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Theorem lebnumlem1 23564
Description: Lemma for lebnum 23567. 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 (𝜑𝐷 ∈ (Met‘𝑋))
lebnum.c (𝜑𝐽 ∈ Comp)
lebnum.s (𝜑𝑈𝐽)
lebnum.u (𝜑𝑋 = 𝑈)
lebnumlem1.u (𝜑𝑈 ∈ Fin)
lebnumlem1.n (𝜑 → ¬ 𝑋𝑈)
lebnumlem1.f 𝐹 = (𝑦𝑋 ↦ Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
Assertion
Ref Expression
lebnumlem1 (𝜑𝐹: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.
StepHypRef Expression
1 lebnumlem1.u . . . . 5 (𝜑𝑈 ∈ Fin)
21adantr 484 . . . 4 ((𝜑𝑦𝑋) → 𝑈 ∈ Fin)
3 lebnum.d . . . . . . . 8 (𝜑𝐷 ∈ (Met‘𝑋))
43ad2antrr 725 . . . . . . 7 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 𝐷 ∈ (Met‘𝑋))
5 difssd 4095 . . . . . . 7 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → (𝑋𝑘) ⊆ 𝑋)
6 lebnum.s . . . . . . . . . . . 12 (𝜑𝑈𝐽)
76adantr 484 . . . . . . . . . . 11 ((𝜑𝑦𝑋) → 𝑈𝐽)
87sselda 3953 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 𝑘𝐽)
9 elssuni 4855 . . . . . . . . . 10 (𝑘𝐽𝑘 𝐽)
108, 9syl 17 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 𝑘 𝐽)
11 metxmet 22939 . . . . . . . . . . . 12 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
123, 11syl 17 . . . . . . . . . . 11 (𝜑𝐷 ∈ (∞Met‘𝑋))
13 lebnum.j . . . . . . . . . . . 12 𝐽 = (MetOpen‘𝐷)
1413mopnuni 23046 . . . . . . . . . . 11 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
1512, 14syl 17 . . . . . . . . . 10 (𝜑𝑋 = 𝐽)
1615ad2antrr 725 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 𝑋 = 𝐽)
1710, 16sseqtrrd 3994 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 𝑘𝑋)
18 lebnumlem1.n . . . . . . . . . . . 12 (𝜑 → ¬ 𝑋𝑈)
19 eleq1 2903 . . . . . . . . . . . . 13 (𝑘 = 𝑋 → (𝑘𝑈𝑋𝑈))
2019notbid 321 . . . . . . . . . . . 12 (𝑘 = 𝑋 → (¬ 𝑘𝑈 ↔ ¬ 𝑋𝑈))
2118, 20syl5ibrcom 250 . . . . . . . . . . 11 (𝜑 → (𝑘 = 𝑋 → ¬ 𝑘𝑈))
2221necon2ad 3029 . . . . . . . . . 10 (𝜑 → (𝑘𝑈𝑘𝑋))
2322adantr 484 . . . . . . . . 9 ((𝜑𝑦𝑋) → (𝑘𝑈𝑘𝑋))
2423imp 410 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 𝑘𝑋)
25 pssdifn0 4308 . . . . . . . 8 ((𝑘𝑋𝑘𝑋) → (𝑋𝑘) ≠ ∅)
2617, 24, 25syl2anc 587 . . . . . . 7 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → (𝑋𝑘) ≠ ∅)
27 eqid 2824 . . . . . . . 8 (𝑦𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) = (𝑦𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
2827metdsre 23456 . . . . . . 7 ((𝐷 ∈ (Met‘𝑋) ∧ (𝑋𝑘) ⊆ 𝑋 ∧ (𝑋𝑘) ≠ ∅) → (𝑦𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶ℝ)
294, 5, 26, 28syl3anc 1368 . . . . . 6 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → (𝑦𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶ℝ)
3027fmpt 6863 . . . . . 6 (∀𝑦𝑋 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ ↔ (𝑦𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶ℝ)
3129, 30sylibr 237 . . . . 5 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → ∀𝑦𝑋 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
32 simplr 768 . . . . 5 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 𝑦𝑋)
33 rsp 3200 . . . . 5 (∀𝑦𝑋 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ → (𝑦𝑋 → inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ))
3431, 32, 33sylc 65 . . . 4 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
352, 34fsumrecl 15089 . . 3 ((𝜑𝑦𝑋) → Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
36 lebnum.u . . . . . . 7 (𝜑𝑋 = 𝑈)
3736eleq2d 2901 . . . . . 6 (𝜑 → (𝑦𝑋𝑦 𝑈))
3837biimpa 480 . . . . 5 ((𝜑𝑦𝑋) → 𝑦 𝑈)
39 eluni2 4829 . . . . 5 (𝑦 𝑈 ↔ ∃𝑚𝑈 𝑦𝑚)
4038, 39sylib 221 . . . 4 ((𝜑𝑦𝑋) → ∃𝑚𝑈 𝑦𝑚)
41 0red 10638 . . . . 5 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 0 ∈ ℝ)
42 simplr 768 . . . . . . 7 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑦𝑋)
43 eqid 2824 . . . . . . . 8 (𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )) = (𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))
4443metdsval 23450 . . . . . . 7 (𝑦𝑋 → ((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) = inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
4542, 44syl 17 . . . . . 6 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → ((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) = inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
463ad2antrr 725 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝐷 ∈ (Met‘𝑋))
47 difssd 4095 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (𝑋𝑚) ⊆ 𝑋)
486ad2antrr 725 . . . . . . . . . . . 12 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑈𝐽)
49 simprl 770 . . . . . . . . . . . 12 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑚𝑈)
5048, 49sseldd 3954 . . . . . . . . . . 11 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑚𝐽)
