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Theorem esumpcvgval 32677
Description: The value of the extended sum when the corresponding series sum is convergent. (Contributed by Thierry Arnoux, 31-Jul-2017.)
Hypotheses
Ref Expression
esumpcvgval.1 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
esumpcvgval.2 (𝑘 = 𝑙𝐴 = 𝐵)
esumpcvgval.3 (𝜑 → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴) ∈ dom ⇝ )
Assertion
Ref Expression
esumpcvgval (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴)
Distinct variable groups:   𝑘,𝑙,𝑛   𝐴,𝑙,𝑛   𝐵,𝑘,𝑛   𝜑,𝑘,𝑛
Allowed substitution hints:   𝜑(𝑙)   𝐴(𝑘)   𝐵(𝑙)

Proof of Theorem esumpcvgval
Dummy variables 𝑠 𝑥 𝑦 𝑧 𝑏 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xrltso 13060 . . . 4 < Or ℝ*
21a1i 11 . . 3 (𝜑 → < Or ℝ*)
3 nnuz 12806 . . . . 5 ℕ = (ℤ‘1)
4 1zzd 12534 . . . . 5 (𝜑 → 1 ∈ ℤ)
5 esumpcvgval.1 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
6 esumpcvgval.2 . . . . . . . . . . . 12 (𝑘 = 𝑙𝐴 = 𝐵)
7 eqcom 2743 . . . . . . . . . . . 12 (𝑘 = 𝑙𝑙 = 𝑘)
8 eqcom 2743 . . . . . . . . . . . 12 (𝐴 = 𝐵𝐵 = 𝐴)
96, 7, 83imtr3i 290 . . . . . . . . . . 11 (𝑙 = 𝑘𝐵 = 𝐴)
109cbvmptv 5218 . . . . . . . . . 10 (𝑙 ∈ ℕ ↦ 𝐵) = (𝑘 ∈ ℕ ↦ 𝐴)
115, 10fmptd 7062 . . . . . . . . 9 (𝜑 → (𝑙 ∈ ℕ ↦ 𝐵):ℕ⟶(0[,)+∞))
1211ffvelcdmda 7035 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞))
13 elrege0 13371 . . . . . . . . 9 (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞) ↔ (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥)))
1413simplbi 498 . . . . . . . 8 (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ ℝ)
1512, 14syl 17 . . . . . . 7 ((𝜑𝑥 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ ℝ)
163, 4, 15serfre 13937 . . . . . 6 (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)):ℕ⟶ℝ)
1711adantr 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑙 ∈ ℕ ↦ 𝐵):ℕ⟶(0[,)+∞))
18 simpr 485 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
1918peano2nnd 12170 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
2017, 19ffvelcdmd 7036 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞))
21 elrege0 13371 . . . . . . . . . 10 (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞) ↔ (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ ℝ ∧ 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
2221simprbi 497 . . . . . . . . 9 (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞) → 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)))
2320, 22syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)))
2416ffvelcdmda 7035 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ∈ ℝ)
2521simplbi 498 . . . . . . . . . 10 (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞) → ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ ℝ)
2620, 25syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ ℝ)
2724, 26addge01d 11743 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ↔ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) + ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)))))
2823, 27mpbid 231 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) + ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
2918, 3eleqtrdi 2848 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
30 seqp1 13921 . . . . . . . 8 (𝑛 ∈ (ℤ‘1) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘(𝑛 + 1)) = ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) + ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
3129, 30syl 17 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘(𝑛 + 1)) = ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) + ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
3228, 31breqtrrd 5133 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘(𝑛 + 1)))
33 simpr 485 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
3410fvmpt2 6959 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ 𝐴 ∈ (0[,)+∞)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
3533, 5, 34syl2anc 584 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
36 rge0ssre 13373 . . . . . . . . 9 (0[,)+∞) ⊆ ℝ
3736, 5sselid 3942 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
3816feqmptd 6910 . . . . . . . . . 10 (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)))
