718	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ran 𝐹 ⊆ ℝ⁺)
21		eqid 2765	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
22		lebnumlem2.k	. . . . . . 7 ⊢ 𝐾 = (topGen‘ran (,))
23	12	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐽 ∈ Comp)
24	10, 11, 12, 13, 14, 15, 16, 17, 22	lebnumlem2 25176	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
25	24	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐹 ∈ (𝐽 Cn 𝐾))
26		metxmet 24546	. . . . . . . . . 10 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
27	10	mopnuni 24653	. . . . . . . . . 10 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
28	11, 26, 27	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
29	28	neeq1d 3019	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ≠ ∅ ↔ ∪ 𝐽 ≠ ∅))
30	29	biimpa 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∪ 𝐽 ≠ ∅)
31	21, 22, 23, 25, 30	evth2 25174	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∃𝑤 ∈ ∪ 𝐽∀𝑥 ∈ ∪ 𝐽(𝐹‘𝑤) ≤ (𝐹‘𝑥))
32	28	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 = ∪ 𝐽)
33		raleq 3322	. . . . . . . 8 ⊢ (𝑋 = ∪ 𝐽 → (∀𝑥 ∈ 𝑋 (𝐹‘𝑤) ≤ (𝐹‘𝑥) ↔ ∀𝑥 ∈ ∪ 𝐽(𝐹‘𝑤) ≤ (𝐹‘𝑥)))
34	33	rexeqbi1dv 3336	. . . . . . 7 ⊢ (𝑋 = ∪ 𝐽 → (∃𝑤 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝐹‘𝑤) ≤ (𝐹‘𝑥) ↔ ∃𝑤 ∈ ∪ 𝐽∀𝑥 ∈ ∪ 𝐽(𝐹‘𝑤) ≤ (𝐹‘𝑥)))
35	32, 34	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (∃𝑤 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝐹‘𝑤) ≤ (𝐹‘𝑥) ↔ ∃𝑤 ∈ ∪ 𝐽∀𝑥 ∈ ∪ 𝐽(𝐹‘𝑤) ≤ (𝐹‘𝑥)))
36	31, 35	mpbird 260	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∃𝑤 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝐹‘𝑤) ≤ (𝐹‘𝑥))
37		ffn 6709	. . . . . 6 ⊢ (𝐹:𝑋⟶ℝ⁺ → 𝐹 Fn 𝑋)
38		breq1 5114	. . . . . . . 8 ⊢ (𝑟 = (𝐹‘𝑤) → (𝑟 ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑤) ≤ (𝐹‘𝑥)))
39	38	ralbidv 3190	. . . . . . 7 ⊢ (𝑟 = (𝐹‘𝑤) → (∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥) ↔ ∀𝑥 ∈ 𝑋 (𝐹‘𝑤) ≤ (𝐹‘𝑥)))
40	39	rexrn 7086	. . . . . 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 19	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (∃𝑟 ∈ ran 𝐹∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥) ↔ ∃𝑤 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝐹‘𝑤) ≤ (𝐹‘𝑥)))
42	36, 41	mpbird 260	. . . 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 66	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∃𝑟 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥))
45		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ⁺)
46	14	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑋 = ∪ 𝑈)
47		simplr 781	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑋 ≠ ∅)
48	46, 47	eqnetrrd 3028	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → ∪ 𝑈 ≠ ∅)
49		unieq 4885	. . . . . . . . . . 11 ⊢ (𝑈 = ∅ → ∪ 𝑈 = ∪ ∅)
50		uni0 4903	. . . . . . . . . . 11 ⊢ ∪ ∅ = ∅
51	49, 50	eqtrdi 2816	. . . . . . . . . 10 ⊢ (𝑈 = ∅ → ∪ 𝑈 = ∅)
52	51	necon3i 2992	. . . . . . . . 9 ⊢ (∪ 𝑈 ≠ ∅ → 𝑈 ≠ ∅)
53	48, 52	syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑈 ≠ ∅)
54	15	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑈 ∈ Fin)
55		hashnncl 14424	. . . . . . . . 9 ⊢ (𝑈 ∈ Fin → ((♯‘𝑈) ∈ ℕ ↔ 𝑈 ≠ ∅))
56	54, 55	syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → ((♯‘𝑈) ∈ ℕ ↔ 𝑈 ≠ ∅))
57	53, 56	mpbird 260	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → (♯‘𝑈) ∈ ℕ)
58	57	nnrpd 13078	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → (♯‘𝑈) ∈ ℝ⁺)
59	45, 58	rpdivcld 13097	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → (𝑟 / (♯‘𝑈)) ∈ ℝ⁺)
60		ralnex 3093	. . . . . . . 8 ⊢ (∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢 ↔ ¬ ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)
61	54	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑈 ∈ Fin)
62	53	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑈 ≠ ∅)
63		simprl 783	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑥 ∈ 𝑋)
64	63	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝑥 ∈ 𝑋)
65		eqid 2765	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < )) = (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ))
66	65	metdsval 25060	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 → ((𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ))‘𝑥) = inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)), ℝ^, < ))
67	64, 66	syl 18	. . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝐷 ∈ (Met‘𝑋))
69	68	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝐷 ∈ (Met‘𝑋))
70		difssd 4091	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ⊆ 𝑋)
