Theorem lebnumlem1 25089

Description: Lemma for lebnum 25092. 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 485	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑈 ∈ Fin)
3		lebnum.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
4	3	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝐷 ∈ (Met‘𝑋))
5		difssd 4097	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ⊆ 𝑋)
6		lebnum.s	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ⊆ 𝐽)
7	6	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑈 ⊆ 𝐽)
8	7	sselda 3943	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝑘 ∈ 𝐽)
9		elssuni 4906	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐽 → 𝑘 ⊆ ∪ 𝐽)
10	8, 9	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝑘 ⊆ ∪ 𝐽)
11		metxmet 24460	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
12	3, 11	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
13		lebnum.j	. . . . . . . . . . . 12 ⊢ 𝐽 = (MetOpen‘𝐷)
14	13	mopnuni 24567	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
15	12, 14	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
16	15	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝑋 = ∪ 𝐽)
17	10, 16	sseqtrrd 3980	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝑘 ⊆ 𝑋)
18		lebnumlem1.n	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)
19		eleq1 2857	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑋 → (𝑘 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
20	19	notbid 321	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑋 → (¬ 𝑘 ∈ 𝑈 ↔ ¬ 𝑋 ∈ 𝑈))
21	18, 20	syl5ibrcom 250	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 = 𝑋 → ¬ 𝑘 ∈ 𝑈))
22	21	necon2ad 2979	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋))
23	22	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋))
24	23	imp 411	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝑘 ≠ 𝑋)
25		pssdifn0 4329	. . . . . . . 8 ⊢ ((𝑘 ⊆ 𝑋 ∧ 𝑘 ≠ 𝑋) → (𝑋 ∖ 𝑘) ≠ ∅)
26	17, 24, 25	syl2anc 595	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ≠ ∅)
27		eqid 2769	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) = (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
28	27	metdsre 24980	. . . . . . 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 1396	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶ℝ)
30	27	fmpt 7106	. . . . . 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 237	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → ∀𝑦 ∈ 𝑋 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
32		simplr 780	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝑦 ∈ 𝑋)
33		rsp 3259	. . . . 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 66	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
35	2, 34	fsumrecl 15785	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
36		lebnum.u	. . . . . . 7 ⊢ (𝜑 → 𝑋 = ∪ 𝑈)
37	36	eleq2d 2855	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝑋 ↔ 𝑦 ∈ ∪ 𝑈))
38	37	biimpa 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ ∪ 𝑈)
39		eluni2 4878	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑈 ↔ ∃𝑚 ∈ 𝑈 𝑦 ∈ 𝑚)
40	38, 39	sylib 221	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ∃𝑚 ∈ 𝑈 𝑦 ∈ 𝑚)
41		0red 11211	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 0 ∈ ℝ)
42		simplr 780	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑦 ∈ 𝑋)
43		eqid 2769	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )) = (𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))
44	43	metdsval 24974	. . . . . . 7 ⊢ (𝑦 ∈ 𝑋 → ((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) = inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
45	42, 44	syl 18	. . . . . 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 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝐷 ∈ (Met‘𝑋))
47		difssd 4097	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑋 ∖ 𝑚) ⊆ 𝑋)
48	6	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑈 ⊆ 𝐽)
49		simprl 782	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑚 ∈ 𝑈)
50	48, 49	sseldd 3944	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑚 ∈ 𝐽)
51		elssuni 4906	. . . . . . . . . . 11 ⊢ (𝑚 ∈ 𝐽 → 𝑚 ⊆ ∪ 𝐽)
52	50, 51	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑚 ⊆ ∪ 𝐽)
53	46, 11, 14	3syl 19	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑋 = ∪ 𝐽)
54	52, 53	sseqtrrd 3980	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑚 ⊆ 𝑋)
55		eleq1 2857	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑋 → (𝑚 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
56	55	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑋 → (¬ 𝑚 ∈ 𝑈 ↔ ¬ 𝑋 ∈ 𝑈))
57	18, 56	syl5ibrcom 250	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑚 = 𝑋 → ¬ 𝑚 ∈ 𝑈))
58	57	necon2ad 2979	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑚 ∈ 𝑈 → 𝑚 ≠ 𝑋))
59	58	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑚 ∈ 𝑈 → 𝑚 ≠ 𝑋))
60	49, 59	mpd 16	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑚 ≠ 𝑋)
61		pssdifn0 4329	. . . . . . . . 9 ⊢ ((𝑚 ⊆ 𝑋 ∧ 𝑚 ≠ 𝑋) → (𝑋 ∖ 𝑚) ≠ ∅)
62	54, 60, 61	syl2anc 595	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑋 ∖ 𝑚) ≠ ∅)
63	43	metdsre 24980	. . . . . . . 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 1396	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )):𝑋⟶ℝ)
65	64, 42	ffvelcdmd 7081	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → ((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ∈ ℝ)
66	45, 65	eqeltrrd 2870	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
67	35	adantr 485	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
68	12	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝐷 ∈ (∞Met‘𝑋))
69	43	metdsf 24975	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑋 ∖ 𝑚) ⊆ 𝑋) → (𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
70	68, 47, 69	syl2anc 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
71	70, 42	ffvelcdmd 7081	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → ((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ∈ (0[,]+∞))
