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Theorem esumpcvgval 34079
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 13183 . . . 4 < Or ℝ*
21a1i 11 . . 3 (𝜑 → < Or ℝ*)
3 nnuz 12921 . . . . 5 ℕ = (ℤ‘1)
4 1zzd 12648 . . . . 5 (𝜑 → 1 ∈ ℤ)
5 esumpcvgval.1 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
6 esumpcvgval.2 . . . . . . . . . . . 12 (𝑘 = 𝑙𝐴 = 𝐵)
7 eqcom 2744 . . . . . . . . . . . 12 (𝑘 = 𝑙𝑙 = 𝑘)
8 eqcom 2744 . . . . . . . . . . . 12 (𝐴 = 𝐵𝐵 = 𝐴)
96, 7, 83imtr3i 291 . . . . . . . . . . 11 (𝑙 = 𝑘𝐵 = 𝐴)
109cbvmptv 5255 . . . . . . . . . 10 (𝑙 ∈ ℕ ↦ 𝐵) = (𝑘 ∈ ℕ ↦ 𝐴)
115, 10fmptd 7134 . . . . . . . . 9 (𝜑 → (𝑙 ∈ ℕ ↦ 𝐵):ℕ⟶(0[,)+∞))
1211ffvelcdmda 7104 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞))
13 elrege0 13494 . . . . . . . . 9 (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞) ↔ (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥)))
1413simplbi 497 . . . . . . . 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 14072 . . . . . 6 (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)):ℕ⟶ℝ)
1711adantr 480 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑙 ∈ ℕ ↦ 𝐵):ℕ⟶(0[,)+∞))
18 simpr 484 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
1918peano2nnd 12283 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
2017, 19ffvelcdmd 7105 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞))
21 elrege0 13494 . . . . . . . . . 10 (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞) ↔ (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ ℝ ∧ 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
2221simprbi 496 . . . . . . . . 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 7104 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ∈ ℝ)
2521simplbi 497 . . . . . . . . . 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 11851 . . . . . . . 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 232 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) + ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
2918, 3eleqtrdi 2851 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
30 seqp1 14057 . . . . . . . 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 5171 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘(𝑛 + 1)))
33 simpr 484 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
3410fvmpt2 7027 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ 𝐴 ∈ (0[,)+∞)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
3533, 5, 34syl2anc 584 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
36 rge0ssre 13496 . . . . . . . . 9 (0[,)+∞) ⊆ ℝ
3736, 5sselid 3981 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
3816feqmptd 6977 . . . . . . . . . 10 (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)))
39 simpll 767 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
40 elfznn 13593 . . . . . . . . . . . . . . 15 (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
4140adantl 481 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
4239, 41, 35syl2anc 584 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
4337recnd 11289 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ ℂ)
4439, 41, 43syl2anc 584 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ ℂ)
4542, 29, 44fsumser 15766 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
4645eqcomd 2743 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) = Σ𝑘 ∈ (1...𝑛)𝐴)
4746mpteq2dva 5242 . . . . . . . . . 10 (𝜑 → (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴))
4838, 47eqtr2d 2778 . . . . . . . . 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 2842 . . . . . . . 8 (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ∈ dom ⇝ )
513, 4, 35, 37, 50isumrecl 15801 . . . . . . 7 (𝜑 → Σ𝑘 ∈ ℕ 𝐴 ∈ ℝ)
52 1zzd 12648 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 1 ∈ ℤ)
53 fzfid 14014 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
54 fzssuz 13605 . . . . . . . . . . . 12 (1...𝑛) ⊆ (ℤ‘1)
5554, 3sseqtrri 4033 . . . . . . . . . . 11 (1...𝑛) ⊆ ℕ
5655a1i 11 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
5735adantlr 715 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
5837adantlr 715 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
595adantlr 715 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
60 elrege0 13494 . . . . . . . . . . . 12 (𝐴 ∈ (0[,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
6160simprbi 496 . . . . . . . . . . 11 (𝐴 ∈ (0[,)+∞) → 0 ≤ 𝐴)
6259, 61syl 17 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 0 ≤ 𝐴)
