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Theorem lebnumlem3 24949

Description: Lemma for lebnum 24950. 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 12938	. . . 4 ⊢ 1 ∈ ℝ⁺
2	1	ne0ii 4273	. . 3 ⊢ ℝ⁺ ≠ ∅
3		ral0 4427	. . . . 5 ⊢ ∀𝑥 ∈ ∅ ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢
4		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑋 = ∅)
5	4	raleqdv 3297	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢 ↔ ∀𝑥 ∈ ∅ ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢))
6	3, 5	mpbiri 259	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
7	6	ralrimivw 3135	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∀𝑑 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
8		r19.2z 4428	. . 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 593	. 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 24947	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ⁺)
19	18	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐹:𝑋⟶ℝ⁺)
20	19	frnd 6664	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ran 𝐹 ⊆ ℝ⁺)
21		eqid 2739	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
22		lebnumlem2.k	. . . . . . 7 ⊢ 𝐾 = (topGen‘ran (,))
23	12	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐽 ∈ Comp)
24	10, 11, 12, 13, 14, 15, 16, 17, 22	lebnumlem2 24948	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
25	24	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐹 ∈ (𝐽 Cn 𝐾))
26		metxmet 24318	. . . . . . . . . 10 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
27	10	mopnuni 24425	. . . . . . . . . 10 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
28	11, 26, 27	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
29	28	neeq1d 2993	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ≠ ∅ ↔ ∪ 𝐽 ≠ ∅))
30	29	biimpa 477	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∪ 𝐽 ≠ ∅)
31	21, 22, 23, 25, 30	evth2 24946	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∃𝑤 ∈ ∪ 𝐽∀𝑥 ∈ ∪ 𝐽(𝐹‘𝑤) ≤ (𝐹‘𝑥))
32	28	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 = ∪ 𝐽)
33		raleq 3294	. . . . . . . 8 ⊢ (𝑋 = ∪ 𝐽 → (∀𝑥 ∈ 𝑋 (𝐹‘𝑤) ≤ (𝐹‘𝑥) ↔ ∀𝑥 ∈ ∪ 𝐽(𝐹‘𝑤) ≤ (𝐹‘𝑥)))
34	33	rexeqbi1dv 3308	. . . . . . 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 258	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∃𝑤 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝐹‘𝑤) ≤ (𝐹‘𝑥))
37		ffn 6656	. . . . . 6 ⊢ (𝐹:𝑋⟶ℝ⁺ → 𝐹 Fn 𝑋)
38		breq1 5076	. . . . . . . 8 ⊢ (𝑟 = (𝐹‘𝑤) → (𝑟 ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑤) ≤ (𝐹‘𝑥)))
39	38	ralbidv 3162	. . . . . . 7 ⊢ (𝑟 = (𝐹‘𝑤) → (∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥) ↔ ∀𝑥 ∈ 𝑋 (𝐹‘𝑤) ≤ (𝐹‘𝑥)))
40	39	rexrn 7029	. . . . . 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 258	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∃𝑟 ∈ ran 𝐹∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥))
43		ssrexv 3985	. . . 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 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ⁺)
46	14	ad2antrr 732	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑋 = ∪ 𝑈)
47		simplr 774	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑋 ≠ ∅)
48	46, 47	eqnetrrd 3002	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → ∪ 𝑈 ≠ ∅)
49		unieq 4850	. . . . . . . . . . 11 ⊢ (𝑈 = ∅ → ∪ 𝑈 = ∪ ∅)
50		uni0 4867	. . . . . . . . . . 11 ⊢ ∪ ∅ = ∅
51	49, 50	eqtrdi 2790	. . . . . . . . . 10 ⊢ (𝑈 = ∅ → ∪ 𝑈 = ∅)
52	51	necon3i 2966	. . . . . . . . 9 ⊢ (∪ 𝑈 ≠ ∅ → 𝑈 ≠ ∅)
53	48, 52	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑈 ≠ ∅)
54	15	ad2antrr 732	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝑈 ∈ Fin)
55		hashnncl 14320	. . . . . . . . 9 ⊢ (𝑈 ∈ Fin → ((♯‘𝑈) ∈ ℕ ↔ 𝑈 ≠ ∅))
56	54, 55	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → ((♯‘𝑈) ∈ ℕ ↔ 𝑈 ≠ ∅))
57	53, 56	mpbird 258	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → (♯‘𝑈) ∈ ℕ)
58	57	nnrpd 12976	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → (♯‘𝑈) ∈ ℝ⁺)
59	45, 58	rpdivcld 12995	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → (𝑟 / (♯‘𝑈)) ∈ ℝ⁺)
60		ralnex 3065	. . . . . . . 8 ⊢ (∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢 ↔ ¬ ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)
61	54	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑈 ∈ Fin)
62	53	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑈 ≠ ∅)
63		simprl 776	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑥 ∈ 𝑋)
64	63	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝑥 ∈ 𝑋)
65		eqid 2739	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < )) = (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ))
66	65	metdsval 24832	. . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → 𝐷 ∈ (Met‘𝑋))
69	68	ad2antrr 732	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝐷 ∈ (Met‘𝑋))