39 simpll 765 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
40 elfznn 13470 . . . . . . . . . . . . . . 15 (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
4140adantl 482 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
4239, 41, 35syl2anc 584 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
4337recnd 11183 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ ℂ)
4439, 41, 43syl2anc 584 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ ℂ)
4542, 29, 44fsumser 15615 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
4645eqcomd 2742 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) = Σ𝑘 ∈ (1...𝑛)𝐴)
4746mpteq2dva 5205 . . . . . . . . . 10 (𝜑 → (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴))
4838, 47eqtr2d 2777 . . . . . . . . 9 (𝜑 → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴) = seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)))
49 esumpcvgval.3 . . . . . . . . 9 (𝜑 → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴) ∈ dom ⇝ )
5048, 49eqeltrrd 2839 . . . . . . . 8 (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ∈ dom ⇝ )
513, 4, 35, 37, 50isumrecl 15650 . . . . . . 7 (𝜑 → Σ𝑘 ∈ ℕ 𝐴 ∈ ℝ)
52 1zzd 12534 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 1 ∈ ℤ)
53 fzfid 13878 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
54 fzssuz 13482 . . . . . . . . . . . 12 (1...𝑛) ⊆ (ℤ‘1)
5554, 3sseqtrri 3981 . . . . . . . . . . 11 (1...𝑛) ⊆ ℕ
5655a1i 11 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
5735adantlr 713 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
5837adantlr 713 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
595adantlr 713 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
60 elrege0 13371 . . . . . . . . . . . 12 (𝐴 ∈ (0[,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
6160simprbi 497 . . . . . . . . . . 11 (𝐴 ∈ (0[,)+∞) → 0 ≤ 𝐴)
6259, 61syl 17 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 0 ≤ 𝐴)
6350adantr 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ∈ dom ⇝ )
643, 52, 53, 56, 57, 58, 62, 63isumless 15730 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 ≤ Σ𝑘 ∈ ℕ 𝐴)
6545, 64eqbrtrrd 5129 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴)
6665ralrimiva 3143 . . . . . . 7 (𝜑 → ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴)
67 brralrspcev 5165 . . . . . . 7 ((Σ𝑘 ∈ ℕ 𝐴 ∈ ℝ ∧ ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴) → ∃𝑠 ∈ ℝ ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠)
6851, 66, 67syl2anc 584 . . . . . 6 (𝜑 → ∃𝑠 ∈ ℝ ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠)
693, 4, 16, 32, 68climsup 15554 . . . . 5 (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⇝ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
703, 4, 69, 24climrecl 15465 . . . 4 (𝜑 → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ)
7170rexrd 11205 . . 3 (𝜑 → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ*)
72 eqid 2736 . . . . . . 7 (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴) = (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)
73 sumex 15572 . . . . . . 7 Σ𝑘𝑏 𝐴 ∈ V
7472, 73elrnmpti 5915 . . . . . 6 (𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘𝑏 𝐴)
75 ssnnssfz 31690 . . . . . . . . . 10 (𝑏 ∈ (𝒫 ℕ ∩ Fin) → ∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚))
76 fzfid 13878 . . . . . . . . . . . . . 14 ((𝜑𝑏 ⊆ (1...𝑚)) → (1...𝑚) ∈ Fin)
77 elfznn 13470 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ)
7877, 5sylan2 593 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (1...𝑚)) → 𝐴 ∈ (0[,)+∞))
7960simplbi 498 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (0[,)+∞) → 𝐴 ∈ ℝ)
8078, 79syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ)
8180adantlr 713 . . . . . . . . . . . . . 14 (((𝜑𝑏 ⊆ (1...𝑚)) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ)
8278, 61syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...𝑚)) → 0 ≤ 𝐴)
8382adantlr 713 . . . . . . . . . . . . . 14 (((𝜑𝑏 ⊆ (1...𝑚)) ∧ 𝑘 ∈ (1...𝑚)) → 0 ≤ 𝐴)
84 simpr 485 . . . . . . . . . . . . . 14 ((𝜑𝑏 ⊆ (1...𝑚)) → 𝑏 ⊆ (1...𝑚))
8576, 81, 83, 84fsumless 15681 . . . . . . . . . . . . 13 ((𝜑𝑏 ⊆ (1...𝑚)) → Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