71		elssuni 4906	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝑈 → 𝑘 ⊆ ∪ 𝑈)
72	71	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝑘 ⊆ ∪ 𝑈)
73	46	ad2antrr 739	. . . . . . . . . . . . . . . . 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 2853	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑋 → (𝑘 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
76	75	notbid 321	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑋 → (¬ 𝑘 ∈ 𝑈 ↔ ¬ 𝑋 ∈ 𝑈))
77	16, 76	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑘 = 𝑋 → ¬ 𝑘 ∈ 𝑈))
78	77	necon2ad 2975	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋))
79	78	ad3antrrr 743	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋))
80	79	imp 412	. . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ≠ ∅)
83	65	metdsre 25066	. . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . 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 7084	. . . . . . . . . . . . 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 2866	. . . . . . . . . . . 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 739	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑟 / (♯‘𝑈)) ∈ ℝ⁺)
88	87	rpred 13080	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑟 / (♯‘𝑈)) ∈ ℝ)
89		simprr 785	. . . . . . . . . . . . . . . 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 321	. . . . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝐷 ∈ (∞Met‘𝑋))
95	87	rpxrd 13081	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑟 / (♯‘𝑈)) ∈ ℝ^*)
96	65	metdsge 25062	. . . . . . . . . . . . . . . . 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 1400	. . . . . . . . . . . . . . . 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 24630	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (𝑟 / (♯‘𝑈)) ∈ ℝ^*) → (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑋)
99	94, 64, 95, 98	syl3anc 1398	. . . . . . . . . . . . . . . . 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 32	. . . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . . 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 201	. . . . . . . . . . . . . 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 11376	. . . . . . . . . . . . . 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 260	. . . . . . . . . . . . 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 5137	. . . . . . . . . . . 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 15879	. . . . . . . . . . 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 7426	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → (𝑦𝐷𝑧) = (𝑥𝐷𝑧))
109	108	mpteq2dv 5207	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)) = (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)))
110	109	rneqd 5930	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)) = ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)))
111	110	infeq1d 9445	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ) = inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)), ℝ^, < ))
112	111	sumeq2sdv 15782	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ) = Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)), ℝ^, < ))
113		sumex 15767	. . . . . . . . . . . . 13 ⊢ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)), ℝ^*, < ) ∈ V
114	112, 17, 113	fvmpt 6993	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 → (𝐹‘𝑥) = Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)), ℝ^*, < ))
115	63, 114	syl 18	. . . . . . . . . . 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 486	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝑟 / (♯‘𝑈)) ∈ ℝ⁺)
117	116	rpcnd 13082	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝑟 / (♯‘𝑈)) ∈ ℂ)
118		fsumconst 15868	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ Fin ∧ (𝑟 / (♯‘𝑈)) ∈ ℂ) → Σ𝑘 ∈ 𝑈 (𝑟 / (♯‘𝑈)) = ((♯‘𝑈) · (𝑟 / (♯‘𝑈))))
119	61, 117, 118	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → Σ𝑘 ∈ 𝑈 (𝑟 / (♯‘𝑈)) = ((♯‘𝑈) · (𝑟 / (♯‘𝑈))))
120		simplr 781	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑟 ∈ ℝ⁺)
121	120	rpcnd 13082	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑟 ∈ ℂ)
122	57	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (♯‘𝑈) ∈ ℕ)
123	122	nncnd 12268	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (♯‘𝑈) ∈ ℂ)
124	122	nnne0d 12305	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (♯‘𝑈) ≠ 0)
125	121, 123, 124	divcan2d 12012	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → ((♯‘𝑈) · (𝑟 / (♯‘𝑈))) = 𝑟)