70		difssd 4068	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ⊆ 𝑋)
71		elssuni 4870	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝑈 → 𝑘 ⊆ ∪ 𝑈)
72	71	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝑘 ⊆ ∪ 𝑈)
73	46	ad2antrr 732	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝑋 = ∪ 𝑈)
74	72, 73	sseqtrrd 3952	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝑘 ⊆ 𝑋)
75		eleq1 2827	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑋 → (𝑘 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
76	75	notbid 319	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑋 → (¬ 𝑘 ∈ 𝑈 ↔ ¬ 𝑋 ∈ 𝑈))
77	16, 76	syl5ibrcom 248	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑘 = 𝑋 → ¬ 𝑘 ∈ 𝑈))
78	77	necon2ad 2949	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋))
79	78	ad3antrrr 736	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋))
80	79	imp 407	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → 𝑘 ≠ 𝑋)
81		pssdifn0 4297	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ⊆ 𝑋 ∧ 𝑘 ≠ 𝑋) → (𝑋 ∖ 𝑘) ≠ ∅)
82	74, 80, 81	syl2anc 590	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ≠ ∅)
83	65	metdsre 24838	. . . . . . . . . . . . . . 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 1379	. . . . . . . . . . . . . 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 7027	. . . . . . . . . . . . 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 2840	. . . . . . . . . . . 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 732	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑟 / (♯‘𝑈)) ∈ ℝ⁺)
88	87	rpred 12978	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑟 / (♯‘𝑈)) ∈ ℝ)
89		simprr 778	. . . . . . . . . . . . . . . 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 3941	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = 𝑘 → ((𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢 ↔ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑘))
91	90	notbid 319	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝑘 → (¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢 ↔ ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑘))
92	91	rspccva 3559	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢 ∧ 𝑘 ∈ 𝑈) → ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑘)
93	89, 92	sylan 586	. . . . . . . . . . . . . . 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 12979	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) ∧ 𝑘 ∈ 𝑈) → (𝑟 / (♯‘𝑈)) ∈ ℝ^*)
96	65	metdsge 24834	. . . . . . . . . . . . . . . . 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 1381	. . . . . . . . . . . . . . . 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 24402	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (𝑟 / (♯‘𝑈)) ∈ ℝ^*) → (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑋)
99	94, 64, 95, 98	syl3anc 1379	. . . . . . . . . . . . . . . . 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 4302	. . . . . . . . . . . . . . . . 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 241	. . . . . . . . . . . . . . 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 199	. . . . . . . . . . . . . 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 11285	. . . . . . . . . . . . . 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 258	. . . . . . . . . . . . 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 5097	. . . . . . . . . . . 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 15755	. . . . . . . . . . 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 7364	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → (𝑦𝐷𝑧) = (𝑥𝐷𝑧))
109	108	mpteq2dv 5167	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)) = (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)))
110	109	rneqd 5881	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)) = ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)))
111	110	infeq1d 9382	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ) = inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)), ℝ^, < ))
112	111	sumeq2sdv 15657	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ) = Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)), ℝ^, < ))
113		sumex 15642	. . . . . . . . . . . . 13 ⊢ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑥𝐷𝑧)), ℝ^*, < ) ∈ V
114	112, 17, 113	fvmpt 6936	. . . . . . . . . . . 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 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝑟 / (♯‘𝑈)) ∈ ℝ⁺)
117	116	rpcnd 12980	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝑟 / (♯‘𝑈)) ∈ ℂ)
118		fsumconst 15744	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ Fin ∧ (𝑟 / (♯‘𝑈)) ∈ ℂ) → Σ𝑘 ∈ 𝑈 (𝑟 / (♯‘𝑈)) = ((♯‘𝑈) · (𝑟 / (♯‘𝑈))))
119	61, 117, 118	syl2anc 590	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → Σ𝑘 ∈ 𝑈 (𝑟 / (♯‘𝑈)) = ((♯‘𝑈) · (𝑟 / (♯‘𝑈))))
120		simplr 774	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑟 ∈ ℝ⁺)