72		elxrge0 13484	. . . . . . . . 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 221	. . . . . . . 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 500	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 0 ≤ ((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦))
75		elndif 4093	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑚 → ¬ 𝑦 ∈ (𝑋 ∖ 𝑚))
76	75	ad2antll 741	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → ¬ 𝑦 ∈ (𝑋 ∖ 𝑚))
77	53	difeq1d 4086	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑋 ∖ 𝑚) = (∪ 𝐽 ∖ 𝑚))
78	13	mopntop 24566	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
79	68, 78	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝐽 ∈ Top)
80		eqid 2769	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
81	80	opncld 23159	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑚) ∈ (Clsd‘𝐽))
82	79, 50, 81	syl2anc 595	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (∪ 𝐽 ∖ 𝑚) ∈ (Clsd‘𝐽))
83	77, 82	eqeltrd 2869	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑋 ∖ 𝑚) ∈ (Clsd‘𝐽))
84		cldcls 23168	. . . . . . . . . 10 ⊢ ((𝑋 ∖ 𝑚) ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘(𝑋 ∖ 𝑚)) = (𝑋 ∖ 𝑚))
85	83, 84	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → ((cls‘𝐽)‘(𝑋 ∖ 𝑚)) = (𝑋 ∖ 𝑚))
86	76, 85	neleqtrrd 2892	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → ¬ 𝑦 ∈ ((cls‘𝐽)‘(𝑋 ∖ 𝑚)))
87	43, 13	metdseq0 24981	. . . . . . . . . 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 1396	. . . . . . . . 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 3000	. . . . . . . 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 260	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → ((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ≠ 0)
91	65, 74, 90	ne0gt0d 11347	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 0 < ((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦))
92	91, 45	breqtrd 5139	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 0 < inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
93	1	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑈 ∈ Fin)
94	34	adantlr 727	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) ∧ 𝑘 ∈ 𝑈) → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
95	12	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝐷 ∈ (∞Met‘𝑋))
96	27	metdsf 24975	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑋 ∖ 𝑘) ⊆ 𝑋) → (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
97	95, 5, 96	syl2anc 595	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
98	27	fmpt 7106	. . . . . . . . . . 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 237	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → ∀𝑦 ∈ 𝑋 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ (0[,]+∞))
100		rsp 3259	. . . . . . . . . 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 66	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ (0[,]+∞))
102		elxrge0 13484	. . . . . . . . 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 221	. . . . . . . 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 500	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 0 ≤ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
105	104	adantlr 727	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) ∧ 𝑘 ∈ 𝑈) → 0 ≤ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
106		difeq2 4081	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (𝑋 ∖ 𝑘) = (𝑋 ∖ 𝑚))
107	106	mpteq1d 5203	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)) = (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑦𝐷𝑧)))
108	107	rneqd 5929	. . . . . . 7 ⊢ (𝑘 = 𝑚 → ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)) = ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑦𝐷𝑧)))
109	108	infeq1d 9438	. . . . . 6 ⊢ (𝑘 = 𝑚 → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) = inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
110	93, 94, 105, 109, 49	fsumge1 15849	. . . . 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 11370	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 0 < Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
112	40, 111	rexlimddv 3178	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 0 < Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
113	35, 112	elrpd 13057	. 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 7110	1 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ+)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095   ∖ cdif 3908   ⊆ wss 3911  ∅c0 4292  ∪ cuni 4874   class class class wbr 5111   ↦ cmpt 5194  ran crn 5663  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411  Fincfn 8943  infcinf 9401  ℝcr 11099  0cc0 11100  +∞cpnf 11240  ℝ^*cxr 11242   < clt 11243   ≤ cle 11244  ℝ⁺crp 13016  [,]cicc 13375  Σcsu 15737  ∞Metcxmet 21476  Metcmet 21477  MetOpencmopn 21481  Topctop 23019  Clsdccld 23142  clsccl 23144  Compccmp 23512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-inf2 9610  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177  ax-pre-sup 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3375  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  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 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8694  df-ec 8696  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-sup 9402  df-inf 9403  df-oi 9472  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-div 11872  df-nn 12234  df-2 12303  df-3 12304  df-n0 12505  df-z 12592  df-uz 12863  df-q 12973  df-rp 13017  df-xneg 13137  df-xadd 13138  df-xmul 13139  df-ico 13378  df-icc 13379  df-fz 13536  df-fzo 13683  df-seq 14038  df-exp 14098  df-hash 14367  df-cj 15150  df-re 15151  df-im 15152  df-sqrt 15286  df-abs 15287  df-clim 15539  df-sum 15738  df-topgen 17496  df-psmet 21483  df-xmet 21484  df-met 21485  df-bl 21486  df-mopn 21487  df-top 23020  df-topon 23037  df-bases 23072  df-cld 23145  df-ntr 23146  df-cls 23147

This theorem is referenced by:  lebnumlem2  25090  lebnumlem3  25091