6350adantr 480 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ∈ dom ⇝ )
643, 52, 53, 56, 57, 58, 62, 63isumless 15881 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 ≤ Σ𝑘 ∈ ℕ 𝐴)
6545, 64eqbrtrrd 5167 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴)
6665ralrimiva 3146 . . . . . . 7 (𝜑 → ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴)
67 brralrspcev 5203 . . . . . . 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 15706 . . . . 5 (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⇝ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
703, 4, 69, 24climrecl 15619 . . . 4 (𝜑 → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ)
7170rexrd 11311 . . 3 (𝜑 → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ*)
72 eqid 2737 . . . . . . 7 (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴) = (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)
73 sumex 15724 . . . . . . 7 Σ𝑘𝑏 𝐴 ∈ V
7472, 73elrnmpti 5973 . . . . . 6 (𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘𝑏 𝐴)
75 ssnnssfz 32789 . . . . . . . . . 10 (𝑏 ∈ (𝒫 ℕ ∩ Fin) → ∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚))
76 fzfid 14014 . . . . . . . . . . . . . 14 ((𝜑𝑏 ⊆ (1...𝑚)) → (1...𝑚) ∈ Fin)
77 elfznn 13593 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ)
7877, 5sylan2 593 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (1...𝑚)) → 𝐴 ∈ (0[,)+∞))
7960simplbi 497 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (0[,)+∞) → 𝐴 ∈ ℝ)
8078, 79syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ)
8180adantlr 715 . . . . . . . . . . . . . 14 (((𝜑𝑏 ⊆ (1...𝑚)) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ)
8278, 61syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...𝑚)) → 0 ≤ 𝐴)
8382adantlr 715 . . . . . . . . . . . . . 14 (((𝜑𝑏 ⊆ (1...𝑚)) ∧ 𝑘 ∈ (1...𝑚)) → 0 ≤ 𝐴)
84 simpr 484 . . . . . . . . . . . . . 14 ((𝜑𝑏 ⊆ (1...𝑚)) → 𝑏 ⊆ (1...𝑚))
8576, 81, 83, 84fsumless 15832 . . . . . . . . . . . . 13 ((𝜑𝑏 ⊆ (1...𝑚)) → Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
8685ex 412 . . . . . . . . . . . 12 (𝜑 → (𝑏 ⊆ (1...𝑚) → Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
8786reximdv 3170 . . . . . . . . . . 11 (𝜑 → (∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚) → ∃𝑚 ∈ ℕ Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
8887imp 406 . . . . . . . . . 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 5146 . . . . . . . . . 10 (𝑥 = Σ𝑘𝑏 𝐴 → (𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴 ↔ Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
9190rexbidv 3179 . . . . . . . . 9 (𝑥 = Σ𝑘𝑏 𝐴 → (∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴 ↔ ∃𝑚 ∈ ℕ Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
9289, 91syl5ibrcom 247 . . . . . . . 8 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (𝑥 = Σ𝑘𝑏 𝐴 → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
9392rexlimdva 3155 . . . . . . 7 (𝜑 → (∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘𝑏 𝐴 → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
9493imp 406 . . . . . 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 484 . . . . . . . . . 10 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘𝑏 𝐴) → 𝑥 = Σ𝑘𝑏 𝐴)
97 inss2 4238 . . . . . . . . . . . . 13 (𝒫 ℕ ∩ Fin) ⊆ Fin
98 simpr 484 . . . . . . . . . . . . 13 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ (𝒫 ℕ ∩ Fin))
9997, 98sselid 3981 . . . . . . . . . . . 12 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ Fin)
100 simpll 767 . . . . . . . . . . . . . 14 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝜑)
101 inss1 4237 . . . . . . . . . . . . . . . . 17 (𝒫 ℕ ∩ Fin) ⊆ 𝒫 ℕ
102 simplr 769 . . . . . . . . . . . . . . . . 17 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝑏 ∈ (𝒫 ℕ ∩ Fin))
103101, 102sselid 3981 . . . . . . . . . . . . . . . 16 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝑏 ∈ 𝒫 ℕ)
104103elpwid 4609 . . . . . . . . . . . . . . 15 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝑏 ⊆ ℕ)
105 simpr 484 . . . . . . . . . . . . . . 15 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝑘𝑏)
106104, 105sseldd 3984 . . . . . . . . . . . . . 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 15770 . . . . . . . . . . 11 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → Σ𝑘𝑏 𝐴 ∈ ℝ)
110109adantr 480 . . . . . . . . . 10 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘𝑏 𝐴) → Σ𝑘𝑏 𝐴 ∈ ℝ)
11196, 110eqeltrd 2841 . . . . . . . . 9 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘𝑏 𝐴) → 𝑥 ∈ ℝ)
