Theorem lebnumlem1 23137

Description: Lemma for lebnum 23140. 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 474	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑈 ∈ Fin)
3		lebnum.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
4	3	ad2antrr 717	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝐷 ∈ (Met‘𝑋))
5		difssd 3967	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ⊆ 𝑋)
6		lebnum.s	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ⊆ 𝐽)
7	6	adantr 474	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑈 ⊆ 𝐽)
8	7	sselda 3827	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝑘 ∈ 𝐽)
9		elssuni 4691	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐽 → 𝑘 ⊆ ∪ 𝐽)
10	8, 9	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝑘 ⊆ ∪ 𝐽)
11		metxmet 22516	. . . . . . . . . . . 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 22623	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
15	12, 14	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
16	15	ad2antrr 717	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝑋 = ∪ 𝐽)
17	10, 16	sseqtr4d 3867	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝑘 ⊆ 𝑋)
18		lebnumlem1.n	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)
19		eleq1 2894	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑋 → (𝑘 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
20	19	notbid 310	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑋 → (¬ 𝑘 ∈ 𝑈 ↔ ¬ 𝑋 ∈ 𝑈))
21	18, 20	syl5ibrcom 239	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 = 𝑋 → ¬ 𝑘 ∈ 𝑈))
22	21	necon2ad 3014	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋))
23	22	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋))
24	23	imp 397	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝑘 ≠ 𝑋)
25		pssdifn0 4175	. . . . . . . 8 ⊢ ((𝑘 ⊆ 𝑋 ∧ 𝑘 ≠ 𝑋) → (𝑋 ∖ 𝑘) ≠ ∅)
26	17, 24, 25	syl2anc 579	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ≠ ∅)
27		eqid 2825	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) = (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
28	27	metdsre 23033	. . . . . . 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 1494	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶ℝ)
30	27	fmpt 6634	. . . . . 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 226	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → ∀𝑦 ∈ 𝑋 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
32		simplr 785	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝑦 ∈ 𝑋)
33		rsp 3138	. . . . 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 14849	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
36		lebnum.u	. . . . . . 7 ⊢ (𝜑 → 𝑋 = ∪ 𝑈)
37	36	eleq2d 2892	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝑋 ↔ 𝑦 ∈ ∪ 𝑈))
38	37	biimpa 470	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ ∪ 𝑈)
39		eluni2 4664	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑈 ↔ ∃𝑚 ∈ 𝑈 𝑦 ∈ 𝑚)
40	38, 39	sylib 210	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ∃𝑚 ∈ 𝑈 𝑦 ∈ 𝑚)
41		0red 10367	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 0 ∈ ℝ)
42		simplr 785	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑦 ∈ 𝑋)
43		eqid 2825	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )) = (𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))
44	43	metdsval 23027	. . . . . . 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 717	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝐷 ∈ (Met‘𝑋))
47		difssd 3967	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑋 ∖ 𝑚) ⊆ 𝑋)
48	6	ad2antrr 717	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑈 ⊆ 𝐽)
49		simprl 787	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑚 ∈ 𝑈)
50	48, 49	sseldd 3828	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑚 ∈ 𝐽)
51		elssuni 4691	. . . . . . . . . . 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	sseqtr4d 3867	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑚 ⊆ 𝑋)
55		eleq1 2894	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑋 → (𝑚 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
56	55	notbid 310	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑋 → (¬ 𝑚 ∈ 𝑈 ↔ ¬ 𝑋 ∈ 𝑈))
57	18, 56	syl5ibrcom 239	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑚 = 𝑋 → ¬ 𝑚 ∈ 𝑈))
58	57	necon2ad 3014	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑚 ∈ 𝑈 → 𝑚 ≠ 𝑋))
59	58	ad2antrr 717	. . . . . . . . . 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 4175	. . . . . . . . 9 ⊢ ((𝑚 ⊆ 𝑋 ∧ 𝑚 ≠ 𝑋) → (𝑋 ∖ 𝑚) ≠ ∅)
62	54, 60, 61	syl2anc 579	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑋 ∖ 𝑚) ≠ ∅)
63	43	metdsre 23033	. . . . . . . 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 1494	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )):𝑋⟶ℝ)
65	64, 42	ffvelrnd 6614	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → ((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ∈ ℝ)
66	45, 65	eqeltrrd 2907	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
67	35	adantr 474	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
68	12	ad2antrr 717	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝐷 ∈ (∞Met‘𝑋))
69	43	metdsf 23028	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑋 ∖ 𝑚) ⊆ 𝑋) → (𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
70	68, 47, 69	syl2anc 579	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
71	70, 42	ffvelrnd 6614	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → ((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ∈ (0[,]+∞))
72		elxrge0 12578	. . . . . . . . 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 210	. . . . . . . 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 491	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 0 ≤ ((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦))
