Theorem lebnumlem1 25012

Description: Lemma for lebnum 25015. The function 𝐹 measures the sum of all of the distances to escape the sets of the cover. Since by assumption it is a cover, there is at least one set which covers a given point, and since it is open, the point is a positive distance from the edge of the set. Thus, the sum is a strictly positive number. (Contributed by Mario Carneiro, 14-Feb-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*, < ))

Assertion

Ref	Expression
lebnumlem1	⊢ (𝜑 → 𝐹:𝑋⟶ℝ+)

Distinct variable groups:   𝑦,𝑘,𝑧,𝐷   𝑘,𝐽,𝑦,𝑧   𝑈,𝑘,𝑦,𝑧   𝜑,𝑘,𝑦,𝑧   𝑘,𝑋,𝑦,𝑧

Allowed substitution hints:   𝐹(𝑦,𝑧,𝑘)

Proof of Theorem lebnumlem1

Dummy variables 𝑚 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lebnumlem1.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ Fin)
2	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑈 ∈ Fin)
3		lebnum.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
4	3	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝐷 ∈ (Met‘𝑋))
5		difssd 4160	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ⊆ 𝑋)
6		lebnum.s	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ⊆ 𝐽)
7	6	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑈 ⊆ 𝐽)
8	7	sselda 4008	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝑘 ∈ 𝐽)
9		elssuni 4961	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐽 → 𝑘 ⊆ ∪ 𝐽)
10	8, 9	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝑘 ⊆ ∪ 𝐽)
11		metxmet 24365	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
12	3, 11	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
13		lebnum.j	. . . . . . . . . . . 12 ⊢ 𝐽 = (MetOpen‘𝐷)
14	13	mopnuni 24472	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
15	12, 14	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
16	15	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝑋 = ∪ 𝐽)
17	10, 16	sseqtrrd 4050	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝑘 ⊆ 𝑋)
18		lebnumlem1.n	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)
19		eleq1 2832	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑋 → (𝑘 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
20	19	notbid 318	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑋 → (¬ 𝑘 ∈ 𝑈 ↔ ¬ 𝑋 ∈ 𝑈))
21	18, 20	syl5ibrcom 247	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 = 𝑋 → ¬ 𝑘 ∈ 𝑈))
22	21	necon2ad 2961	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋))
23	22	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋))
24	23	imp 406	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝑘 ≠ 𝑋)
25		pssdifn0 4391	. . . . . . . 8 ⊢ ((𝑘 ⊆ 𝑋 ∧ 𝑘 ≠ 𝑋) → (𝑋 ∖ 𝑘) ≠ ∅)
26	17, 24, 25	syl2anc 583	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ≠ ∅)
27		eqid 2740	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) = (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
28	27	metdsre 24894	. . . . . . 7 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝑋 ∖ 𝑘) ⊆ 𝑋 ∧ (𝑋 ∖ 𝑘) ≠ ∅) → (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶ℝ)
29	4, 5, 26, 28	syl3anc 1371	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶ℝ)
30	27	fmpt 7144	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑋 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ ↔ (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶ℝ)
31	29, 30	sylibr 234	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → ∀𝑦 ∈ 𝑋 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
32		simplr 768	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝑦 ∈ 𝑋)
33		rsp 3253	. . . . 5 ⊢ (∀𝑦 ∈ 𝑋 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ → (𝑦 ∈ 𝑋 → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ))
34	31, 32, 33	sylc 65	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
35	2, 34	fsumrecl 15782	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
36		lebnum.u	. . . . . . 7 ⊢ (𝜑 → 𝑋 = ∪ 𝑈)
37	36	eleq2d 2830	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝑋 ↔ 𝑦 ∈ ∪ 𝑈))
38	37	biimpa 476	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ ∪ 𝑈)
39		eluni2 4935	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑈 ↔ ∃𝑚 ∈ 𝑈 𝑦 ∈ 𝑚)
40	38, 39	sylib 218	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ∃𝑚 ∈ 𝑈 𝑦 ∈ 𝑚)
41		0red 11293	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 0 ∈ ℝ)
42		simplr 768	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑦 ∈ 𝑋)
43		eqid 2740	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )) = (𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))
44	43	metdsval 24888	. . . . . . 7 ⊢ (𝑦 ∈ 𝑋 → ((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) = inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
