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Theorem lebnumlem1 22741
 Description: Lemma for lebnum 22744. 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 481 . . . 4 ((𝜑𝑦𝑋) → 𝑈 ∈ Fin)
3 lebnum.d . . . . . . . 8 (𝜑𝐷 ∈ (Met‘𝑋))
43ad2antrr 761 . . . . . . 7 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 𝐷 ∈ (Met‘𝑋))
5 difssd 3730 . . . . . . 7 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → (𝑋𝑘) ⊆ 𝑋)
6 lebnum.s . . . . . . . . . . . 12 (𝜑𝑈𝐽)
76adantr 481 . . . . . . . . . . 11 ((𝜑𝑦𝑋) → 𝑈𝐽)
87sselda 3595 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 𝑘𝐽)
9 elssuni 4458 . . . . . . . . . 10 (𝑘𝐽𝑘 𝐽)
108, 9syl 17 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 𝑘 𝐽)
11 metxmet 22120 . . . . . . . . . . . 12 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
123, 11syl 17 . . . . . . . . . . 11 (𝜑𝐷 ∈ (∞Met‘𝑋))
13 lebnum.j . . . . . . . . . . . 12 𝐽 = (MetOpen‘𝐷)
1413mopnuni 22227 . . . . . . . . . . 11 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
1512, 14syl 17 . . . . . . . . . 10 (𝜑𝑋 = 𝐽)
1615ad2antrr 761 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 𝑋 = 𝐽)
1710, 16sseqtr4d 3634 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 𝑘𝑋)
18 lebnumlem1.n . . . . . . . . . . . 12 (𝜑 → ¬ 𝑋𝑈)
19 eleq1 2687 . . . . . . . . . . . . 13 (𝑘 = 𝑋 → (𝑘𝑈𝑋𝑈))
2019notbid 308 . . . . . . . . . . . 12 (𝑘 = 𝑋 → (¬ 𝑘𝑈 ↔ ¬ 𝑋𝑈))
2118, 20syl5ibrcom 237 . . . . . . . . . . 11 (𝜑 → (𝑘 = 𝑋 → ¬ 𝑘𝑈))
2221necon2ad 2806 . . . . . . . . . 10 (𝜑 → (𝑘𝑈𝑘𝑋))
2322adantr 481 . . . . . . . . 9 ((𝜑𝑦𝑋) → (𝑘𝑈𝑘𝑋))
2423imp 445 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 𝑘𝑋)
25 pssdifn0 3935 . . . . . . . 8 ((𝑘𝑋𝑘𝑋) → (𝑋𝑘) ≠ ∅)
2617, 24, 25syl2anc 692 . . . . . . 7 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → (𝑋𝑘) ≠ ∅)
27 eqid 2620 . . . . . . . 8 (𝑦𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) = (𝑦𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
2827metdsre 22637 . . . . . . 7 ((𝐷 ∈ (Met‘𝑋) ∧ (𝑋𝑘) ⊆ 𝑋 ∧ (𝑋𝑘) ≠ ∅) → (𝑦𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶ℝ)
294, 5, 26, 28syl3anc 1324 . . . . . 6 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → (𝑦𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶ℝ)
3027fmpt 6367 . . . . . 6 (∀𝑦𝑋 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ ↔ (𝑦𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶ℝ)
3129, 30sylibr 224 . . . . 5 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → ∀𝑦𝑋 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
32 simplr 791 . . . . 5 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 𝑦𝑋)
33 rsp 2926 . . . . 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 14446 . . 3 ((𝜑𝑦𝑋) → Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
36 lebnum.u . . . . . . 7 (𝜑𝑋 = 𝑈)
3736eleq2d 2685 . . . . . 6 (𝜑 → (𝑦𝑋𝑦 𝑈))
3837biimpa 501 . . . . 5 ((𝜑𝑦𝑋) → 𝑦 𝑈)
39 eluni2 4431 . . . . 5 (𝑦 𝑈 ↔ ∃𝑚𝑈 𝑦𝑚)
4038, 39sylib 208 . . . 4 ((𝜑𝑦𝑋) → ∃𝑚𝑈 𝑦𝑚)
41 0red 10026 . . . . 5 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 0 ∈ ℝ)
42 simplr 791 . . . . . . 7 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑦𝑋)
43 eqid 2620 . . . . . . . 8 (𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )) = (𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))
4443metdsval 22631 . . . . . . 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 761 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝐷 ∈ (Met‘𝑋))
47 difssd 3730 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (𝑋𝑚) ⊆ 𝑋)
486ad2antrr 761 . . . . . . . . . . . 12 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑈𝐽)
49 simprl 793 . . . . . . . . . . . 12 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑚𝑈)
5048, 49sseldd 3596 . . . . . . . . . . 11 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑚𝐽)
51 elssuni 4458 . . . . . . . . . . 11 (𝑚𝐽𝑚 𝐽)
5250, 51syl 17 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑚 𝐽)
5346, 11, 143syl 18 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑋 = 𝐽)
5452, 53sseqtr4d 3634 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑚𝑋)