51 elssuni 4855 . . . . . . . . . . 11 (𝑚𝐽𝑚 𝐽)
5250, 51syl 17 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑚 𝐽)
5346, 11, 143syl 18 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑋 = 𝐽)
5452, 53sseqtrrd 3994 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑚𝑋)
55 eleq1 2903 . . . . . . . . . . . . . 14 (𝑚 = 𝑋 → (𝑚𝑈𝑋𝑈))
5655notbid 321 . . . . . . . . . . . . 13 (𝑚 = 𝑋 → (¬ 𝑚𝑈 ↔ ¬ 𝑋𝑈))
5718, 56syl5ibrcom 250 . . . . . . . . . . . 12 (𝜑 → (𝑚 = 𝑋 → ¬ 𝑚𝑈))
5857necon2ad 3029 . . . . . . . . . . 11 (𝜑 → (𝑚𝑈𝑚𝑋))
5958ad2antrr 725 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (𝑚𝑈𝑚𝑋))
6049, 59mpd 15 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑚𝑋)
61 pssdifn0 4308 . . . . . . . . 9 ((𝑚𝑋𝑚𝑋) → (𝑋𝑚) ≠ ∅)
6254, 60, 61syl2anc 587 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (𝑋𝑚) ≠ ∅)
6343metdsre 23456 . . . . . . . 8 ((𝐷 ∈ (Met‘𝑋) ∧ (𝑋𝑚) ⊆ 𝑋 ∧ (𝑋𝑚) ≠ ∅) → (𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )):𝑋⟶ℝ)
6446, 47, 62, 63syl3anc 1368 . . . . . . 7 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )):𝑋⟶ℝ)
6564, 42ffvelrnd 6841 . . . . . 6 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → ((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ∈ ℝ)
6645, 65eqeltrrd 2917 . . . . 5 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
6735adantr 484 . . . . 5 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
6812ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝐷 ∈ (∞Met‘𝑋))
6943metdsf 23451 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑋𝑚) ⊆ 𝑋) → (𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
7068, 47, 69syl2anc 587 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
7170, 42ffvelrnd 6841 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → ((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ∈ (0[,]+∞))
72 elxrge0 12842 . . . . . . . . 9 (((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ∈ (0[,]+∞) ↔ (((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ∈ ℝ* ∧ 0 ≤ ((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦)))
7371, 72sylib 221 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ∈ ℝ* ∧ 0 ≤ ((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦)))
7473simprd 499 . . . . . . 7 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 0 ≤ ((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦))
75 elndif 4091 . . . . . . . . . 10 (𝑦𝑚 → ¬ 𝑦 ∈ (𝑋𝑚))
7675ad2antll 728 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → ¬ 𝑦 ∈ (𝑋𝑚))
7753difeq1d 4084 . . . . . . . . . . 11 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (𝑋𝑚) = ( 𝐽𝑚))
7813mopntop 23045 . . . . . . . . . . . . 13 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
7968, 78syl 17 . . . . . . . . . . . 12 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝐽 ∈ Top)
80 eqid 2824 . . . . . . . . . . . . 13 𝐽 = 𝐽
8180opncld 21636 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑚𝐽) → ( 𝐽𝑚) ∈ (Clsd‘𝐽))
8279, 50, 81syl2anc 587 . . . . . . . . . . 11 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → ( 𝐽𝑚) ∈ (Clsd‘𝐽))
8377, 82eqeltrd 2916 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (𝑋𝑚) ∈ (Clsd‘𝐽))
84 cldcls 21645 . . . . . . . . . 10 ((𝑋𝑚) ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘(𝑋𝑚)) = (𝑋𝑚))
8583, 84syl 17 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → ((cls‘𝐽)‘(𝑋𝑚)) = (𝑋𝑚))
8676, 85neleqtrrd 2938 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → ¬ 𝑦 ∈ ((cls‘𝐽)‘(𝑋𝑚)))
8743, 13metdseq0 23457 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑋𝑚) ⊆ 𝑋𝑦𝑋) → (((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) = 0 ↔ 𝑦 ∈ ((cls‘𝐽)‘(𝑋𝑚))))
8868, 47, 42, 87syl3anc 1368 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) = 0 ↔ 𝑦 ∈ ((cls‘𝐽)‘(𝑋𝑚))))
8988necon3abid 3050 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ≠ 0 ↔ ¬ 𝑦 ∈ ((cls‘𝐽)‘(𝑋𝑚))))
9086, 89mpbird 260 . . . . . . 7 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → ((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ≠ 0)
9165, 74, 90ne0gt0d 10771 . . . . . 6 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 0 < ((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦))
9291, 45breqtrd 5079 . . . . 5 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 0 < inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
931ad2antrr 725 . . . . . 6 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑈 ∈ Fin)
9434adantlr 714 . . . . . 6 ((((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) ∧ 𝑘𝑈) → inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