45	42, 44	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → ((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) = inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
46	3	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝐷 ∈ (Met‘𝑋))
47		difssd 4160	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑋 ∖ 𝑚) ⊆ 𝑋)
48	6	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑈 ⊆ 𝐽)
49		simprl 770	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑚 ∈ 𝑈)
50	48, 49	sseldd 4009	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑚 ∈ 𝐽)
51		elssuni 4961	. . . . . . . . . . 11 ⊢ (𝑚 ∈ 𝐽 → 𝑚 ⊆ ∪ 𝐽)
52	50, 51	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑚 ⊆ ∪ 𝐽)
53	46, 11, 14	3syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑋 = ∪ 𝐽)
54	52, 53	sseqtrrd 4050	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑚 ⊆ 𝑋)
55		eleq1 2832	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑋 → (𝑚 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
56	55	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑋 → (¬ 𝑚 ∈ 𝑈 ↔ ¬ 𝑋 ∈ 𝑈))
57	18, 56	syl5ibrcom 247	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑚 = 𝑋 → ¬ 𝑚 ∈ 𝑈))
58	57	necon2ad 2961	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑚 ∈ 𝑈 → 𝑚 ≠ 𝑋))
59	58	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑚 ∈ 𝑈 → 𝑚 ≠ 𝑋))
60	49, 59	mpd 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑚 ≠ 𝑋)
61		pssdifn0 4391	. . . . . . . . 9 ⊢ ((𝑚 ⊆ 𝑋 ∧ 𝑚 ≠ 𝑋) → (𝑋 ∖ 𝑚) ≠ ∅)
62	54, 60, 61	syl2anc 583	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑋 ∖ 𝑚) ≠ ∅)
63	43	metdsre 24894	. . . . . . . 8 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝑋 ∖ 𝑚) ⊆ 𝑋 ∧ (𝑋 ∖ 𝑚) ≠ ∅) → (𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )):𝑋⟶ℝ)
64	46, 47, 62, 63	syl3anc 1371	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )):𝑋⟶ℝ)
65	64, 42	ffvelcdmd 7119	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → ((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ∈ ℝ)
66	45, 65	eqeltrrd 2845	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
67	35	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
68	12	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝐷 ∈ (∞Met‘𝑋))
69	43	metdsf 24889	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑋 ∖ 𝑚) ⊆ 𝑋) → (𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
70	68, 47, 69	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
71	70, 42	ffvelcdmd 7119	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → ((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ∈ (0[,]+∞))
72		elxrge0 13517	. . . . . . . . 9 ⊢ (((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ∈ (0[,]+∞) ↔ (((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ∈ ℝ* ∧ 0 ≤ ((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦)))
73	71, 72	sylib 218	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ∈ ℝ* ∧ 0 ≤ ((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦)))
74	73	simprd 495	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 0 ≤ ((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦))
75		elndif 4156	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑚 → ¬ 𝑦 ∈ (𝑋 ∖ 𝑚))
76	75	ad2antll 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → ¬ 𝑦 ∈ (𝑋 ∖ 𝑚))
77	53	difeq1d 4148	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑋 ∖ 𝑚) = (∪ 𝐽 ∖ 𝑚))
78	13	mopntop 24471	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
79	68, 78	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝐽 ∈ Top)
80		eqid 2740	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
81	80	opncld 23062	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑚) ∈ (Clsd‘𝐽))
82	79, 50, 81	syl2anc 583	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (∪ 𝐽 ∖ 𝑚) ∈ (Clsd‘𝐽))
83	77, 82	eqeltrd 2844	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑋 ∖ 𝑚) ∈ (Clsd‘𝐽))
84		cldcls 23071	. . . . . . . . . 10 ⊢ ((𝑋 ∖ 𝑚) ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘(𝑋 ∖ 𝑚)) = (𝑋 ∖ 𝑚))
85	83, 84	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → ((cls‘𝐽)‘(𝑋 ∖ 𝑚)) = (𝑋 ∖ 𝑚))
86	76, 85	neleqtrrd 2867	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → ¬ 𝑦 ∈ ((cls‘𝐽)‘(𝑋 ∖ 𝑚)))
87	43, 13	metdseq0 24895	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑋 ∖ 𝑚) ⊆ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) = 0 ↔ 𝑦 ∈ ((cls‘𝐽)‘(𝑋 ∖ 𝑚))))
88	68, 47, 42, 87	syl3anc 1371	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) = 0 ↔ 𝑦 ∈ ((cls‘𝐽)‘(𝑋 ∖ 𝑚))))
89	88	necon3abid 2983	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ≠ 0 ↔ ¬ 𝑦 ∈ ((cls‘𝐽)‘(𝑋 ∖ 𝑚))))