8685ex 413 . . . . . . . . . . . 12 (𝜑 → (𝑏 ⊆ (1...𝑚) → Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
8786reximdv 3167 . . . . . . . . . . 11 (𝜑 → (∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚) → ∃𝑚 ∈ ℕ Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
8887imp 407 . . . . . . . . . 10 ((𝜑 ∧ ∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚)) → ∃𝑚 ∈ ℕ Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
8975, 88sylan2 593 . . . . . . . . 9 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → ∃𝑚 ∈ ℕ Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
90 breq1 5108 . . . . . . . . . 10 (𝑥 = Σ𝑘𝑏 𝐴 → (𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴 ↔ Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
9190rexbidv 3175 . . . . . . . . 9 (𝑥 = Σ𝑘𝑏 𝐴 → (∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴 ↔ ∃𝑚 ∈ ℕ Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
9289, 91syl5ibrcom 246 . . . . . . . 8 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (𝑥 = Σ𝑘𝑏 𝐴 → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
9392rexlimdva 3152 . . . . . . 7 (𝜑 → (∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘𝑏 𝐴 → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
9493imp 407 . . . . . 6 ((𝜑 ∧ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘𝑏 𝐴) → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
9574, 94sylan2b 594 . . . . 5 ((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
96 simpr 485 . . . . . . . . . 10 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘𝑏 𝐴) → 𝑥 = Σ𝑘𝑏 𝐴)
97 inss2 4189 . . . . . . . . . . . . 13 (𝒫 ℕ ∩ Fin) ⊆ Fin
98 simpr 485 . . . . . . . . . . . . 13 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ (𝒫 ℕ ∩ Fin))
9997, 98sselid 3942 . . . . . . . . . . . 12 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ Fin)
100 simpll 765 . . . . . . . . . . . . . 14 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝜑)
101 inss1 4188 . . . . . . . . . . . . . . . . 17 (𝒫 ℕ ∩ Fin) ⊆ 𝒫 ℕ
102 simplr 767 . . . . . . . . . . . . . . . . 17 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝑏 ∈ (𝒫 ℕ ∩ Fin))
103101, 102sselid 3942 . . . . . . . . . . . . . . . 16 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝑏 ∈ 𝒫 ℕ)
104103elpwid 4569 . . . . . . . . . . . . . . 15 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝑏 ⊆ ℕ)
105 simpr 485 . . . . . . . . . . . . . . 15 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝑘𝑏)
106104, 105sseldd 3945 . . . . . . . . . . . . . 14 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝑘 ∈ ℕ)
107100, 106, 5syl2anc 584 . . . . . . . . . . . . 13 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝐴 ∈ (0[,)+∞))
108107, 79syl 17 . . . . . . . . . . . 12 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝐴 ∈ ℝ)
10999, 108fsumrecl 15619 . . . . . . . . . . 11 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → Σ𝑘𝑏 𝐴 ∈ ℝ)
110109adantr 481 . . . . . . . . . 10 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘𝑏 𝐴) → Σ𝑘𝑏 𝐴 ∈ ℝ)
11196, 110eqeltrd 2838 . . . . . . . . 9 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘𝑏 𝐴) → 𝑥 ∈ ℝ)
112111r19.29an 3155 . . . . . . . 8 ((𝜑 ∧ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘𝑏 𝐴) → 𝑥 ∈ ℝ)
11374, 112sylan2b 594 . . . . . . 7 ((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) → 𝑥 ∈ ℝ)
114113adantr 481 . . . . . 6 (((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → 𝑥 ∈ ℝ)
115 fzfid 13878 . . . . . . . 8 (𝜑 → (1...𝑚) ∈ Fin)
116115, 80fsumrecl 15619 . . . . . . 7 (𝜑 → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ℝ)
117116ad2antrr 724 . . . . . 6 (((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ℝ)
11870ad2antrr 724 . . . . . 6 (((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ)
119 simprr 771 . . . . . 6 (((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
12016frnd 6676 . . . . . . . 8 (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ)
121120ad2antrr 724 . . . . . . 7 (((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ)
122 1nn 12164 . . . . . . . . . 10 1 ∈ ℕ
123122ne0ii 4297 . . . . . . . . 9 ℕ ≠ ∅
124 dm0rn0 5880 . . . . . . . . . . 11 (dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅)