112111r19.29an 3158 . . . . . . . 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 480 . . . . . 6 (((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → 𝑥 ∈ ℝ)
115 fzfid 14014 . . . . . . . 8 (𝜑 → (1...𝑚) ∈ Fin)
116115, 80fsumrecl 15770 . . . . . . 7 (𝜑 → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ℝ)
117116ad2antrr 726 . . . . . 6 (((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ℝ)
11870ad2antrr 726 . . . . . 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 773 . . . . . 6 (((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
12016frnd 6744 . . . . . . . 8 (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ)
121120ad2antrr 726 . . . . . . 7 (((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ)
122 1nn 12277 . . . . . . . . . 10 1 ∈ ℕ
123122ne0ii 4344 . . . . . . . . 9 ℕ ≠ ∅
124 dm0rn0 5935 . . . . . . . . . . 11 (dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅)
12516fdmd 6746 . . . . . . . . . . . 12 (𝜑 → dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ℕ)
126125eqeq1d 2739 . . . . . . . . . . 11 (𝜑 → (dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ℕ = ∅))
127124, 126bitr3id 285 . . . . . . . . . 10 (𝜑 → (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ℕ = ∅))
128127necon3bid 2985 . . . . . . . . 9 (𝜑 → (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅ ↔ ℕ ≠ ∅))
129123, 128mpbiri 258 . . . . . . . 8 (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅)
130129ad2antrr 726 . . . . . . 7 (((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅)
131 1z 12647 . . . . . . . . . . . . . . . 16 1 ∈ ℤ
132 seqfn 14054 . . . . . . . . . . . . . . . 16 (1 ∈ ℤ → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn (ℤ‘1))
133131, 132ax-mp 5 . . . . . . . . . . . . . . 15 seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn (ℤ‘1)
1343fneq2i 6666 . . . . . . . . . . . . . . 15 (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ ↔ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn (ℤ‘1))
135133, 134mpbir 231 . . . . . . . . . . . . . 14 seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ
136 dffn5 6967 . . . . . . . . . . . . . 14 (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ ↔ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)))
137135, 136mpbi 230 . . . . . . . . . . . . 13 seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
138 fvex 6919 . . . . . . . . . . . . 13 (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ∈ V
139137, 138elrnmpti 5973 . . . . . . . . . . . 12 (𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
140 r19.29 3114 . . . . . . . . . . . . 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 5146 . . . . . . . . . . . . . . 15 (𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → (𝑧𝑠 ↔ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠))
142141biimparc 479 . . . . . . . . . . . . . 14 (((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑧𝑠)
143142rexlimivw 3151 . . . . . . . . . . . . 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 3146 . . . . . . . . . 10 (∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 → ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧𝑠)
147146reximi 3084 . . . . . . . . 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 726 . . . . . . 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 484 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
151 simpll 767 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝜑)
15277adantl 481 . . . . . . . . . . . 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 2851 . . . . . . . . . . 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 11289 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℂ)
158153, 154, 157fsumser 15766 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚))
159 fveq2 6906 . . . . . . . . . . 11 (𝑛 = 𝑚 → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚))
160159rspceeqv 3645 . . . . . . . . . 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 5973 . . . . . . . . 9 𝑘 ∈ (1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
163161, 162sylibr 234 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)))
164163ad2ant2r 747 . . . . . . 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 12229 . . . . . . 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 1375 . . . . . 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 11418 . . . . 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 3161 . . . 4 ((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) → 𝑥 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
