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Theorem lebnumlem1 22496
Description: Lemma for lebnum 22499. 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 479 . . . 4 ((𝜑𝑦𝑋) → 𝑈 ∈ Fin)
3 lebnum.d . . . . . . . 8 (𝜑𝐷 ∈ (Met‘𝑋))
43ad2antrr 757 . . . . . . 7 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 𝐷 ∈ (Met‘𝑋))
5 difssd 3696 . . . . . . 7 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → (𝑋𝑘) ⊆ 𝑋)
6 lebnum.s . . . . . . . . . . . 12 (𝜑𝑈𝐽)
76adantr 479 . . . . . . . . . . 11 ((𝜑𝑦𝑋) → 𝑈𝐽)
87sselda 3564 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 𝑘𝐽)
9 elssuni 4394 . . . . . . . . . 10 (𝑘𝐽𝑘 𝐽)
108, 9syl 17 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 𝑘 𝐽)
11 metxmet 21887 . . . . . . . . . . . 12 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
123, 11syl 17 . . . . . . . . . . 11 (𝜑𝐷 ∈ (∞Met‘𝑋))
13 lebnum.j . . . . . . . . . . . 12 𝐽 = (MetOpen‘𝐷)
1413mopnuni 21994 . . . . . . . . . . 11 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
1512, 14syl 17 . . . . . . . . . 10 (𝜑𝑋 = 𝐽)
1615ad2antrr 757 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 𝑋 = 𝐽)
1710, 16sseqtr4d 3601 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 𝑘𝑋)
18 lebnumlem1.n . . . . . . . . . . . 12 (𝜑 → ¬ 𝑋𝑈)
19 eleq1 2672 . . . . . . . . . . . . 13 (𝑘 = 𝑋 → (𝑘𝑈𝑋𝑈))
2019notbid 306 . . . . . . . . . . . 12 (𝑘 = 𝑋 → (¬ 𝑘𝑈 ↔ ¬ 𝑋𝑈))
2118, 20syl5ibrcom 235 . . . . . . . . . . 11 (𝜑 → (𝑘 = 𝑋 → ¬ 𝑘𝑈))
2221necon2ad 2793 . . . . . . . . . 10 (𝜑 → (𝑘𝑈𝑘𝑋))
2322adantr 479 . . . . . . . . 9 ((𝜑𝑦𝑋) → (𝑘𝑈𝑘𝑋))
2423imp 443 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 𝑘𝑋)
25 pssdifn0 3894 . . . . . . . 8 ((𝑘𝑋𝑘𝑋) → (𝑋𝑘) ≠ ∅)
2617, 24, 25syl2anc 690 . . . . . . 7 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → (𝑋𝑘) ≠ ∅)
27 eqid 2606 . . . . . . . 8 (𝑦𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) = (𝑦𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
2827metdsre 22392 . . . . . . 7 ((𝐷 ∈ (Met‘𝑋) ∧ (𝑋𝑘) ⊆ 𝑋 ∧ (𝑋𝑘) ≠ ∅) → (𝑦𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶ℝ)
294, 5, 26, 28syl3anc 1317 . . . . . 6 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → (𝑦𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶ℝ)
3027fmpt 6271 . . . . . 6 (∀𝑦𝑋 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ ↔ (𝑦𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶ℝ)
3129, 30sylibr 222 . . . . 5 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → ∀𝑦𝑋 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
32 simplr 787 . . . . 5 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 𝑦𝑋)
33 rsp 2909 . . . . 5 (∀𝑦𝑋 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ → (𝑦𝑋 → inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ))
3431, 32, 33sylc 62 . . . 4 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
352, 34fsumrecl 14255 . . 3 ((𝜑𝑦𝑋) → Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
36 lebnum.u . . . . . . 7 (𝜑𝑋 = 𝑈)
3736eleq2d 2669 . . . . . 6 (𝜑 → (𝑦𝑋𝑦 𝑈))
3837biimpa 499 . . . . 5 ((𝜑𝑦𝑋) → 𝑦 𝑈)
39 eluni2 4367 . . . . 5 (𝑦 𝑈 ↔ ∃𝑚𝑈 𝑦𝑚)
4038, 39sylib 206 . . . 4 ((𝜑𝑦𝑋) → ∃𝑚𝑈 𝑦𝑚)
41 0red 9894 . . . . 5 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 0 ∈ ℝ)
42 simplr 787 . . . . . . 7 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑦𝑋)
43 eqid 2606 . . . . . . . 8 (𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )) = (𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))
4443metdsval 22386 . . . . . . 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 757 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝐷 ∈ (Met‘𝑋))
47 difssd 3696 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (𝑋𝑚) ⊆ 𝑋)
486ad2antrr 757 . . . . . . . . . . . 12 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑈𝐽)
49 simprl 789 . . . . . . . . . . . 12 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑚𝑈)
5048, 49sseldd 3565 . . . . . . . . . . 11 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑚𝐽)
51 elssuni 4394 . . . . . . . . . . 11 (𝑚𝐽𝑚 𝐽)
5250, 51syl 17 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑚 𝐽)
5346, 11, 143syl 18 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑋 = 𝐽)
5452, 53sseqtr4d 3601 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑚𝑋)
55 eleq1 2672 . . . . . . . . . . . . . 14 (𝑚 = 𝑋 → (𝑚𝑈𝑋𝑈))
