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Theorem lebnumlem3 24860

Description: Lemma for lebnum 24861. 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 12897	. . . 4 ⊢ 1 ∈ ℝ⁺
2	1	ne0ii 4295	. . 3 ⊢ ℝ⁺ ≠ ∅
3		ral0 4464	. . . . 5 ⊢ ∀𝑥 ∈ ∅ ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢
4		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑋 = ∅)
5	4	raleqdv 3289	. . . . 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 3125	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∀𝑑 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
8		r19.2z 4446	. . 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 587	. 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 24858	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ⁺)
19	18	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐹:𝑋⟶ℝ⁺)
20	19	frnd 6660	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ran 𝐹 ⊆ ℝ⁺)
21		eqid 2729	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
22		lebnumlem2.k	. . . . . . 7 ⊢ 𝐾 = (topGen‘ran (,))
23	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐽 ∈ Comp)
24	10, 11, 12, 13, 14, 15, 16, 17, 22	lebnumlem2 24859	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
25	24	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐹 ∈ (𝐽 Cn 𝐾))
26		metxmet 24220	. . . . . . . . . 10 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
27	10	mopnuni 24327	. . . . . . . . . 10 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
28	11, 26, 27	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
29	28	neeq1d 2984	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ≠ ∅ ↔ ∪ 𝐽 ≠ ∅))
30	29	biimpa 476	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∪ 𝐽 ≠ ∅)
31	21, 22, 23, 25, 30	evth2 24857	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∃𝑤 ∈ ∪ 𝐽∀𝑥 ∈ ∪ 𝐽(𝐹‘𝑤) ≤ (𝐹‘𝑥))
32	28	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 = ∪ 𝐽)
33		raleq 3286	. . . . . . . 8 ⊢ (𝑋 = ∪ 𝐽 → (∀𝑥 ∈ 𝑋 (𝐹‘𝑤) ≤ (𝐹‘𝑥) ↔ ∀𝑥 ∈ ∪ 𝐽(𝐹‘𝑤) ≤ (𝐹‘𝑥)))
34	33	rexeqbi1dv 3302	. . . . . . 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 6652	. . . . . 6 ⊢ (𝐹:𝑋⟶ℝ⁺ → 𝐹 Fn 𝑋)
38		breq1 5095	. . . . . . . 8 ⊢ (𝑟 = (𝐹‘𝑤) → (𝑟 ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑤) ≤ (𝐹‘𝑥)))
39	38	ralbidv 3152	. . . . . . 7 ⊢ (𝑟 = (𝐹‘𝑤) → (∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥) ↔ ∀𝑥 ∈ 𝑋 (𝐹‘𝑤) ≤ (𝐹‘𝑥)))
40	39	rexrn 7021	. . . . . 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 4005	. . . 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 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ⁺)
46	14	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑋 = ∪ 𝑈)
47		simplr 768	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑋 ≠ ∅)
48	46, 47	eqnetrrd 2993	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → ∪ 𝑈 ≠ ∅)
49		unieq 4869	. . . . . . . . . . 11 ⊢ (𝑈 = ∅ → ∪ 𝑈 = ∪ ∅)
50		uni0 4886	. . . . . . . . . . 11 ⊢ ∪ ∅ = ∅
51	49, 50	eqtrdi 2780	. . . . . . . . . 10 ⊢ (𝑈 = ∅ → ∪ 𝑈 = ∅)
52	51	necon3i 2957	. . . . . . . . 9 ⊢ (∪ 𝑈 ≠ ∅ → 𝑈 ≠ ∅)
53	48, 52	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑈 ≠ ∅)
54	15	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑈 ∈ Fin)
55		hashnncl 14273	. . . . . . . . 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 12935	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → (♯‘𝑈) ∈ ℝ⁺)
59	45, 58	rpdivcld 12954	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → (𝑟 / (♯‘𝑈)) ∈ ℝ⁺)
60		ralnex 3055	. . . . . . . 8 ⊢ (∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢 ↔ ¬ ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)
61	54	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑈 ∈ Fin)
62	53	adantr 480	. . . . . . . . . . . 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 480	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝑥 ∈ 𝑋)
65		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < )) = (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ))
66	65	metdsval 24734	. . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝐷 ∈ (Met‘𝑋))
69	68	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝐷 ∈ (Met‘𝑋))
70		difssd 4088	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ⊆ 𝑋)
71		elssuni 4888	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝑈 → 𝑘 ⊆ ∪ 𝑈)
72	71	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝑘 ⊆ ∪ 𝑈)
73	46	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝑋 = ∪ 𝑈)
74	72, 73	sseqtrrd 3973	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝑘 ⊆ 𝑋)
75		eleq1 2816	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑋 → (𝑘 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
76	75	notbid 318	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑋 → (¬ 𝑘 ∈ 𝑈 ↔ ¬ 𝑋 ∈ 𝑈))
77	16, 76	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑘 = 𝑋 → ¬ 𝑘 ∈ 𝑈))
78	77	necon2ad 2940	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋))
79	78	ad3antrrr 730	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋))
80	79	imp 406	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝑘 ≠ 𝑋)
81		pssdifn0 4319	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ⊆ 𝑋 ∧ 𝑘 ≠ 𝑋) → (𝑋 ∖ 𝑘) ≠ ∅)
82	74, 80, 81	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ≠ ∅)
83	65	metdsre 24740	. . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . 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 7019	. . . . . . . . . . . . 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 2829	. . . . . . . . . . . 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 726	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑟 / (♯‘𝑈)) ∈ ℝ⁺)
88	87	rpred 12937	. . . . . . . . . . . 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 3962	. . . . . . . . . . . . . . . . . 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 3576	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢 ∧ 𝑘 ∈ 𝑈) → ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑘)
93	89, 92	sylan 580	. . . . . . . . . . . . . . 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 12938	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑟 / (♯‘𝑈)) ∈ ℝ^*)