12516fdmd 6679 . . . . . . . . . . . 12 (𝜑 → dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ℕ)
126125eqeq1d 2738 . . . . . . . . . . 11 (𝜑 → (dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ℕ = ∅))
127124, 126bitr3id 284 . . . . . . . . . 10 (𝜑 → (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ℕ = ∅))
128127necon3bid 2988 . . . . . . . . 9 (𝜑 → (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅ ↔ ℕ ≠ ∅))
129123, 128mpbiri 257 . . . . . . . 8 (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅)
130129ad2antrr 724 . . . . . . 7 (((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅)
131 1z 12533 . . . . . . . . . . . . . . . 16 1 ∈ ℤ
132 seqfn 13918 . . . . . . . . . . . . . . . 16 (1 ∈ ℤ → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn (ℤ‘1))
133131, 132ax-mp 5 . . . . . . . . . . . . . . 15 seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn (ℤ‘1)
1343fneq2i 6600 . . . . . . . . . . . . . . 15 (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ ↔ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn (ℤ‘1))
135133, 134mpbir 230 . . . . . . . . . . . . . 14 seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ
136 dffn5 6901 . . . . . . . . . . . . . 14 (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ ↔ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)))
137135, 136mpbi 229 . . . . . . . . . . . . 13 seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
138 fvex 6855 . . . . . . . . . . . . 13 (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ∈ V
139137, 138elrnmpti 5915 . . . . . . . . . . . 12 (𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
140 r19.29 3117 . . . . . . . . . . . . 13 ((∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ ∃𝑛 ∈ ℕ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → ∃𝑛 ∈ ℕ ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)))
141 breq1 5108 . . . . . . . . . . . . . . 15 (𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → (𝑧𝑠 ↔ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠))
142141biimparc 480 . . . . . . . . . . . . . 14 (((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑧𝑠)
143142rexlimivw 3148 . . . . . . . . . . . . 13 (∃𝑛 ∈ ℕ ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑧𝑠)
144140, 143syl 17 . . . . . . . . . . . 12 ((∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ ∃𝑛 ∈ ℕ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑧𝑠)
145139, 144sylan2b 594 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))) → 𝑧𝑠)
146145ralrimiva 3143 . . . . . . . . . 10 (∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 → ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧𝑠)
147146reximi 3087 . . . . . . . . 9 (∃𝑠 ∈ ℝ ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧𝑠)
14868, 147syl 17 . . . . . . . 8 (𝜑 → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧𝑠)
149148ad2antrr 724 . . . . . . 7 (((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧𝑠)
150 simpr 485 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
151 simpll 765 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝜑)
15277adantl 482 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ ℕ)
153151, 152, 35syl2anc 584 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
154150, 3eleqtrdi 2848 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ (ℤ‘1))
155151, 152, 5syl2anc 584 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ (0[,)+∞))
156155, 79syl 17 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ)
157156recnd 11183 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℂ)
158153, 154, 157fsumser 15615 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚))
159 fveq2 6842 . . . . . . . . . . 11 (𝑛 = 𝑚 → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚))
160159rspceeqv 3595 . . . . . . . . . 10 ((𝑚 ∈ ℕ ∧ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚)) → ∃𝑛 ∈ ℕ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
161150, 158, 160syl2anc 584 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → ∃𝑛 ∈ ℕ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
162137, 138elrnmpti 5915 . . . . . . . . 9 𝑘 ∈ (1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
