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Theorem lebnumlem1 24468

Description: Lemma for lebnum 24471. 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 481	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑈 ∈ Fin)
3		lebnum.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
4	3	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝐷 ∈ (Met‘𝑋))
5		difssd 4131	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ⊆ 𝑋)
6		lebnum.s	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ⊆ 𝐽)
7	6	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑈 ⊆ 𝐽)
8	7	sselda 3981	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝑘 ∈ 𝐽)
9		elssuni 4940	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐽 → 𝑘 ⊆ ∪ 𝐽)
10	8, 9	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝑘 ⊆ ∪ 𝐽)
11		metxmet 23831	. . . . . . . . . . . 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 23938	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
15	12, 14	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
16	15	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝑋 = ∪ 𝐽)
17	10, 16	sseqtrrd 4022	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝑘 ⊆ 𝑋)
18		lebnumlem1.n	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)
19		eleq1 2821	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑋 → (𝑘 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
20	19	notbid 317	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑋 → (¬ 𝑘 ∈ 𝑈 ↔ ¬ 𝑋 ∈ 𝑈))
21	18, 20	syl5ibrcom 246	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 = 𝑋 → ¬ 𝑘 ∈ 𝑈))
22	21	necon2ad 2955	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋))
23	22	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋))
24	23	imp 407	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝑘 ≠ 𝑋)
25		pssdifn0 4364	. . . . . . . 8 ⊢ ((𝑘 ⊆ 𝑋 ∧ 𝑘 ≠ 𝑋) → (𝑋 ∖ 𝑘) ≠ ∅)
26	17, 24, 25	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ≠ ∅)
27		eqid 2732	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < )) = (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ))
28	27	metdsre 24360	. . . . . . 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 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 233	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → ∀𝑦 ∈ 𝑋 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < ) ∈ ℝ)
32		simplr 767	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝑦 ∈ 𝑋)
33		rsp 3244	. . . . 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 15676	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < ) ∈ ℝ)
36		lebnum.u	. . . . . . 7 ⊢ (𝜑 → 𝑋 = ∪ 𝑈)
37	36	eleq2d 2819	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝑋 ↔ 𝑦 ∈ ∪ 𝑈))
38	37	biimpa 477	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ ∪ 𝑈)
39		eluni2 4911	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑈 ↔ ∃𝑚 ∈ 𝑈 𝑦 ∈ 𝑚)
40	38, 39	sylib 217	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ∃𝑚 ∈ 𝑈 𝑦 ∈ 𝑚)
41		0red 11213	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 0 ∈ ℝ)
42		simplr 767	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑦 ∈ 𝑋)
43		eqid 2732	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ^, < )) = (𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ^, < ))
44	43	metdsval 24354	. . . . . . 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 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝐷 ∈ (Met‘𝑋))
47		difssd 4131	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑋 ∖ 𝑚) ⊆ 𝑋)
48	6	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑈 ⊆ 𝐽)
49		simprl 769	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑚 ∈ 𝑈)
50	48, 49	sseldd 3982	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑚 ∈ 𝐽)
51		elssuni 4940	. . . . . . . . . . 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 4022	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑚 ⊆ 𝑋)
55		eleq1 2821	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑋 → (𝑚 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
56	55	notbid 317	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑋 → (¬ 𝑚 ∈ 𝑈 ↔ ¬ 𝑋 ∈ 𝑈))
57	18, 56	syl5ibrcom 246	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑚 = 𝑋 → ¬ 𝑚 ∈ 𝑈))
58	57	necon2ad 2955	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑚 ∈ 𝑈 → 𝑚 ≠ 𝑋))
59	58	ad2antrr 724	. . . . . . . . . 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 4364	. . . . . . . . 9 ⊢ ((𝑚 ⊆ 𝑋 ∧ 𝑚 ≠ 𝑋) → (𝑋 ∖ 𝑚) ≠ ∅)
62	54, 60, 61	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑋 ∖ 𝑚) ≠ ∅)
63	43	metdsre 24360	. . . . . . . 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 7084	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → ((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ^*, < ))‘𝑦) ∈ ℝ)
66	45, 65	eqeltrrd 2834	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑦𝐷𝑧)), ℝ^*, < ) ∈ ℝ)
67	35	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < ) ∈ ℝ)