9512ad2antrr 725 . . . . . . . . . . . 12 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 𝐷 ∈ (∞Met‘𝑋))
9627metdsf 23451 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑋𝑘) ⊆ 𝑋) → (𝑦𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
9795, 5, 96syl2anc 587 . . . . . . . . . . 11 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → (𝑦𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
9827fmpt 6863 . . . . . . . . . . 11 (∀𝑦𝑋 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ (0[,]+∞) ↔ (𝑦𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
9997, 98sylibr 237 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → ∀𝑦𝑋 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ (0[,]+∞))
100 rsp 3200 . . . . . . . . . 10 (∀𝑦𝑋 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ (0[,]+∞) → (𝑦𝑋 → inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ (0[,]+∞)))
10199, 32, 100sylc 65 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ (0[,]+∞))
102 elxrge0 12842 . . . . . . . . 9 (inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ (0[,]+∞) ↔ (inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ* ∧ 0 ≤ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )))
103101, 102sylib 221 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → (inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ* ∧ 0 ≤ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )))
104103simprd 499 . . . . . . 7 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 0 ≤ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
105104adantlr 714 . . . . . 6 ((((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) ∧ 𝑘𝑈) → 0 ≤ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
106 difeq2 4079 . . . . . . . . 9 (𝑘 = 𝑚 → (𝑋𝑘) = (𝑋𝑚))
107106mpteq1d 5142 . . . . . . . 8 (𝑘 = 𝑚 → (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)) = (𝑧 ∈ (𝑋𝑚) ↦ (𝑦𝐷𝑧)))
108107rneqd 5796 . . . . . . 7 (𝑘 = 𝑚 → ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)) = ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑦𝐷𝑧)))
109108infeq1d 8934 . . . . . 6 (𝑘 = 𝑚 → inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) = inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
11093, 94, 105, 109, 49fsumge1 15150 . . . . 5 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ≤ Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
11141, 66, 67, 92, 110ltletrd 10794 . . . 4 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 0 < Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
11240, 111rexlimddv 3284 . . 3 ((𝜑𝑦𝑋) → 0 < Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
11335, 112elrpd 12423 . 2 ((𝜑𝑦𝑋) → Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ+)
114 lebnumlem1.f . 2 𝐹 = (𝑦𝑋 ↦ Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
115113, 114fmptd 6867 1 (𝜑𝐹:𝑋⟶ℝ+)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wne 3014  wral 3133  wrex 3134  cdif 3916  wss 3919  c0 4276   cuni 4825   class class class wbr 5053  cmpt 5133  ran crn 5544  wf 6340  cfv 6344  (class class class)co 7146  Fincfn 8501  infcinf 8898  cr 10530  0cc0 10531  +∞cpnf 10666  *cxr 10668   < clt 10669  cle 10670  +crp 12384  [,]cicc 12736  Σcsu 15040  ∞Metcxmet 20525  Metcmet 20526  MetOpencmopn 20530  Topctop 21496  Clsdccld 21619  clsccl 21621  Compccmp 21989
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452  ax-inf2 9097  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4826  df-int 4864  df-iun 4908  df-iin 4909  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-se 5503  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6136  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-isom 6353  df-riota 7104  df-ov 7149  df-oprab 7150  df-mpo 7151  df-om 7572  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-ec 8283  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-sup 8899  df-inf 8900  df-oi 8967  df-card 9361  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11695  df-3 11696  df-n0 11893  df-z 11977  df-uz 12239  df-q 12344  df-rp 12385  df-xneg 12502  df-xadd 12503  df-xmul 12504  df-ico 12739  df-icc 12740  df-fz 12893  df-fzo 13036  df-seq 13372  df-exp 13433  df-hash 13694  df-cj 14456  df-re 14457  df-im 14458  df-sqrt 14592  df-abs 14593  df-clim 14843  df-sum 15041  df-topgen 16715  df-psmet 20532  df-xmet 20533  df-met 20534  df-bl 20535  df-mopn 20536  df-top 21497  df-topon 21514  df-bases 21549  df-cld 21622  df-ntr 21623  df-cls 21624
This theorem is referenced by:  lebnumlem2  23565  lebnumlem3  23566
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