163161, 162sylibr 233 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)))
164163ad2ant2r 745 . . . . . . 7 (((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)))
165 suprub 12116 . . . . . . 7 (((ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ ∧ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅ ∧ ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧𝑠) ∧ Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))) → Σ𝑘 ∈ (1...𝑚)𝐴 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
166121, 130, 149, 164, 165syl31anc 1373 . . . . . 6 (((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → Σ𝑘 ∈ (1...𝑚)𝐴 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
167114, 117, 118, 119, 166letrd 11312 . . . . 5 (((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → 𝑥 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
16895, 167rexlimddv 3158 . . . 4 ((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) → 𝑥 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
16970adantr 481 . . . . 5 ((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ)
170113, 169lenltd 11301 . . . 4 ((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) → (𝑥 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ ¬ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) < 𝑥))
171168, 170mpbid 231 . . 3 ((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) → ¬ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) < 𝑥)
172 simpr1r 1231 . . . . . . 7 ((𝜑 ∧ ((𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0 ≤ 𝑥𝑥 = +∞)) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
1731723anassrs 1360 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
17471ad3antrrr 728 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ*)
175 pnfnlt 13049 . . . . . . . 8 (sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ* → ¬ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
176174, 175syl 17 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → ¬ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
177 breq1 5108 . . . . . . . . 9 (𝑥 = +∞ → (𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )))
178177notbid 317 . . . . . . . 8 (𝑥 = +∞ → (¬ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ ¬ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )))
179178adantl 482 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → (¬ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ ¬ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )))
180176, 179mpbird 256 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → ¬ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
181173, 180pm2.21dd 194 . . . . 5 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)𝑥 < 𝑦)
182 simplll 773 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝜑)
183 simpr1l 1230 . . . . . . . 8 ((𝜑 ∧ ((𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0 ≤ 𝑥𝑥 < +∞)) → 𝑥 ∈ ℝ*)
1841833anassrs 1360 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 ∈ ℝ*)
185 simplr 767 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 0 ≤ 𝑥)
186 simpr 485 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 < +∞)
187 0xr 11202 . . . . . . . 8 0 ∈ ℝ*
188 pnfxr 11209 . . . . . . . 8 +∞ ∈ ℝ*
189 elico1 13307 . . . . . . . 8 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥𝑥 < +∞)))
190187, 188, 189mp2an 690 . . . . . . 7 (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥𝑥 < +∞))
191184, 185, 186, 190syl3anbrc 1343 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 ∈ (0[,)+∞))
192 simpr1r 1231 . . . . . . 7 ((𝜑 ∧ ((𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0 ≤ 𝑥𝑥 < +∞)) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
1931923anassrs 1360 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
194120adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ)
195129adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅)
196148adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧𝑠)
197194, 195, 1963jca 1128 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ ∧ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅ ∧ ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧𝑠))
198 simprl 769 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → 𝑥 ∈ (0[,)+∞))
