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Theorem sge0cl 46337
Description: The arbitrary sum of nonnegative extended reals is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
sge0cl.x (𝜑𝑋𝑉)
sge0cl.f (𝜑𝐹:𝑋⟶(0[,]+∞))
Assertion
Ref Expression
sge0cl (𝜑 → (Σ^𝐹) ∈ (0[,]+∞))

Proof of Theorem sge0cl
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6907 . . . . 5 (𝐹 = ∅ → (Σ^𝐹) = (Σ^‘∅))
2 sge00 46332 . . . . . 6 ^‘∅) = 0
32a1i 11 . . . . 5 (𝐹 = ∅ → (Σ^‘∅) = 0)
41, 3eqtrd 2775 . . . 4 (𝐹 = ∅ → (Σ^𝐹) = 0)
5 0e0iccpnf 13496 . . . . 5 0 ∈ (0[,]+∞)
65a1i 11 . . . 4 (𝐹 = ∅ → 0 ∈ (0[,]+∞))
74, 6eqeltrd 2839 . . 3 (𝐹 = ∅ → (Σ^𝐹) ∈ (0[,]+∞))
87adantl 481 . 2 ((𝜑𝐹 = ∅) → (Σ^𝐹) ∈ (0[,]+∞))
9 sge0cl.x . . . . . . 7 (𝜑𝑋𝑉)
109adantr 480 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝑋𝑉)
11 sge0cl.f . . . . . . 7 (𝜑𝐹:𝑋⟶(0[,]+∞))
1211adantr 480 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
13 simpr 484 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
1410, 12, 13sge0pnfval 46329 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) = +∞)
15 pnfel0pnf 45481 . . . . . 6 +∞ ∈ (0[,]+∞)
1615a1i 11 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ (0[,]+∞))
1714, 16eqeltrd 2839 . . . 4 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) ∈ (0[,]+∞))
1817adantlr 715 . . 3 (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) ∈ (0[,]+∞))
19 simpll 767 . . . 4 (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 𝜑)
20 neqne 2946 . . . . 5 𝐹 = ∅ → 𝐹 ≠ ∅)
2120ad2antlr 727 . . . 4 (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹 ≠ ∅)
22 simpr 484 . . . 4 (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
23 0xr 11306 . . . . . 6 0 ∈ ℝ*
2423a1i 11 . . . . 5 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 0 ∈ ℝ*)
25 pnfxr 11313 . . . . . 6 +∞ ∈ ℝ*
2625a1i 11 . . . . 5 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → +∞ ∈ ℝ*)
279adantr 480 . . . . . . . 8 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝑋𝑉)
2811adantr 480 . . . . . . . . 9 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
29 simpr 484 . . . . . . . . 9 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
3028, 29fge0iccico 46326 . . . . . . . 8 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,)+∞))
3127, 30sge0reval 46328 . . . . . . 7 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
32 elinel2 4212 . . . . . . . . . . . . 13 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
3332adantl 481 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
3411ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝐹:𝑋⟶(0[,]+∞))
35 elinel1 4211 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ 𝒫 𝑋)
36 elpwi 4612 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
3735, 36syl 17 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥𝑋)
3837adantl 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥𝑋)
3938adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑥𝑋)
40 simpr 484 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑥)
4139, 40sseldd 3996 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑋)
4234, 41ffvelcdmd 7105 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,]+∞))
4342adantllr 719 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,]+∞))
44 nne 2942 . . . . . . . . . . . . . . . . . 18 (¬ (𝐹𝑦) ≠ +∞ ↔ (𝐹𝑦) = +∞)
4544biimpi 216 . . . . . . . . . . . . . . . . 17 (¬ (𝐹𝑦) ≠ +∞ → (𝐹𝑦) = +∞)