55 eleq1 2687 . . . . . . . . . . . . . 14 (𝑚 = 𝑋 → (𝑚𝑈𝑋𝑈))
5655notbid 308 . . . . . . . . . . . . 13 (𝑚 = 𝑋 → (¬ 𝑚𝑈 ↔ ¬ 𝑋𝑈))
5718, 56syl5ibrcom 237 . . . . . . . . . . . 12 (𝜑 → (𝑚 = 𝑋 → ¬ 𝑚𝑈))
5857necon2ad 2806 . . . . . . . . . . 11 (𝜑 → (𝑚𝑈𝑚𝑋))
5958ad2antrr 761 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (𝑚𝑈𝑚𝑋))
6049, 59mpd 15 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑚𝑋)
61 pssdifn0 3935 . . . . . . . . 9 ((𝑚𝑋𝑚𝑋) → (𝑋𝑚) ≠ ∅)
6254, 60, 61syl2anc 692 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (𝑋𝑚) ≠ ∅)
6343metdsre 22637 . . . . . . . 8 ((𝐷 ∈ (Met‘𝑋) ∧ (𝑋𝑚) ⊆ 𝑋 ∧ (𝑋𝑚) ≠ ∅) → (𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )):𝑋⟶ℝ)
6446, 47, 62, 63syl3anc 1324 . . . . . . 7 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )):𝑋⟶ℝ)
6564, 42ffvelrnd 6346 . . . . . 6 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → ((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ∈ ℝ)
6645, 65eqeltrrd 2700 . . . . 5 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
6735adantr 481 . . . . 5 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
6812ad2antrr 761 . . . . . . . . . . 11 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝐷 ∈ (∞Met‘𝑋))
6943metdsf 22632 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑋𝑚) ⊆ 𝑋) → (𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
7068, 47, 69syl2anc 692 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
7170, 42ffvelrnd 6346 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → ((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ∈ (0[,]+∞))
72 elxrge0 12266 . . . . . . . . 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 208 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ∈ ℝ* ∧ 0 ≤ ((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦)))
7473simprd 479 . . . . . . 7 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 0 ≤ ((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦))
75 elndif 3726 . . . . . . . . . 10 (𝑦𝑚 → ¬ 𝑦 ∈ (𝑋𝑚))
7675ad2antll 764 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → ¬ 𝑦 ∈ (𝑋𝑚))
7753difeq1d 3719 . . . . . . . . . . 11 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (𝑋𝑚) = ( 𝐽𝑚))
7813mopntop 22226 . . . . . . . . . . . . 13 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
7968, 78syl 17 . . . . . . . . . . . 12 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝐽 ∈ Top)
80 eqid 2620 . . . . . . . . . . . . 13 𝐽 = 𝐽
8180opncld 20818 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑚𝐽) → ( 𝐽𝑚) ∈ (Clsd‘𝐽))
8279, 50, 81syl2anc 692 . . . . . . . . . . 11 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → ( 𝐽𝑚) ∈ (Clsd‘𝐽))
8377, 82eqeltrd 2699 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (𝑋𝑚) ∈ (Clsd‘𝐽))
84 cldcls 20827 . . . . . . . . . 10 ((𝑋𝑚) ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘(𝑋𝑚)) = (𝑋𝑚))
8583, 84syl 17 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → ((cls‘𝐽)‘(𝑋𝑚)) = (𝑋𝑚))
8676, 85neleqtrrd 2721 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → ¬ 𝑦 ∈ ((cls‘𝐽)‘(𝑋𝑚)))
8743, 13metdseq0 22638 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑋𝑚) ⊆ 𝑋𝑦𝑋) → (((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) = 0 ↔ 𝑦 ∈ ((cls‘𝐽)‘(𝑋𝑚))))
8868, 47, 42, 87syl3anc 1324 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) = 0 ↔ 𝑦 ∈ ((cls‘𝐽)‘(𝑋𝑚))))
8988necon3abid 2827 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ≠ 0 ↔ ¬ 𝑦 ∈ ((cls‘𝐽)‘(𝑋𝑚))))
9086, 89mpbird 247 . . . . . . 7 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → ((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ≠ 0)
9165, 74, 90ne0gt0d 10159 . . . . . 6 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 0 < ((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦))
9291, 45breqtrd 4670 . . . . 5 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 0 < inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
931ad2antrr 761 . . . . . 6 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑈 ∈ Fin)
9434adantlr 750 . . . . . 6 ((((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) ∧ 𝑘𝑈) → inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
9512ad2antrr 761 . . . . . . . . . . . 12 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 𝐷 ∈ (∞Met‘𝑋))
9627metdsf 22632 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑋𝑘) ⊆ 𝑋) → (𝑦𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
9795, 5, 96syl2anc 692 . . . . . . . . . . 11 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → (𝑦𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
9827fmpt 6367 . . . . . . . . . . 11 (∀𝑦𝑋 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ (0[,]+∞) ↔ (𝑦𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
9997, 98sylibr 224 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → ∀𝑦𝑋 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ (0[,]+∞))
100 rsp 2926 . . . . . . . . . 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 12266 . . . . . . . . 9 (inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ (0[,]+∞) ↔ (inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ* ∧ 0 ≤ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )))
103101, 102sylib 208 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → (inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ* ∧ 0 ≤ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )))
104103simprd 479 . . . . . . 7 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 0 ≤ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
105104adantlr 750 . . . . . 6 ((((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) ∧ 𝑘𝑈) → 0 ≤ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
106 difeq2 3714 . . . . . . . . 9 (𝑘 = 𝑚 → (𝑋𝑘) = (𝑋𝑚))
107106mpteq1d 4729 . . . . . . . 8 (𝑘 = 𝑚 → (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)) = (𝑧 ∈ (𝑋𝑚) ↦ (𝑦𝐷𝑧)))
108107rneqd 5342 . . . . . . 7 (𝑘 = 𝑚 → ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)) = ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑦𝐷𝑧)))
109108infeq1d 8368 . . . . . 6 (𝑘 = 𝑚 → inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) = inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
11093, 94, 105, 109, 49fsumge1 14510 . . . . 5 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ≤ Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
11141, 66, 67, 92, 110ltletrd 10182 . . . 4 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 0 < Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
11240, 111rexlimddv 3031 . . 3 ((𝜑𝑦𝑋) → 0 < Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
11335, 112elrpd 11854 . 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 6371 1 (𝜑𝐹:𝑋⟶ℝ+)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1481   ∈ wcel 1988   ≠ wne 2791  ∀wral 2909  ∃wrex 2910   ∖ cdif 3564   ⊆ wss 3567  ∅c0 3907  ∪ cuni 4427   class class class wbr 4644   ↦ cmpt 4720  ran crn 5105  ⟶wf 5872  ‘cfv 5876  (class class class)co 6635  Fincfn 7940  infcinf 8332  ℝcr 9920  0cc0 9921  +∞cpnf 10056  ℝ*cxr 10058   < clt 10059   ≤ cle 10060  ℝ+crp 11817  [,]cicc 12163  Σcsu 14397  ∞Metcxmt 19712  Metcme 19713  MetOpencmopn 19717  Topctop 20679  Clsdccld 20801  clsccl 20803  Compccmp 21170 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-iin 4514  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-ec 7729  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-sup 8333  df-inf 8334  df-oi 8400  df-card 8750  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-n0 11278  df-z 11363  df-uz 11673  df-q 11774  df-rp 11818  df-xneg 11931  df-xadd 11932  df-xmul 11933  df-ico 12166  df-icc 12167  df-fz 12312  df-fzo 12450  df-seq 12785  df-exp 12844  df-hash 13101  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-abs 13957  df-clim 14200  df-sum 14398  df-topgen 16085  df-psmet 19719  df-xmet 19720  df-met 19721  df-bl 19722  df-mopn 19723  df-top 20680  df-topon 20697  df-bases 20731  df-cld 20804  df-ntr 20805  df-cls 20806 This theorem is referenced by:  lebnumlem2  22742  lebnumlem3  22743
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