75		elndif 3963	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑚 → ¬ 𝑦 ∈ (𝑋 ∖ 𝑚))
76	75	ad2antll 720	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → ¬ 𝑦 ∈ (𝑋 ∖ 𝑚))
77	53	difeq1d 3956	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑋 ∖ 𝑚) = (∪ 𝐽 ∖ 𝑚))
78	13	mopntop 22622	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
79	68, 78	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝐽 ∈ Top)
80		eqid 2825	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
81	80	opncld 21215	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑚) ∈ (Clsd‘𝐽))
82	79, 50, 81	syl2anc 579	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (∪ 𝐽 ∖ 𝑚) ∈ (Clsd‘𝐽))
83	77, 82	eqeltrd 2906	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑋 ∖ 𝑚) ∈ (Clsd‘𝐽))
84		cldcls 21224	. . . . . . . . . 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 2928	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → ¬ 𝑦 ∈ ((cls‘𝐽)‘(𝑋 ∖ 𝑚)))
87	43, 13	metdseq0 23034	. . . . . . . . . 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 1494	. . . . . . . . 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 3035	. . . . . . . 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 249	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → ((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ≠ 0)
91	65, 74, 90	ne0gt0d 10500	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 0 < ((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦))
92	91, 45	breqtrd 4901	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 0 < inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
93	1	ad2antrr 717	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑈 ∈ Fin)
94	34	adantlr 706	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) ∧ 𝑘 ∈ 𝑈) → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
95	12	ad2antrr 717	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝐷 ∈ (∞Met‘𝑋))
96	27	metdsf 23028	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑋 ∖ 𝑘) ⊆ 𝑋) → (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
97	95, 5, 96	syl2anc 579	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
98	27	fmpt 6634	. . . . . . . . . . 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 226	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → ∀𝑦 ∈ 𝑋 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ (0[,]+∞))
100		rsp 3138	. . . . . . . . . 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 12578	. . . . . . . . 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 210	. . . . . . . 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 491	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 0 ≤ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
105	104	adantlr 706	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) ∧ 𝑘 ∈ 𝑈) → 0 ≤ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
106		difeq2 3951	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (𝑋 ∖ 𝑘) = (𝑋 ∖ 𝑚))
107	106	mpteq1d 4963	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)) = (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑦𝐷𝑧)))
108	107	rneqd 5589	. . . . . . 7 ⊢ (𝑘 = 𝑚 → ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)) = ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑦𝐷𝑧)))
109	108	infeq1d 8658	. . . . . 6 ⊢ (𝑘 = 𝑚 → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) = inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
110	93, 94, 105, 109, 49	fsumge1 14910	. . . . 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 10523	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 0 < Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
112	40, 111	rexlimddv 3245	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 0 < Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
113	35, 112	elrpd 12160	. 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 6638	1 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ+)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164   ≠ wne 2999  ∀wral 3117  ∃wrex 3118   ∖ cdif 3795   ⊆ wss 3798  ∅c0 4146  ∪ cuni 4660   class class class wbr 4875   ↦ cmpt 4954  ran crn 5347  ⟶wf 6123  ‘cfv 6127  (class class class)co 6910  Fincfn 8228  infcinf 8622  ℝcr 10258  0cc0 10259  +∞cpnf 10395  ℝ^*cxr 10397   < clt 10398   ≤ cle 10399  ℝ⁺crp 12119  [,]cicc 12473  Σcsu 14800  ∞Metcxmet 20098  Metcmet 20099  MetOpencmopn 20103  Topctop 21075  Clsdccld 21198  clsccl 21200  Compccmp 21567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-inf2 8822  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336  ax-pre-sup 10337

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-iin 4745  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-se 5306  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-isom 6136  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-oadd 7835  df-er 8014  df-ec 8016  df-map 8129  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-sup 8623  df-inf 8624  df-oi 8691  df-card 9085  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-div 11017  df-nn 11358  df-2 11421  df-3 11422  df-n0 11626  df-z 11712  df-uz 11976  df-q 12079  df-rp 12120  df-xneg 12239  df-xadd 12240  df-xmul 12241  df-ico 12476  df-icc 12477  df-fz 12627  df-fzo 12768  df-seq 13103  df-exp 13162  df-hash 13418  df-cj 14223  df-re 14224  df-im 14225  df-sqrt 14359  df-abs 14360  df-clim 14603  df-sum 14801  df-topgen 16464  df-psmet 20105  df-xmet 20106  df-met 20107  df-bl 20108  df-mopn 20109  df-top 21076  df-topon 21093  df-bases 21128  df-cld 21201  df-ntr 21202  df-cls 21203

This theorem is referenced by:  lebnumlem2  23138  lebnumlem3  23139