4645eqcomd 2741 . . . . . . . . . . . . . . . 16 (¬ (𝐹𝑦) ≠ +∞ → +∞ = (𝐹𝑦))
4746adantl 481 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ ¬ (𝐹𝑦) ≠ +∞) → +∞ = (𝐹𝑦))
4811ffund 6741 . . . . . . . . . . . . . . . . . 18 (𝜑 → Fun 𝐹)
49483ad2ant1 1132 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → Fun 𝐹)
50413impa 1109 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → 𝑦𝑋)
5111fdmd 6747 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → dom 𝐹 = 𝑋)
5251eqcomd 2741 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑋 = dom 𝐹)
53523ad2ant1 1132 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → 𝑋 = dom 𝐹)
5450, 53eleqtrd 2841 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → 𝑦 ∈ dom 𝐹)
55 fvelrn 7096 . . . . . . . . . . . . . . . . 17 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (𝐹𝑦) ∈ ran 𝐹)
5649, 54, 55syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ran 𝐹)
5756ad5ant134 1366 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ ¬ (𝐹𝑦) ≠ +∞) → (𝐹𝑦) ∈ ran 𝐹)
5847, 57eqeltrd 2839 . . . . . . . . . . . . . 14 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ ¬ (𝐹𝑦) ≠ +∞) → +∞ ∈ ran 𝐹)
5929ad3antrrr 730 . . . . . . . . . . . . . 14 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ ¬ (𝐹𝑦) ≠ +∞) → ¬ +∞ ∈ ran 𝐹)
6058, 59condan 818 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ≠ +∞)
61 ge0xrre 45484 . . . . . . . . . . . . 13 (((𝐹𝑦) ∈ (0[,]+∞) ∧ (𝐹𝑦) ≠ +∞) → (𝐹𝑦) ∈ ℝ)
6243, 60, 61syl2anc 584 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℝ)
6333, 62fsumrecl 15767 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) ∈ ℝ)
6463ralrimiva 3144 . . . . . . . . . 10 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)Σ𝑦𝑥 (𝐹𝑦) ∈ ℝ)
65 eqid 2735 . . . . . . . . . . 11 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
6665rnmptss 7143 . . . . . . . . . 10 (∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)Σ𝑦𝑥 (𝐹𝑦) ∈ ℝ → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ)
6764, 66syl 17 . . . . . . . . 9 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ)
68 ressxr 11303 . . . . . . . . . 10 ℝ ⊆ ℝ*
6968a1i 11 . . . . . . . . 9 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ℝ ⊆ ℝ*)
7067, 69sstrd 4006 . . . . . . . 8 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ*)
71 supxrcl 13354 . . . . . . . 8 (ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ* → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) ∈ ℝ*)
7270, 71syl 17 . . . . . . 7 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) ∈ ℝ*)
7331, 72eqeltrd 2839 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) ∈ ℝ*)
7473adantlr 715 . . . . 5 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) ∈ ℝ*)
7552adantr 480 . . . . . . . . 9 ((𝜑𝐹 ≠ ∅) → 𝑋 = dom 𝐹)
76 neneq 2944 . . . . . . . . . . . 12 (𝐹 ≠ ∅ → ¬ 𝐹 = ∅)
7776adantl 481 . . . . . . . . . . 11 ((𝜑𝐹 ≠ ∅) → ¬ 𝐹 = ∅)
78 frel 6742 . . . . . . . . . . . . . 14 (𝐹:𝑋⟶(0[,]+∞) → Rel 𝐹)
7911, 78syl 17 . . . . . . . . . . . . 13 (𝜑 → Rel 𝐹)
8079adantr 480 . . . . . . . . . . . 12 ((𝜑𝐹 ≠ ∅) → Rel 𝐹)
81 reldm0 5941 . . . . . . . . . . . 12 (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
8280, 81syl 17 . . . . . . . . . . 11 ((𝜑𝐹 ≠ ∅) → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
8377, 82mtbid 324 . . . . . . . . . 10 ((𝜑𝐹 ≠ ∅) → ¬ dom 𝐹 = ∅)
8483neqned 2945 . . . . . . . . 9 ((𝜑𝐹 ≠ ∅) → dom 𝐹 ≠ ∅)