16970adantr 480 . . . . 5 ((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ)
170113, 169lenltd 11407 . . . 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 232 . . 3 ((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) → ¬ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) < 𝑥)
172 simpr1r 1232 . . . . . . 7 ((𝜑 ∧ ((𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0 ≤ 𝑥𝑥 = +∞)) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
1731723anassrs 1361 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
17471ad3antrrr 730 . . . . . . . 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 13170 . . . . . . . 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 5146 . . . . . . . . 9 (𝑥 = +∞ → (𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )))
178177notbid 318 . . . . . . . 8 (𝑥 = +∞ → (¬ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ ¬ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )))
179178adantl 481 . . . . . . 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 257 . . . . . 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 195 . . . . 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 775 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝜑)
183 simpr1l 1231 . . . . . . . 8 ((𝜑 ∧ ((𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0 ≤ 𝑥𝑥 < +∞)) → 𝑥 ∈ ℝ*)
1841833anassrs 1361 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 ∈ ℝ*)
185 simplr 769 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 0 ≤ 𝑥)
186 simpr 484 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 < +∞)
187 0xr 11308 . . . . . . . 8 0 ∈ ℝ*
188 pnfxr 11315 . . . . . . . 8 +∞ ∈ ℝ*
189 elico1 13430 . . . . . . . 8 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥𝑥 < +∞)))
190187, 188, 189mp2an 692 . . . . . . 7 (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥𝑥 < +∞))
191184, 185, 186, 190syl3anbrc 1344 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 ∈ (0[,)+∞))
192 simpr1r 1232 . . . . . . 7 ((𝜑 ∧ ((𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0 ≤ 𝑥𝑥 < +∞)) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
1931923anassrs 1361 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
194120adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ)
195129adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅)
196148adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧𝑠)
197194, 195, 1963jca 1129 . . . . . . . 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 771 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → 𝑥 ∈ (0[,)+∞))
19936, 198sselid 3981 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → 𝑥 ∈ ℝ)
200 simprr 773 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
201 suprlub 12232 . . . . . . . . 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 476 . . . . . . . 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 838 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦)
20440ssriv 3987 . . . . . . . . . . . . . . . . 17 (1...𝑛) ⊆ ℕ
205 ovex 7464 . . . . . . . . . . . . . . . . . 18 (1...𝑛) ∈ V
206205elpw 4604 . . . . . . . . . . . . . . . . 17 ((1...𝑛) ∈ 𝒫 ℕ ↔ (1...𝑛) ⊆ ℕ)
207204, 206mpbir 231 . . . . . . . . . . . . . . . 16 (1...𝑛) ∈ 𝒫 ℕ
208 fzfi 14013 . . . . . . . . . . . . . . . 16 (1...𝑛) ∈ Fin
209 elin 3967 . . . . . . . . . . . . . . . 16 ((1...𝑛) ∈ (𝒫 ℕ ∩ Fin) ↔ ((1...𝑛) ∈ 𝒫 ℕ ∧ (1...𝑛) ∈ Fin))
210207, 208, 209mpbir2an 711 . . . . . . . . . . . . . . 15 (1...𝑛) ∈ (𝒫 ℕ ∩ Fin)
211210a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → (1...𝑛) ∈ (𝒫 ℕ ∩ Fin))
212 simpr 484 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
21345adantr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → Σ𝑘 ∈ (1...𝑛)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
214212, 213eqtr4d 2780 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑦 = Σ𝑘 ∈ (1...𝑛)𝐴)
215 sumeq1 15725 . . . . . . . . . . . . . . 15 (𝑏 = (1...𝑛) → Σ𝑘𝑏 𝐴 = Σ𝑘 ∈ (1...𝑛)𝐴)
216215rspceeqv 3645 . . . . . . . . . . . . . 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 412 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘𝑏 𝐴))
219218rexlimdva 3155 . . . . . . . . . . 11 (𝜑 → (∃𝑛 ∈ ℕ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘𝑏 𝐴))
220137, 138elrnmpti 5973 . . . . . . . . . . 11 (𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
22172, 73elrnmpti 5973 . . . . . . . . . . 11 (𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘𝑏 𝐴)