90	86, 89	mpbird 257	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → ((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ≠ 0)
91	65, 74, 90	ne0gt0d 11427	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 0 < ((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦))
92	91, 45	breqtrd 5192	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 0 < inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
93	1	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑈 ∈ Fin)
94	34	adantlr 714	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) ∧ 𝑘 ∈ 𝑈) → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
95	12	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝐷 ∈ (∞Met‘𝑋))
96	27	metdsf 24889	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑋 ∖ 𝑘) ⊆ 𝑋) → (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
97	95, 5, 96	syl2anc 583	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
98	27	fmpt 7144	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝑋 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ (0[,]+∞) ↔ (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
99	97, 98	sylibr 234	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → ∀𝑦 ∈ 𝑋 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ (0[,]+∞))
100		rsp 3253	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑋 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ (0[,]+∞) → (𝑦 ∈ 𝑋 → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ (0[,]+∞)))
101	99, 32, 100	sylc 65	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ (0[,]+∞))
102		elxrge0 13517	. . . . . . . . 9 ⊢ (inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ (0[,]+∞) ↔ (inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ* ∧ 0 ≤ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )))
103	101, 102	sylib 218	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → (inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ* ∧ 0 ≤ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )))
104	103	simprd 495	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 0 ≤ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
105	104	adantlr 714	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) ∧ 𝑘 ∈ 𝑈) → 0 ≤ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
106		difeq2 4143	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (𝑋 ∖ 𝑘) = (𝑋 ∖ 𝑚))
107	106	mpteq1d 5261	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)) = (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑦𝐷𝑧)))
108	107	rneqd 5963	. . . . . . 7 ⊢ (𝑘 = 𝑚 → ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)) = ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑦𝐷𝑧)))
109	108	infeq1d 9546	. . . . . 6 ⊢ (𝑘 = 𝑚 → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) = inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
110	93, 94, 105, 109, 49	fsumge1 15845	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ≤ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
111	41, 66, 67, 92, 110	ltletrd 11450	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 0 < Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
112	40, 111	rexlimddv 3167	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 0 < Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
113	35, 112	elrpd 13096	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ+)
114		lebnumlem1.f	. 2 ⊢ 𝐹 = (𝑦 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
115	113, 114	fmptd 7148	1 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ+)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076   ∖ cdif 3973   ⊆ wss 3976  ∅c0 4352  ∪ cuni 4931   class class class wbr 5166   ↦ cmpt 5249  ran crn 5701  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  Fincfn 9003  infcinf 9510  ℝcr 11183  0cc0 11184  +∞cpnf 11321  ℝ^*cxr 11323   < clt 11324   ≤ cle 11325  ℝ⁺crp 13057  [,]cicc 13410  Σcsu 15734  ∞Metcxmet 21372  Metcmet 21373  MetOpencmopn 21377  Topctop 22920  Clsdccld 23045  clsccl 23047  Compccmp 23415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-ec 8765  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-q 13014  df-rp 13058  df-xneg 13175  df-xadd 13176  df-xmul 13177  df-ico 13413  df-icc 13414  df-fz 13568  df-fzo 13712  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-sum 15735  df-topgen 17503  df-psmet 21379  df-xmet 21380  df-met 21381  df-bl 21382  df-mopn 21383  df-top 22921  df-topon 22938  df-bases 22974  df-cld 23048  df-ntr 23049  df-cls 23050

This theorem is referenced by:  lebnumlem2  25013  lebnumlem3  25014