8575, 84eqnetrd 3006 . . . . . . . 8 ((𝜑𝐹 ≠ ∅) → 𝑋 ≠ ∅)
86 n0 4359 . . . . . . . 8 (𝑋 ≠ ∅ ↔ ∃𝑧 𝑧𝑋)
8785, 86sylib 218 . . . . . . 7 ((𝜑𝐹 ≠ ∅) → ∃𝑧 𝑧𝑋)
8887adantr 480 . . . . . 6 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → ∃𝑧 𝑧𝑋)
8923a1i 11 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → 0 ∈ ℝ*)
9011ffvelcdmda 7104 . . . . . . . . . . . . 13 ((𝜑𝑧𝑋) → (𝐹𝑧) ∈ (0[,]+∞))
9190adantlr 715 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ (0[,]+∞))
92 nne 2942 . . . . . . . . . . . . . . . . 17 (¬ (𝐹𝑧) ≠ +∞ ↔ (𝐹𝑧) = +∞)
9392biimpi 216 . . . . . . . . . . . . . . . 16 (¬ (𝐹𝑧) ≠ +∞ → (𝐹𝑧) = +∞)
9493eqcomd 2741 . . . . . . . . . . . . . . 15 (¬ (𝐹𝑧) ≠ +∞ → +∞ = (𝐹𝑧))
9594adantl 481 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) ∧ ¬ (𝐹𝑧) ≠ +∞) → +∞ = (𝐹𝑧))
9611adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑋) → 𝐹:𝑋⟶(0[,]+∞))
9796ffund 6741 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝑋) → Fun 𝐹)
98 simpr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑋) → 𝑧𝑋)
9952adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑋) → 𝑋 = dom 𝐹)
10098, 99eleqtrd 2841 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝑋) → 𝑧 ∈ dom 𝐹)
101 fvelrn 7096 . . . . . . . . . . . . . . . . 17 ((Fun 𝐹𝑧 ∈ dom 𝐹) → (𝐹𝑧) ∈ ran 𝐹)
10297, 100, 101syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑋) → (𝐹𝑧) ∈ ran 𝐹)
103102adantlr 715 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ran 𝐹)
104103adantr 480 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) ∧ ¬ (𝐹𝑧) ≠ +∞) → (𝐹𝑧) ∈ ran 𝐹)
10595, 104eqeltrd 2839 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) ∧ ¬ (𝐹𝑧) ≠ +∞) → +∞ ∈ ran 𝐹)
10629ad2antrr 726 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) ∧ ¬ (𝐹𝑧) ≠ +∞) → ¬ +∞ ∈ ran 𝐹)
107105, 106condan 818 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ≠ +∞)
108 ge0xrre 45484 . . . . . . . . . . . 12 (((𝐹𝑧) ∈ (0[,]+∞) ∧ (𝐹𝑧) ≠ +∞) → (𝐹𝑧) ∈ ℝ)
10991, 107, 108syl2anc 584 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ℝ)
110109rexrd 11309 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ℝ*)
11173adantr 480 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (Σ^𝐹) ∈ ℝ*)
11223a1i 11 . . . . . . . . . . . 12 ((𝜑𝑧𝑋) → 0 ∈ ℝ*)
11325a1i 11 . . . . . . . . . . . 12 ((𝜑𝑧𝑋) → +∞ ∈ ℝ*)
114 iccgelb 13440 . . . . . . . . . . . 12 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝐹𝑧) ∈ (0[,]+∞)) → 0 ≤ (𝐹𝑧))
115112, 113, 90, 114syl3anc 1370 . . . . . . . . . . 11 ((𝜑𝑧𝑋) → 0 ≤ (𝐹𝑧))
116115adantlr 715 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → 0 ≤ (𝐹𝑧))
11770adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ*)
118 snelpwi 5454 . . . . . . . . . . . . . . . 16 (𝑧𝑋 → {𝑧} ∈ 𝒫 𝑋)
119 snfi 9082 . . . . . . . . . . . . . . . . 17 {𝑧} ∈ Fin
120119a1i 11 . . . . . . . . . . . . . . . 16 (𝑧𝑋 → {𝑧} ∈ Fin)
121118, 120elind 4210 . . . . . . . . . . . . . . 15 (𝑧𝑋 → {𝑧} ∈ (𝒫 𝑋 ∩ Fin))
122121adantl 481 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → {𝑧} ∈ (𝒫 𝑋 ∩ Fin))
123 simpr 484 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → 𝑧𝑋)
124109recnd 11287 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ℂ)
125 fveq2 6907 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝐹𝑦) = (𝐹𝑧))
126125sumsn 15779 . . . . . . . . . . . . . . . 16 ((𝑧𝑋 ∧ (𝐹𝑧) ∈ ℂ) → Σ𝑦 ∈ {𝑧} (𝐹𝑦) = (𝐹𝑧))
