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Theorem lebnumlem3 24471

Description: Lemma for lebnum 24472. 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 12975	. . . 4 ⊢ 1 ∈ ℝ⁺
2	1	ne0ii 4337	. . 3 ⊢ ℝ⁺ ≠ ∅
3		ral0 4512	. . . . 5 ⊢ ∀𝑥 ∈ ∅ ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢
4		simpr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑋 = ∅)
5	4	raleqdv 3326	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢 ↔ ∀𝑥 ∈ ∅ ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢))
6	3, 5	mpbiri 258	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
7	6	ralrimivw 3151	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∀𝑑 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
8		r19.2z 4494	. . 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 588	. 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 24469	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ⁺)
19	18	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐹:𝑋⟶ℝ⁺)
20	19	frnd 6723	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ran 𝐹 ⊆ ℝ⁺)
21		eqid 2733	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
22		lebnumlem2.k	. . . . . . 7 ⊢ 𝐾 = (topGen‘ran (,))
23	12	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐽 ∈ Comp)
24	10, 11, 12, 13, 14, 15, 16, 17, 22	lebnumlem2 24470	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
25	24	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐹 ∈ (𝐽 Cn 𝐾))
26		metxmet 23832	. . . . . . . . . 10 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
27	10	mopnuni 23939	. . . . . . . . . 10 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
28	11, 26, 27	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
29	28	neeq1d 3001	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ≠ ∅ ↔ ∪ 𝐽 ≠ ∅))
30	29	biimpa 478	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∪ 𝐽 ≠ ∅)
31	21, 22, 23, 25, 30	evth2 24468	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∃𝑤 ∈ ∪ 𝐽∀𝑥 ∈ ∪ 𝐽(𝐹‘𝑤) ≤ (𝐹‘𝑥))
32	28	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 = ∪ 𝐽)
33		raleq 3323	. . . . . . . 8 ⊢ (𝑋 = ∪ 𝐽 → (∀𝑥 ∈ 𝑋 (𝐹‘𝑤) ≤ (𝐹‘𝑥) ↔ ∀𝑥 ∈ ∪ 𝐽(𝐹‘𝑤) ≤ (𝐹‘𝑥)))
34	33	rexeqbi1dv 3335	. . . . . . 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 257	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∃𝑤 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝐹‘𝑤) ≤ (𝐹‘𝑥))
37		ffn 6715	. . . . . 6 ⊢ (𝐹:𝑋⟶ℝ⁺ → 𝐹 Fn 𝑋)
38		breq1 5151	. . . . . . . 8 ⊢ (𝑟 = (𝐹‘𝑤) → (𝑟 ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑤) ≤ (𝐹‘𝑥)))
39	38	ralbidv 3178	. . . . . . 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 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (∃𝑟 ∈ ran 𝐹∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥) ↔ ∃𝑤 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝐹‘𝑤) ≤ (𝐹‘𝑥)))
42	36, 41	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∃𝑟 ∈ ran 𝐹∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥))
43		ssrexv 4051	. . . 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 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ⁺)
46	14	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑋 = ∪ 𝑈)
47		simplr 768	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑋 ≠ ∅)
48	46, 47	eqnetrrd 3010	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → ∪ 𝑈 ≠ ∅)
49		unieq 4919	. . . . . . . . . . 11 ⊢ (𝑈 = ∅ → ∪ 𝑈 = ∪ ∅)
50		uni0 4939	. . . . . . . . . . 11 ⊢ ∪ ∅ = ∅
51	49, 50	eqtrdi 2789	. . . . . . . . . 10 ⊢ (𝑈 = ∅ → ∪ 𝑈 = ∅)
52	51	necon3i 2974	. . . . . . . . 9 ⊢ (∪ 𝑈 ≠ ∅ → 𝑈 ≠ ∅)
53	48, 52	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑈 ≠ ∅)
54	15	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑈 ∈ Fin)
55		hashnncl 14323	. . . . . . . . 9 ⊢ (𝑈 ∈ Fin → ((♯‘𝑈) ∈ ℕ ↔ 𝑈 ≠ ∅))
56	54, 55	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → ((♯‘𝑈) ∈ ℕ ↔ 𝑈 ≠ ∅))
57	53, 56	mpbird 257	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → (♯‘𝑈) ∈ ℕ)
58	57	nnrpd 13011	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → (♯‘𝑈) ∈ ℝ⁺)
59	45, 58	rpdivcld 13030	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → (𝑟 / (♯‘𝑈)) ∈ ℝ⁺)
60		ralnex 3073	. . . . . . . 8 ⊢ (∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢 ↔ ¬ ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)
61	54	adantr 482	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑈 ∈ Fin)
62	53	adantr 482	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑈 ≠ ∅)
63		simprl 770	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑥 ∈ 𝑋)
64	63	adantr 482	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝑥 ∈ 𝑋)