121	120	rpcnd 12980	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑟 ∈ ℂ)
122	57	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (♯‘𝑈) ∈ ℕ)
123	122	nncnd 12182	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (♯‘𝑈) ∈ ℂ)
124	122	nnne0d 12219	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (♯‘𝑈) ≠ 0)
125	121, 123, 124	divcan2d 11925	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → ((♯‘𝑈) · (𝑟 / (♯‘𝑈))) = 𝑟)
126	119, 125	eqtr2d 2775	. . . . . . . . . . 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 5105	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝐹‘𝑥) < 𝑟)
128	19	ad2antrr 732	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝐹:𝑋⟶ℝ⁺)
129	128, 63	ffvelcdmd 7027	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝐹‘𝑥) ∈ ℝ⁺)
130	129	rpred 12978	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → (𝐹‘𝑥) ∈ ℝ)
131	120	rpred 12978	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → 𝑟 ∈ ℝ)
132	130, 131	ltnled 11285	. . . . . . . . . 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 233	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢)) → ¬ 𝑟 ≤ (𝐹‘𝑥))
134	133	expr 457	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝑋) → (∀𝑢 ∈ 𝑈 ¬ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢 → ¬ 𝑟 ≤ (𝐹‘𝑥)))
135	60, 134	biimtrrid 244	. . . . . . 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 3151	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → (∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥) → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢))
138		oveq2 7365	. . . . . . . . 9 ⊢ (𝑑 = (𝑟 / (♯‘𝑈)) → (𝑥(ball‘𝐷)𝑑) = (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))))
139	138	sseq1d 3946	. . . . . . . 8 ⊢ (𝑑 = (𝑟 / (♯‘𝑈)) → ((𝑥(ball‘𝐷)𝑑) ⊆ 𝑢 ↔ (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢))
140	139	rexbidv 3163	. . . . . . 7 ⊢ (𝑑 = (𝑟 / (♯‘𝑈)) → (∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢 ↔ ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)(𝑟 / (♯‘𝑈))) ⊆ 𝑢))
141	140	ralbidv 3162	. . . . . 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 3560	. . . . 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 690	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ 𝑟 ∈ ℝ⁺) → (∀𝑥 ∈ 𝑋 𝑟 ≤ (𝐹‘𝑥) → ∃𝑑 ∈ ℝ⁺ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢))
144	143	rexlimdva 3140	. . 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 3021	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 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∀wral 3053 ∃wrex 3063 ∖ cdif 3880 ∩ cin 3882 ⊆ wss 3883 ∅c0 4262 ∪ cuni 4839 class class class wbr 5073 ↦ cmpt 5154 ran crn 5620 Fn wfn 6481 ⟶wf 6482 ‘cfv 6486 (class class class)co 7357 Fincfn 8884 infcinf 9345 ℂcc 11028 ℝcr 11029 1c1 11031 · cmul 11035 ℝ^*cxr 11170 < clt 11171 ≤ cle 11172 / cdiv 11799 ℕcn 12166 ℝ⁺crp 12934 (,)cioo 13290 ♯chash 14284 Σcsu 15640 topGenctg 17392 ∞Metcxmet 21333 Metcmet 21334 ballcbl 21335 MetOpencmopn 21338 Cn ccn 23208 Compccmp 23370

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-inf2 9554 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4840 df-int 4879 df-iun 4924 df-iin 4925 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 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 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7621 df-om 7808 df-1st 7932 df-2nd 7933 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-ec 8636 df-map 8766 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-fi 9315 df-sup 9346 df-inf 9347 df-oi 9416 df-card 9855 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-div 11800 df-nn 12167 df-2 12236 df-3 12237 df-4 12238 df-5 12239 df-6 12240 df-7 12241 df-8 12242 df-9 12243 df-n0 12430 df-z 12517 df-dec 12637 df-uz 12781 df-q 12891 df-rp 12935 df-xneg 13055 df-xadd 13056 df-xmul 13057 df-ioo 13294 df-ico 13296 df-icc 13297 df-fz 13454 df-fzo 13601 df-seq 13956 df-exp 14016 df-hash 14285 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15442 df-sum 15641 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17172 df-ress 17193 df-plusg 17225 df-mulr 17226 df-starv 17227 df-sca 17228 df-vsca 17229 df-ip 17230 df-tset 17231 df-ple 17232 df-ds 17234 df-unif 17235 df-hom 17236 df-cco 17237 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17458 df-qtop 17463 df-imas 17464 df-xps 17466 df-mre 17540 df-mrc 17541 df-acs 17543 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18744 df-mulg 19036 df-cntz 19284 df-cmn 19749 df-psmet 21340 df-xmet 21341 df-met 21342 df-bl 21343 df-mopn 21344 df-cnfld 21349 df-top 22878 df-topon 22895 df-topsp 22917 df-bases 22930 df-cld 23003 df-ntr 23004 df-cls 23005 df-cn 23211 df-cnp 23212 df-cmp 23371 df-tx 23546 df-hmeo 23739 df-xms 24304 df-ms 24305 df-tms 24306

This theorem is referenced by: lebnum 24950

W3C validator