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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  cr 9920  0cc0 9921  +∞cpnf 10056  *cxr 10058   < clt 10059  cle 10060  +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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