19936, 198sselid 3942 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → 𝑥 ∈ ℝ)
200 simprr 771 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
201 suprlub 12119 . . . . . . . . 9 (((ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ ∧ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅ ∧ ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧𝑠) ∧ 𝑥 ∈ ℝ) → (𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ ∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦))
202201biimpa 477 . . . . . . . 8 ((((ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ ∧ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅ ∧ ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧𝑠) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) → ∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦)
203197, 199, 200, 202syl21anc 836 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦)
20440ssriv 3948 . . . . . . . . . . . . . . . . 17 (1...𝑛) ⊆ ℕ
205 ovex 7390 . . . . . . . . . . . . . . . . . 18 (1...𝑛) ∈ V
206205elpw 4564 . . . . . . . . . . . . . . . . 17 ((1...𝑛) ∈ 𝒫 ℕ ↔ (1...𝑛) ⊆ ℕ)
207204, 206mpbir 230 . . . . . . . . . . . . . . . 16 (1...𝑛) ∈ 𝒫 ℕ
208 fzfi 13877 . . . . . . . . . . . . . . . 16 (1...𝑛) ∈ Fin
209 elin 3926 . . . . . . . . . . . . . . . 16 ((1...𝑛) ∈ (𝒫 ℕ ∩ Fin) ↔ ((1...𝑛) ∈ 𝒫 ℕ ∧ (1...𝑛) ∈ Fin))
210207, 208, 209mpbir2an 709 . . . . . . . . . . . . . . 15 (1...𝑛) ∈ (𝒫 ℕ ∩ Fin)
211210a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → (1...𝑛) ∈ (𝒫 ℕ ∩ Fin))
212 simpr 485 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
21345adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → Σ𝑘 ∈ (1...𝑛)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
214212, 213eqtr4d 2779 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑦 = Σ𝑘 ∈ (1...𝑛)𝐴)
215 sumeq1 15573 . . . . . . . . . . . . . . 15 (𝑏 = (1...𝑛) → Σ𝑘𝑏 𝐴 = Σ𝑘 ∈ (1...𝑛)𝐴)
216215rspceeqv 3595 . . . . . . . . . . . . . 14 (((1...𝑛) ∈ (𝒫 ℕ ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ (1...𝑛)𝐴) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘𝑏 𝐴)
217211, 214, 216syl2anc 584 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘𝑏 𝐴)
218217ex 413 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘𝑏 𝐴))
219218rexlimdva 3152 . . . . . . . . . . 11 (𝜑 → (∃𝑛 ∈ ℕ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘𝑏 𝐴))
220137, 138elrnmpti 5915 . . . . . . . . . . 11 (𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
22172, 73elrnmpti 5915 . . . . . . . . . . 11 (𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘𝑏 𝐴)
222219, 220, 2213imtr4g 295 . . . . . . . . . 10 (𝜑 → (𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) → 𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)))
223222ssrdv 3950 . . . . . . . . 9 (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴))
224 ssrexv 4011 . . . . . . . . 9 (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴) → (∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦 → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)𝑥 < 𝑦))
225223, 224syl 17 . . . . . . . 8 (𝜑 → (∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦 → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)𝑥 < 𝑦))
226225imp 407 . . . . . . 7 ((𝜑 ∧ ∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)𝑥 < 𝑦)
227203, 226syldan 591 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)𝑥 < 𝑦)
228182, 191, 193, 227syl12anc 835 . . . . 5 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)𝑥 < 𝑦)
229 simplrl 775 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) → 𝑥 ∈ ℝ*)
230 xrlelttric 31657 . . . . . . . 8 ((+∞ ∈ ℝ*𝑥 ∈ ℝ*) → (+∞ ≤ 𝑥𝑥 < +∞))
231188, 230mpan 688 . . . . . . 7 (𝑥 ∈ ℝ* → (+∞ ≤ 𝑥𝑥 < +∞))
232 xgepnf 13084 . . . . . . . 8 (𝑥 ∈ ℝ* → (+∞ ≤ 𝑥𝑥 = +∞))
233232orbi1d 915 . . . . . . 7 (𝑥 ∈ ℝ* → ((+∞ ≤ 𝑥𝑥 < +∞) ↔ (𝑥 = +∞ ∨ 𝑥 < +∞)))
234231, 233mpbid 231 . . . . . 6 (𝑥 ∈ ℝ* → (𝑥 = +∞ ∨ 𝑥 < +∞))
235229, 234syl 17 . . . . 5 (((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) → (𝑥 = +∞ ∨ 𝑥 < +∞))
236181, 228, 235mpjaodan 957 . . . 4 (((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)𝑥 < 𝑦)
237 0elpw 5311 . . . . . . . . 9 ∅ ∈ 𝒫 ℕ
238 0fin 9115 . . . . . . . . 9 ∅ ∈ Fin
239 elin 3926 . . . . . . . . 9 (∅ ∈ (𝒫 ℕ ∩ Fin) ↔ (∅ ∈ 𝒫 ℕ ∧ ∅ ∈ Fin))