5655notbid 306 . . . . . . . . . . . . 13 (𝑚 = 𝑋 → (¬ 𝑚𝑈 ↔ ¬ 𝑋𝑈))
5718, 56syl5ibrcom 235 . . . . . . . . . . . 12 (𝜑 → (𝑚 = 𝑋 → ¬ 𝑚𝑈))
5857necon2ad 2793 . . . . . . . . . . 11 (𝜑 → (𝑚𝑈𝑚𝑋))
5958ad2antrr 757 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (𝑚𝑈𝑚𝑋))
6049, 59mpd 15 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑚𝑋)
61 pssdifn0 3894 . . . . . . . . 9 ((𝑚𝑋𝑚𝑋) → (𝑋𝑚) ≠ ∅)
6254, 60, 61syl2anc 690 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (𝑋𝑚) ≠ ∅)
6343metdsre 22392 . . . . . . . 8 ((𝐷 ∈ (Met‘𝑋) ∧ (𝑋𝑚) ⊆ 𝑋 ∧ (𝑋𝑚) ≠ ∅) → (𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )):𝑋⟶ℝ)
6446, 47, 62, 63syl3anc 1317 . . . . . . 7 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )):𝑋⟶ℝ)
6564, 42ffvelrnd 6250 . . . . . 6 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → ((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ∈ ℝ)
6645, 65eqeltrrd 2685 . . . . 5 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
6735adantr 479 . . . . 5 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
6812ad2antrr 757 . . . . . . . . . . 11 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝐷 ∈ (∞Met‘𝑋))
6943metdsf 22387 . . . . . . . . . . 11 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑋𝑚) ⊆ 𝑋) → (𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
7068, 47, 69syl2anc 690 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
7170, 42ffvelrnd 6250 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → ((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ∈ (0[,]+∞))
72 elxrge0 12105 . . . . . . . . 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 206 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ∈ ℝ* ∧ 0 ≤ ((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦)))
7473simprd 477 . . . . . . 7 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 0 ≤ ((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦))
75 elndif 3692 . . . . . . . . . 10 (𝑦𝑚 → ¬ 𝑦 ∈ (𝑋𝑚))
7675ad2antll 760 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → ¬ 𝑦 ∈ (𝑋𝑚))
7753difeq1d 3685 . . . . . . . . . . 11 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (𝑋𝑚) = ( 𝐽𝑚))
7813mopntop 21993 . . . . . . . . . . . . 13 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
7968, 78syl 17 . . . . . . . . . . . 12 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝐽 ∈ Top)
80 eqid 2606 . . . . . . . . . . . . 13 𝐽 = 𝐽
8180opncld 20586 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑚𝐽) → ( 𝐽𝑚) ∈ (Clsd‘𝐽))
8279, 50, 81syl2anc 690 . . . . . . . . . . 11 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → ( 𝐽𝑚) ∈ (Clsd‘𝐽))
8377, 82eqeltrd 2684 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (𝑋𝑚) ∈ (Clsd‘𝐽))
84 cldcls 20595 . . . . . . . . . 10 ((𝑋𝑚) ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘(𝑋𝑚)) = (𝑋𝑚))
8583, 84syl 17 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → ((cls‘𝐽)‘(𝑋𝑚)) = (𝑋𝑚))
8676, 85neleqtrrd 2706 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → ¬ 𝑦 ∈ ((cls‘𝐽)‘(𝑋𝑚)))
8743, 13metdseq0 22393 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑋𝑚) ⊆ 𝑋𝑦𝑋) → (((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) = 0 ↔ 𝑦 ∈ ((cls‘𝐽)‘(𝑋𝑚))))
8868, 47, 42, 87syl3anc 1317 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) = 0 ↔ 𝑦 ∈ ((cls‘𝐽)‘(𝑋𝑚))))
8988necon3abid 2814 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → (((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ≠ 0 ↔ ¬ 𝑦 ∈ ((cls‘𝐽)‘(𝑋𝑚))))
9086, 89mpbird 245 . . . . . . 7 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → ((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦) ≠ 0)
9165, 74, 90ne0gt0d 10022 . . . . . 6 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 0 < ((𝑤𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑤𝐷𝑧)), ℝ*, < ))‘𝑦))
9291, 45breqtrd 4600 . . . . 5 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 0 < inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
931ad2antrr 757 . . . . . 6 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 𝑈 ∈ Fin)
9434adantlr 746 . . . . . 6 ((((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) ∧ 𝑘𝑈) → inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ)
9512ad2antrr 757 . . . . . . . . . . . 12 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 𝐷 ∈ (∞Met‘𝑋))