127123, 124, 126syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → Σ𝑦 ∈ {𝑧} (𝐹𝑦) = (𝐹𝑧))
128127eqcomd 2741 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) = Σ𝑦 ∈ {𝑧} (𝐹𝑦))
129 sumeq1 15722 . . . . . . . . . . . . . . 15 (𝑥 = {𝑧} → Σ𝑦𝑥 (𝐹𝑦) = Σ𝑦 ∈ {𝑧} (𝐹𝑦))
130129rspceeqv 3645 . . . . . . . . . . . . . 14 (({𝑧} ∈ (𝒫 𝑋 ∩ Fin) ∧ (𝐹𝑧) = Σ𝑦 ∈ {𝑧} (𝐹𝑦)) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐹𝑧) = Σ𝑦𝑥 (𝐹𝑦))
131122, 128, 130syl2anc 584 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐹𝑧) = Σ𝑦𝑥 (𝐹𝑦))
13265elrnmpt 5972 . . . . . . . . . . . . . 14 ((𝐹𝑧) ∈ (0[,]+∞) → ((𝐹𝑧) ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐹𝑧) = Σ𝑦𝑥 (𝐹𝑦)))
13391, 132syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐹𝑧) = Σ𝑦𝑥 (𝐹𝑦)))
134131, 133mpbird 257 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
135 supxrub 13363 . . . . . . . . . . . 12 ((ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ* ∧ (𝐹𝑧) ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))) → (𝐹𝑧) ≤ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
136117, 134, 135syl2anc 584 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ≤ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
13731eqcomd 2741 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) = (Σ^𝐹))
138137adantr 480 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) = (Σ^𝐹))
139136, 138breqtrd 5174 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → (𝐹𝑧) ≤ (Σ^𝐹))
14089, 110, 111, 116, 139xrletrd 13201 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑧𝑋) → 0 ≤ (Σ^𝐹))
141140ex 412 . . . . . . . 8 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (𝑧𝑋 → 0 ≤ (Σ^𝐹)))
142141adantlr 715 . . . . . . 7 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (𝑧𝑋 → 0 ≤ (Σ^𝐹)))
143142exlimdv 1931 . . . . . 6 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (∃𝑧 𝑧𝑋 → 0 ≤ (Σ^𝐹)))
14488, 143mpd 15 . . . . 5 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → 0 ≤ (Σ^𝐹))
145 pnfge 13170 . . . . . . 7 ((Σ^𝐹) ∈ ℝ* → (Σ^𝐹) ≤ +∞)
14673, 145syl 17 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) ≤ +∞)
147146adantlr 715 . . . . 5 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) ≤ +∞)
14824, 26, 74, 144, 147eliccxrd 45480 . . . 4 (((𝜑𝐹 ≠ ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) ∈ (0[,]+∞))
14919, 21, 22, 148syl21anc 838 . . 3 (((𝜑 ∧ ¬ 𝐹 = ∅) ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) ∈ (0[,]+∞))
15018, 149pm2.61dan 813 . 2 ((𝜑 ∧ ¬ 𝐹 = ∅) → (Σ^𝐹) ∈ (0[,]+∞))
1518, 150pm2.61dan 813 1 (𝜑 → (Σ^𝐹) ∈ (0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  wne 2938  wral 3059  wrex 3068  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631   class class class wbr 5148  cmpt 5231  dom cdm 5689  ran crn 5690  Rel wrel 5694  Fun wfun 6557  wf 6559  cfv 6563  (class class class)co 7431  Fincfn 8984  supcsup 9478  cc 11151  cr 11152  0cc0 11153  +∞cpnf 11290  *cxr 11292   < clt 11293  cle 11294  [,]cicc 13387  Σcsu 15719  Σ^csumge0 46318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-sum 15720  df-sumge0 46319
This theorem is referenced by:  sge0ge0  46340  sge0xrcl  46341  sge0split  46365  sge0iunmptlemre  46371  sge0iunmpt  46374  sge0nemnf  46376  sge0clmpt  46381  sge0isum  46383  psmeasure  46427  ovnsupge0  46513  ovnsubaddlem1  46526  sge0hsphoire  46545  hoidmvlelem1  46551  hspmbllem2  46583
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