65		eqid 2733	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < )) = (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ))
66	65	metdsval 24355	. . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝐷 ∈ (Met‘𝑋))
69	68	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝐷 ∈ (Met‘𝑋))
70		difssd 4132	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ⊆ 𝑋)
71		elssuni 4941	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝑈 → 𝑘 ⊆ ∪ 𝑈)
72	71	adantl 483	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝑘 ⊆ ∪ 𝑈)
73	46	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝑋 = ∪ 𝑈)
74	72, 73	sseqtrrd 4023	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝑘 ⊆ 𝑋)
75		eleq1 2822	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑋 → (𝑘 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
76	75	notbid 318	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑋 → (¬ 𝑘 ∈ 𝑈 ↔ ¬ 𝑋 ∈ 𝑈))
77	16, 76	syl5ibrcom 246	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑘 = 𝑋 → ¬ 𝑘 ∈ 𝑈))
78	77	necon2ad 2956	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋))
79	78	ad3antrrr 729	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋))
80	79	imp 408	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝑘 ≠ 𝑋)
81		pssdifn0 4365	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ⊆ 𝑋 ∧ 𝑘 ≠ 𝑋) → (𝑋 ∖ 𝑘) ≠ ∅)
82	74, 80, 81	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ≠ ∅)
83	65	metdsre 24361	. . . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . . 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 7085	. . . . . . . . . . . . 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 2835	. . . . . . . . . . . 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 725	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑟 / (♯‘𝑈)) ∈ ℝ⁺)
88	87	rpred 13013	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑟 / (♯‘𝑈)) ∈ ℝ)
89		simprr 772	. . . . . . . . . . . . . . . 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 4008	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = 𝑘 → ((𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢 ↔ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑘))
91	90	notbid 318	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝑘 → (¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢 ↔ ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑘))
92	91	rspccva 3612	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢 ∧ 𝑘 ∈ 𝑈) → ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑘)
93	89, 92	sylan 581	. . . . . . . . . . . . . . 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 13014	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑟 / (♯‘𝑈)) ∈ ℝ^*)
96	65	metdsge 24357	. . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . 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 23916	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (𝑟 / (♯‘𝑈)) ∈ ℝ^*) → (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑋)
99	94, 64, 95, 98	syl3anc 1372	. . . . . . . . . . . . . . . . 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 4368	. . . . . . . . . . . . . . . . 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 239	. . . . . . . . . . . . . . 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 197	. . . . . . . . . . . . . 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 11358	. . . . . . . . . . . . . 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 257	. . . . . . . . . . . . 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 5172	. . . . . . . . . . . 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 15743	. . . . . . . . . . 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 7413	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → (𝑦𝐷𝑧) = (𝑥𝐷𝑧))
109	108	mpteq2dv 5250	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)) = (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)))
110	109	rneqd 5936	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)) = ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)))
111	110	infeq1d 9469	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ) = inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)), ℝ^, < ))
112	111	sumeq2sdv 15647	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ) = Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)), ℝ^, < ))
113		sumex 15631	. . . . . . . . . . . . 13 ⊢ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)), ℝ^*, < ) ∈ V
114	112, 17, 113	fvmpt 6996	. . . . . . . . . . . 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 482	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝑟 / (♯‘𝑈)) ∈ ℝ⁺)