9627metdsf 22387 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑋𝑘) ⊆ 𝑋) → (𝑦𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
9795, 5, 96syl2anc 690 . . . . . . . . . . 11 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → (𝑦𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
9827fmpt 6271 . . . . . . . . . . 11 (∀𝑦𝑋 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ (0[,]+∞) ↔ (𝑦𝑋 ↦ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )):𝑋⟶(0[,]+∞))
9997, 98sylibr 222 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → ∀𝑦𝑋 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ (0[,]+∞))
100 rsp 2909 . . . . . . . . . 10 (∀𝑦𝑋 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ (0[,]+∞) → (𝑦𝑋 → inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ (0[,]+∞)))
10199, 32, 100sylc 62 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ (0[,]+∞))
102 elxrge0 12105 . . . . . . . . 9 (inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ (0[,]+∞) ↔ (inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ* ∧ 0 ≤ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )))
103101, 102sylib 206 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → (inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ∈ ℝ* ∧ 0 ≤ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )))
104103simprd 477 . . . . . . 7 (((𝜑𝑦𝑋) ∧ 𝑘𝑈) → 0 ≤ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
105104adantlr 746 . . . . . 6 ((((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) ∧ 𝑘𝑈) → 0 ≤ inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
106 difeq2 3680 . . . . . . . . 9 (𝑘 = 𝑚 → (𝑋𝑘) = (𝑋𝑚))
107106mpteq1d 4657 . . . . . . . 8 (𝑘 = 𝑚 → (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)) = (𝑧 ∈ (𝑋𝑚) ↦ (𝑦𝐷𝑧)))
108107rneqd 5258 . . . . . . 7 (𝑘 = 𝑚 → ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)) = ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑦𝐷𝑧)))
109108infeq1d 8240 . . . . . 6 (𝑘 = 𝑚 → inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ) = inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
11093, 94, 105, 109, 49fsumge1 14313 . . . . 5 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → inf(ran (𝑧 ∈ (𝑋𝑚) ↦ (𝑦𝐷𝑧)), ℝ*, < ) ≤ Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
11141, 66, 67, 92, 110ltletrd 10045 . . . 4 (((𝜑𝑦𝑋) ∧ (𝑚𝑈𝑦𝑚)) → 0 < Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
11240, 111rexlimddv 3013 . . 3 ((𝜑𝑦𝑋) → 0 < Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))
11335, 112elrpd 11698 . 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 6274 1 (𝜑𝐹:𝑋⟶ℝ+)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wne 2776  wral 2892  wrex 2893  cdif 3533  wss 3536  c0 3870   cuni 4363   class class class wbr 4574  cmpt 4634  ran crn 5026  wf 5783  cfv 5787  (class class class)co 6524  Fincfn 7815  infcinf 8204  cr 9788  0cc0 9789  +∞cpnf 9924  *cxr 9926   < clt 9927  cle 9928  +crp 11661  [,]cicc 12002  Σcsu 14207  ∞Metcxmt 19495  Metcme 19496  MetOpencmopn 19500  Topctop 20456  Clsdccld 20569  clsccl 20571  Compccmp 20938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-inf2 8395  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866  ax-pre-sup 9867
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-iin 4449  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-se 4985  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-isom 5796  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-oadd 7425  df-er 7603  df-ec 7605  df-map 7720  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-sup 8205  df-inf 8206  df-oi 8272  df-card 8622  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-div 10531  df-nn 10865  df-2 10923  df-3 10924  df-n0 11137  df-z 11208  df-uz 11517  df-q 11618  df-rp 11662  df-xneg 11775  df-xadd 11776  df-xmul 11777  df-ico 12005  df-icc 12006  df-fz 12150  df-fzo 12287  df-seq 12616  df-exp 12675  df-hash 12932  df-cj 13630  df-re 13631  df-im 13632  df-sqrt 13766  df-abs 13767  df-clim 14010  df-sum 14208  df-topgen 15870  df-psmet 19502  df-xmet 19503  df-met 19504  df-bl 19505  df-mopn 19506  df-top 20460  df-bases 20461  df-topon 20462  df-cld 20572  df-ntr 20573  df-cls 20574
This theorem is referenced by:  lebnumlem2  22497  lebnumlem3  22498
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