96	65	metdsge 24736	. . . . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . . . . 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 24304	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (𝑟 / (♯‘𝑈)) ∈ ℝ^*) → (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑋)
99	94, 64, 95, 98	syl3anc 1373	. . . . . . . . . . . . . . . . 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 4324	. . . . . . . . . . . . . . . . 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 240	. . . . . . . . . . . . . . 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 198	. . . . . . . . . . . . . 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 11263	. . . . . . . . . . . . . 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 5116	. . . . . . . . . . . 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 15707	. . . . . . . . . . 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 7356	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → (𝑦𝐷𝑧) = (𝑥𝐷𝑧))
109	108	mpteq2dv 5186	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)) = (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)))
110	109	rneqd 5880	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)) = ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)))
111	110	infeq1d 9368	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ) = inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)), ℝ^, < ))
112	111	sumeq2sdv 15610	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ) = Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)), ℝ^, < ))
113		sumex 15595	. . . . . . . . . . . . 13 ⊢ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)), ℝ^*, < ) ∈ V
114	112, 17, 113	fvmpt 6930	. . . . . . . . . . . 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 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝑟 / (♯‘𝑈)) ∈ ℝ⁺)
117	116	rpcnd 12939	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝑟 / (♯‘𝑈)) ∈ ℂ)
118		fsumconst 15697	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ Fin ∧ (𝑟 / (♯‘𝑈)) ∈ ℂ) → Σ𝑘 ∈ 𝑈 (𝑟 / (♯‘𝑈)) = ((♯‘𝑈) · (𝑟 / (♯‘𝑈))))
119	61, 117, 118	syl2anc 584	. . . . . . . . . . . 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 12939	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑟 ∈ ℂ)
122	57	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (♯‘𝑈) ∈ ℕ)
123	122	nncnd 12144	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (♯‘𝑈) ∈ ℂ)
124	122	nnne0d 12178	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (♯‘𝑈) ≠ 0)
125	121, 123, 124	divcan2d 11902	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → ((♯‘𝑈) · (𝑟 / (♯‘𝑈))) = 𝑟)
126	119, 125	eqtr2d 2765	. . . . . . . . . . 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 5124	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝐹‘𝑥) < 𝑟)
128	19	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝐹:𝑋⟶ℝ⁺)
129	128, 63	ffvelcdmd 7019	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝐹‘𝑥) ∈ ℝ⁺)
130	129	rpred 12937	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝐹‘𝑥) ∈ ℝ)
131	120	rpred 12937	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑟 ∈ ℝ)
132	130, 131	ltnled 11263	. . . . . . . . . 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 232	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → ¬ 𝑟 ≤ (𝐹‘𝑥))
134	133	expr 456	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝑋) → (∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢 → ¬ 𝑟 ≤ (𝐹‘𝑥)))
135	60, 134	biimtrrid 243	. . . . . . 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 3141	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → (∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥) → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢))
138		oveq2 7357	. . . . . . . . 9 ⊢ (𝑑 = (𝑟 / (♯‘𝑈)) → (𝑥(ball‘𝐷)𝑑) = (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))))
139	138	sseq1d 3967	. . . . . . . 8 ⊢ (𝑑 = (𝑟 / (♯‘𝑈)) → ((𝑥(ball‘𝐷)𝑑) ⊆ 𝑢 ↔ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢))
140	139	rexbidv 3153	. . . . . . 7 ⊢ (𝑑 = (𝑟 / (♯‘𝑈)) → (∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢 ↔ ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢))
141	140	ralbidv 3152	. . . . . 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 3577	. . . . 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 684	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → (∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥) → ∃𝑑 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢))
144	143	rexlimdva 3130	. . 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 3012	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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 ∖ cdif 3900 ∩ cin 3902 ⊆ wss 3903 ∅c0 4284 ∪ cuni 4858 class class class wbr 5092 ↦ cmpt 5173 ran crn 5620 Fn wfn 6477 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 Fincfn 8872 infcinf 9331 ℂcc 11007 ℝcr 11008 1c1 11010 · cmul 11014 ℝ^*cxr 11148 < clt 11149 ≤ cle 11150 / cdiv 11777 ℕcn 12128 ℝ⁺crp 12893 (,)cioo 13248 ♯chash 14237 Σcsu 15593 topGenctg 17341 ∞Metcxmet 21246 Metcmet 21247 ballcbl 21248 MetOpencmopn 21251 Cn ccn 23109 Compccmp 23271

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-inf2 9537 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 ax-addf 11088

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-tp 4582 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-se 5573 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-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-ec 8627 df-map 8755 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-card 9835 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-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-sum 15594 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-cn 23112 df-cnp 23113 df-cmp 23272 df-tx 23447 df-hmeo 23640 df-xms 24206 df-ms 24207 df-tms 24208

This theorem is referenced by: lebnum 24861

W3C validator