126	119, 125	eqtr2d 2801	. . . . . . . . . . 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 5145	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝐹‘𝑥) < 𝑟)
128	19	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝐹:𝑋⟶ℝ⁺)
129	128, 63	ffvelcdmd 7084	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝐹‘𝑥) ∈ ℝ⁺)
130	129	rpred 13080	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝐹‘𝑥) ∈ ℝ)
131	120	rpred 13080	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑟 ∈ ℝ)
132	130, 131	ltnled 11376	. . . . . . . . . 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 235	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → ¬ 𝑟 ≤ (𝐹‘𝑥))
134	133	expr 462	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝑋) → (∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢 → ¬ 𝑟 ≤ (𝐹‘𝑥)))
135	60, 134	biimtrrid 246	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝑋) → (¬ ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢 → ¬ 𝑟 ≤ (𝐹‘𝑥)))
136	135	con4d 116	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝑋) → (𝑟 ≤ (𝐹‘𝑥) → ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢))
137	136	ralimdva 3179	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → (∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥) → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢))
138		oveq2 7427	. . . . . . . . 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 3191	. . . . . . 7 ⊢ (𝑑 = (𝑟 / (♯‘𝑈)) → (∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢 ↔ ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢))
141	140	ralbidv 3190	. . . . . 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 697	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → (∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥) → ∃𝑑 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢))
144	143	rexlimdva 3168	. . 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 16	. 2 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∃𝑑 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
146	9, 145	pm2.61dane 3047	1 ⊢ (𝜑 → ∃𝑑 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∀wral 3081 ∃wrex 3091 ∖ cdif 3903 ∩ cin 3905 ⊆ wss 3906 ∅c0 4286 ∪ cuni 4874 class class class wbr 5111 ↦ cmpt 5194 ran crn 5664 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 Fincfn 8949 infcinf 9408 ℂcc 11117 ℝcr 11118 1c1 11120 · cmul 11124 ℝ^*cxr 11261 < clt 11262 ≤ cle 11263 / cdiv 11890 ℕcn 12252 ℝ⁺crp 13036 (,)cioo 13392 ♯chash 14388 Σcsu 15765 topGenctg 17516 ∞Metcxmet 21561 Metcmet 21562 ballcbl 21563 MetOpencmopn 21566 Cn ccn 23435 Compccmp 23597

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11175 ax-resscn 11176 ax-1cn 11177 ax-icn 11178 ax-addcl 11179 ax-addrcl 11180 ax-mulcl 11181 ax-mulrcl 11182 ax-mulcom 11183 ax-addass 11184 ax-mulass 11185 ax-distr 11186 ax-i2m1 11187 ax-1ne0 11188 ax-1rid 11189 ax-rnegex 11190 ax-rrecex 11191 ax-cnre 11192 ax-pre-lttri 11193 ax-pre-lttrn 11194 ax-pre-ltadd 11195 ax-pre-mulgt0 11196 ax-pre-sup 11197 ax-addf 11198

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-ec 8702 df-map 8832 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-fi 9378 df-sup 9409 df-inf 9410 df-oi 9479 df-card 9941 df-pnf 11264 df-mnf 11265 df-xr 11266 df-ltxr 11267 df-le 11268 df-sub 11462 df-neg 11463 df-div 11891 df-nn 12253 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-7 12327 df-8 12328 df-9 12329 df-n0 12524 df-z 12611 df-dec 12732 df-uz 12883 df-q 12993 df-rp 13037 df-xneg 13157 df-xadd 13158 df-xmul 13159 df-ioo 13396 df-ico 13398 df-icc 13399 df-fz 13556 df-fzo 13704 df-seq 14060 df-exp 14120 df-hash 14389 df-cj 15178 df-re 15179 df-im 15180 df-sqrt 15314 df-abs 15315 df-clim 15567 df-sum 15766 df-struct 17233 df-sets 17250 df-slot 17268 df-ndx 17280 df-base 17296 df-ress 17317 df-plusg 17349 df-mulr 17350 df-starv 17351 df-sca 17352 df-vsca 17353 df-ip 17354 df-tset 17355 df-ple 17356 df-ds 17358 df-unif 17359 df-hom 17360 df-cco 17361 df-rest 17501 df-topn 17502 df-0g 17520 df-gsum 17521 df-topgen 17522 df-pt 17523 df-prds 17526 df-xrs 17582 df-qtop 17587 df-imas 17588 df-xps 17590 df-mre 17664 df-mrc 17665 df-acs 17667 df-mgm 18724 df-sgrp 18813 df-mnd 18829 df-submnd 18883 df-mulg 19182 df-cntz 19435 df-cmn 19900 df-psmet 21568 df-xmet 21569 df-met 21570 df-bl 21571 df-mopn 21572 df-cnfld 21577 df-top 23105 df-topon 23122 df-topsp 23144 df-bases 23157 df-cld 23230 df-ntr 23231 df-cls 23232 df-cn 23438 df-cnp 23439 df-cmp 23598 df-tx 23774 df-hmeo 23967 df-xms 24532 df-ms 24533 df-tms 24534

This theorem is used by: lebnum 25178

W3C validator