68	12	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝐷 ∈ (∞Met‘𝑋))
69	43	metdsf 24355	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑋 ∖ 𝑚) ⊆ 𝑋) → (𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ^*, < )):𝑋⟶(0[,]+∞))
70	68, 47, 69	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ^*, < )):𝑋⟶(0[,]+∞))
71	70, 42	ffvelcdmd 7084	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → ((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ^*, < ))‘𝑦) ∈ (0[,]+∞))
72		elxrge0 13430	. . . . . . . . 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 217	. . . . . . . 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 496	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 0 ≤ ((𝑤 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑤𝐷𝑧)), ℝ^*, < ))‘𝑦))
75		elndif 4127	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑚 → ¬ 𝑦 ∈ (𝑋 ∖ 𝑚))
76	75	ad2antll 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → ¬ 𝑦 ∈ (𝑋 ∖ 𝑚))
77	53	difeq1d 4120	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑋 ∖ 𝑚) = (∪ 𝐽 ∖ 𝑚))
78	13	mopntop 23937	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
79	68, 78	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝐽 ∈ Top)
80		eqid 2732	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
81	80	opncld 22528	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑚) ∈ (Clsd‘𝐽))
82	79, 50, 81	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (∪ 𝐽 ∖ 𝑚) ∈ (Clsd‘𝐽))
83	77, 82	eqeltrd 2833	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → (𝑋 ∖ 𝑚) ∈ (Clsd‘𝐽))
84		cldcls 22537	. . . . . . . . . 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 2856	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → ¬ 𝑦 ∈ ((cls‘𝐽)‘(𝑋 ∖ 𝑚)))
87	43, 13	metdseq0 24361	. . . . . . . . . 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 2977	. . . . . . . 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 256	. . . . . . 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 5173	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 0 < inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑦𝐷𝑧)), ℝ^*, < ))
93	1	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) → 𝑈 ∈ Fin)
94	34	adantlr 713	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) ∧ 𝑘 ∈ 𝑈) → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < ) ∈ ℝ)
95	12	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 𝐷 ∈ (∞Met‘𝑋))
96	27	metdsf 24355	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑋 ∖ 𝑘) ⊆ 𝑋) → (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < )):𝑋⟶(0[,]+∞))
97	95, 5, 96	syl2anc 584	. . . . . . . . . . 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 233	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → ∀𝑦 ∈ 𝑋 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < ) ∈ (0[,]+∞))
100		rsp 3244	. . . . . . . . . 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 13430	. . . . . . . . 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 217	. . . . . . . 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 496	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑘 ∈ 𝑈) → 0 ≤ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < ))
105	104	adantlr 713	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑈 ∧ 𝑦 ∈ 𝑚)) ∧ 𝑘 ∈ 𝑈) → 0 ≤ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < ))
106		difeq2 4115	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (𝑋 ∖ 𝑘) = (𝑋 ∖ 𝑚))
107	106	mpteq1d 5242	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)) = (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑦𝐷𝑧)))
108	107	rneqd 5935	. . . . . . 7 ⊢ (𝑘 = 𝑚 → ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)) = ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑦𝐷𝑧)))
109	108	infeq1d 9468	. . . . . 6 ⊢ (𝑘 = 𝑚 → inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ) = inf(ran (𝑧 ∈ (𝑋 ∖ 𝑚) ↦ (𝑦𝐷𝑧)), ℝ^, < ))
110	93, 94, 105, 109, 49	fsumge1 15739	. . . . 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 3161	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 0 < Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < ))
113	35, 112	elrpd 13009	. 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⁺)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ∖ cdif 3944 ⊆ wss 3947 ∅c0 4321 ∪ cuni 4907 class class class wbr 5147 ↦ cmpt 5230 ran crn 5676 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 Fincfn 8935 infcinf 9432 ℝcr 11105 0cc0 11106 +∞cpnf 11241 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 ℝ⁺crp 12970 [,]cicc 13323 Σcsu 15628 ∞Metcxmet 20921 Metcmet 20922 MetOpencmopn 20926 Topctop 22386 Clsdccld 22511 clsccl 22513 Compccmp 22881

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-ec 8701 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-topgen 17385 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-top 22387 df-topon 22404 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516

This theorem is referenced by: lebnumlem2 24469 lebnumlem3 24470

W3C validator