117	116	rpcnd 13015	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝑟 / (♯‘𝑈)) ∈ ℂ)
118		fsumconst 15733	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ Fin ∧ (𝑟 / (♯‘𝑈)) ∈ ℂ) → Σ𝑘 ∈ 𝑈 (𝑟 / (♯‘𝑈)) = ((♯‘𝑈) · (𝑟 / (♯‘𝑈))))
119	61, 117, 118	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → Σ𝑘 ∈ 𝑈 (𝑟 / (♯‘𝑈)) = ((♯‘𝑈) · (𝑟 / (♯‘𝑈))))
120		simplr 768	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑟 ∈ ℝ⁺)
121	120	rpcnd 13015	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑟 ∈ ℂ)
122	57	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (♯‘𝑈) ∈ ℕ)
123	122	nncnd 12225	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (♯‘𝑈) ∈ ℂ)
124	122	nnne0d 12259	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (♯‘𝑈) ≠ 0)
125	121, 123, 124	divcan2d 11989	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → ((♯‘𝑈) · (𝑟 / (♯‘𝑈))) = 𝑟)
126	119, 125	eqtr2d 2774	. . . . . . . . . . 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 5180	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝐹‘𝑥) < 𝑟)
128	19	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝐹:𝑋⟶ℝ⁺)
129	128, 63	ffvelcdmd 7085	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝐹‘𝑥) ∈ ℝ⁺)
130	129	rpred 13013	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝐹‘𝑥) ∈ ℝ)
131	120	rpred 13013	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑟 ∈ ℝ)
132	130, 131	ltnled 11358	. . . . . . . . . 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 231	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → ¬ 𝑟 ≤ (𝐹‘𝑥))
134	133	expr 458	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝑋) → (∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢 → ¬ 𝑟 ≤ (𝐹‘𝑥)))
135	60, 134	biimtrrid 242	. . . . . . 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 3168	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → (∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥) → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢))
138		oveq2 7414	. . . . . . . . 9 ⊢ (𝑑 = (𝑟 / (♯‘𝑈)) → (𝑥(ball‘𝐷)𝑑) = (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))))
139	138	sseq1d 4013	. . . . . . . 8 ⊢ (𝑑 = (𝑟 / (♯‘𝑈)) → ((𝑥(ball‘𝐷)𝑑) ⊆ 𝑢 ↔ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢))
140	139	rexbidv 3179	. . . . . . 7 ⊢ (𝑑 = (𝑟 / (♯‘𝑈)) → (∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢 ↔ ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢))
141	140	ralbidv 3178	. . . . . 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 3613	. . . . 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 683	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → (∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥) → ∃𝑑 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢))
144	143	rexlimdva 3156	. . 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 3030	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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 ∖ cdif 3945 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 ∪ cuni 4908 class class class wbr 5148 ↦ cmpt 5231 ran crn 5677 Fn wfn 6536 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 Fincfn 8936 infcinf 9433 ℂcc 11105 ℝcr 11106 1c1 11108 · cmul 11112 ℝ^*cxr 11244 < clt 11245 ≤ cle 11246 / cdiv 11868 ℕcn 12209 ℝ⁺crp 12971 (,)cioo 13321 ♯chash 14287 Σcsu 15629 topGenctg 17380 ∞Metcxmet 20922 Metcmet 20923 ballcbl 20924 MetOpencmopn 20927 Cn ccn 22720 Compccmp 22882

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-supp 8144 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-2o 8464 df-er 8700 df-ec 8702 df-map 8819 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-seq 13964 df-exp 14025 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-clim 15429 df-sum 15630 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-starv 17209 df-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ds 17216 df-unif 17217 df-hom 17218 df-cco 17219 df-rest 17365 df-topn 17366 df-0g 17384 df-gsum 17385 df-topgen 17386 df-pt 17387 df-prds 17390 df-xrs 17445 df-qtop 17450 df-imas 17451 df-xps 17453 df-mre 17527 df-mrc 17528 df-acs 17530 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-submnd 18669 df-mulg 18946 df-cntz 19176 df-cmn 19645 df-psmet 20929 df-xmet 20930 df-met 20931 df-bl 20932 df-mopn 20933 df-cnfld 20938 df-top 22388 df-topon 22405 df-topsp 22427 df-bases 22441 df-cld 22515 df-ntr 22516 df-cls 22517 df-cn 22723 df-cnp 22724 df-cmp 22883 df-tx 23058 df-hmeo 23251 df-xms 23818 df-ms 23819 df-tms 23820

This theorem is referenced by: lebnum 24472

W3C validator