240237, 238, 239mpbir2an 709 . . . . . . . 8 ∅ ∈ (𝒫 ℕ ∩ Fin)
241 sum0 15606 . . . . . . . . 9 Σ𝑘 ∈ ∅ 𝐴 = 0
242241eqcomi 2745 . . . . . . . 8 0 = Σ𝑘 ∈ ∅ 𝐴
243 sumeq1 15573 . . . . . . . . 9 (𝑏 = ∅ → Σ𝑘𝑏 𝐴 = Σ𝑘 ∈ ∅ 𝐴)
244243rspceeqv 3595 . . . . . . . 8 ((∅ ∈ (𝒫 ℕ ∩ Fin) ∧ 0 = Σ𝑘 ∈ ∅ 𝐴) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘𝑏 𝐴)
245240, 242, 244mp2an 690 . . . . . . 7 𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘𝑏 𝐴
24672, 73elrnmpti 5915 . . . . . . 7 (0 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘𝑏 𝐴)
247245, 246mpbir 230 . . . . . 6 0 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)
248 breq2 5109 . . . . . . 7 (𝑦 = 0 → (𝑥 < 𝑦𝑥 < 0))
249248rspcev 3581 . . . . . 6 ((0 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴) ∧ 𝑥 < 0) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)𝑥 < 𝑦)
250247, 249mpan 688 . . . . 5 (𝑥 < 0 → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)𝑥 < 𝑦)
251250adantl 482 . . . 4 (((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 𝑥 < 0) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)𝑥 < 𝑦)
252 xrlelttric 31657 . . . . . 6 ((0 ∈ ℝ*𝑥 ∈ ℝ*) → (0 ≤ 𝑥𝑥 < 0))
253187, 252mpan 688 . . . . 5 (𝑥 ∈ ℝ* → (0 ≤ 𝑥𝑥 < 0))
254253ad2antrl 726 . . . 4 ((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → (0 ≤ 𝑥𝑥 < 0))
255236, 251, 254mpjaodan 957 . . 3 ((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)𝑥 < 𝑦)
2562, 71, 171, 255eqsupd 9393 . 2 (𝜑 → sup(ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴), ℝ*, < ) = sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
257 nfv 1917 . . 3 𝑘𝜑
258 nfcv 2907 . . 3 𝑘
259 nnex 12159 . . . 4 ℕ ∈ V
260259a1i 11 . . 3 (𝜑 → ℕ ∈ V)
261 icossicc 13353 . . . 4 (0[,)+∞) ⊆ (0[,]+∞)
262261, 5sselid 3942 . . 3 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
263 elex 3463 . . . . . 6 (𝑏 ∈ (𝒫 ℕ ∩ Fin) → 𝑏 ∈ V)
264263adantl 482 . . . . 5 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ V)
265107fmpttd 7063 . . . . 5 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (𝑘𝑏𝐴):𝑏⟶(0[,)+∞))
266 esumpfinvallem 32673 . . . . 5 ((𝑏 ∈ V ∧ (𝑘𝑏𝐴):𝑏⟶(0[,)+∞)) → (ℂfld Σg (𝑘𝑏𝐴)) = ((ℝ*𝑠s (0[,]+∞)) Σg (𝑘𝑏𝐴)))
267264, 265, 266syl2anc 584 . . . 4 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (ℂfld Σg (𝑘𝑏𝐴)) = ((ℝ*𝑠s (0[,]+∞)) Σg (𝑘𝑏𝐴)))
268108recnd 11183 . . . . 5 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝐴 ∈ ℂ)
26999, 268gsumfsum 20864 . . . 4 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (ℂfld Σg (𝑘𝑏𝐴)) = Σ𝑘𝑏 𝐴)
270267, 269eqtr3d 2778 . . 3 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → ((ℝ*𝑠s (0[,]+∞)) Σg (𝑘𝑏𝐴)) = Σ𝑘𝑏 𝐴)
271257, 258, 260, 262, 270esumval 32645 . 2 (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = sup(ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴), ℝ*, < ))
2723, 4, 35, 43, 69isumclim 15642 . 2 (𝜑 → Σ𝑘 ∈ ℕ 𝐴 = sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
273256, 271, 2723eqtr4d 2786 1 (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  Vcvv 3445  cin 3909  wss 3910  c0 4282  𝒫 cpw 4560   class class class wbr 5105  cmpt 5188   Or wor 5544  dom cdm 5633  ran crn 5634   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  Fincfn 8883  supcsup 9376  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054  +∞cpnf 11186  *cxr 11188   < clt 11189  cle 11190  cn 12153  cz 12499  cuz 12763  [,)cico 13266  [,]cicc 13267  ...cfz 13424  seqcseq 13906  cli 15366  Σcsu 15570  s cress 17112   Σg cgsu 17322  *𝑠cxrs 17382  fldccnfld 20796  Σ*cesum 32626
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xadd 13034  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-ordt 17383  df-xrs 17384  df-mre 17466  df-mrc 17467  df-acs 17469  df-ps 18455  df-tsr 18456  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-grp 18751  df-minusg 18752  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-fbas 20793  df-fg 20794  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-ntr 22371  df-nei 22449  df-cn 22578  df-haus 22666  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-tsms 23478  df-esum 32627
This theorem is referenced by:  esumcvg  32685  esumcvgsum  32687
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