222219, 220, 2213imtr4g 296 . . . . . . . . . 10 (𝜑 → (𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) → 𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)))
223222ssrdv 3989 . . . . . . . . 9 (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴))
224 ssrexv 4053 . . . . . . . . 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 406 . . . . . . 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 837 . . . . 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 777 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) → 𝑥 ∈ ℝ*)
230 xrlelttric 32756 . . . . . . . 8 ((+∞ ∈ ℝ*𝑥 ∈ ℝ*) → (+∞ ≤ 𝑥𝑥 < +∞))
231188, 230mpan 690 . . . . . . 7 (𝑥 ∈ ℝ* → (+∞ ≤ 𝑥𝑥 < +∞))
232 xgepnf 13207 . . . . . . . 8 (𝑥 ∈ ℝ* → (+∞ ≤ 𝑥𝑥 = +∞))
233232orbi1d 917 . . . . . . 7 (𝑥 ∈ ℝ* → ((+∞ ≤ 𝑥𝑥 < +∞) ↔ (𝑥 = +∞ ∨ 𝑥 < +∞)))
234231, 233mpbid 232 . . . . . 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 961 . . . 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 5356 . . . . . . . . 9 ∅ ∈ 𝒫 ℕ
238 0fi 9082 . . . . . . . . 9 ∅ ∈ Fin
239 elin 3967 . . . . . . . . 9 (∅ ∈ (𝒫 ℕ ∩ Fin) ↔ (∅ ∈ 𝒫 ℕ ∧ ∅ ∈ Fin))
240237, 238, 239mpbir2an 711 . . . . . . . 8 ∅ ∈ (𝒫 ℕ ∩ Fin)
241 sum0 15757 . . . . . . . . 9 Σ𝑘 ∈ ∅ 𝐴 = 0
242241eqcomi 2746 . . . . . . . 8 0 = Σ𝑘 ∈ ∅ 𝐴
243 sumeq1 15725 . . . . . . . . 9 (𝑏 = ∅ → Σ𝑘𝑏 𝐴 = Σ𝑘 ∈ ∅ 𝐴)
244243rspceeqv 3645 . . . . . . . 8 ((∅ ∈ (𝒫 ℕ ∩ Fin) ∧ 0 = Σ𝑘 ∈ ∅ 𝐴) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘𝑏 𝐴)
245240, 242, 244mp2an 692 . . . . . . 7 𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘𝑏 𝐴
24672, 73elrnmpti 5973 . . . . . . 7 (0 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘𝑏 𝐴)
247245, 246mpbir 231 . . . . . 6 0 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)
248 breq2 5147 . . . . . . 7 (𝑦 = 0 → (𝑥 < 𝑦𝑥 < 0))
249248rspcev 3622 . . . . . 6 ((0 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴) ∧ 𝑥 < 0) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)𝑥 < 𝑦)
250247, 249mpan 690 . . . . 5 (𝑥 < 0 → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)𝑥 < 𝑦)
251250adantl 481 . . . 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 32756 . . . . . 6 ((0 ∈ ℝ*𝑥 ∈ ℝ*) → (0 ≤ 𝑥𝑥 < 0))
253187, 252mpan 690 . . . . 5 (𝑥 ∈ ℝ* → (0 ≤ 𝑥𝑥 < 0))
254253ad2antrl 728 . . . 4 ((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → (0 ≤ 𝑥𝑥 < 0))
255236, 251, 254mpjaodan 961 . . 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 9497 . 2 (𝜑 → sup(ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴), ℝ*, < ) = sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
257 nfv 1914 . . 3 𝑘𝜑
258 nfcv 2905 . . 3 𝑘
259 nnex 12272 . . . 4 ℕ ∈ V
260259a1i 11 . . 3 (𝜑 → ℕ ∈ V)
261 icossicc 13476 . . . 4 (0[,)+∞) ⊆ (0[,]+∞)
262261, 5sselid 3981 . . 3 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
263 elex 3501 . . . . . 6 (𝑏 ∈ (𝒫 ℕ ∩ Fin) → 𝑏 ∈ V)
264263adantl 481 . . . . 5 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ V)
265107fmpttd 7135 . . . . 5 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (𝑘𝑏𝐴):𝑏⟶(0[,)+∞))
266 esumpfinvallem 34075 . . . . 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 11289 . . . . 5 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝐴 ∈ ℂ)
26999, 268gsumfsum 21452 . . . 4 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (ℂfld Σg (𝑘𝑏𝐴)) = Σ𝑘𝑏 𝐴)
270267, 269eqtr3d 2779 . . 3 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → ((ℝ*𝑠s (0[,]+∞)) Σg (𝑘𝑏𝐴)) = Σ𝑘𝑏 𝐴)
271257, 258, 260, 262, 270esumval 34047 . 2 (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = sup(ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴), ℝ*, < ))
2723, 4, 35, 43, 69isumclim 15793 . 2 (𝜑 → Σ𝑘 ∈ ℕ 𝐴 = sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
273256, 271, 2723eqtr4d 2787 1 (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  Vcvv 3480  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600   class class class wbr 5143  cmpt 5225   Or wor 5591  dom cdm 5685  ran crn 5686   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  Fincfn 8985  supcsup 9480  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158  +∞cpnf 11292  *cxr 11294   < clt 11295  cle 11296  cn 12266  cz 12613  cuz 12878  [,)cico 13389  [,]cicc 13390  ...cfz 13547  seqcseq 14042  cli 15520  Σcsu 15722  s cress 17274   Σg cgsu 17485  *𝑠cxrs 17545  fldccnfld 21364  Σ*cesum 34028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xadd 13155  df-ioo 13391  df-ioc 13392  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-rest 17467  df-topn 17468  df-0g 17486  df-gsum 17487  df-topgen 17488  df-ordt 17546  df-xrs 17547  df-mre 17629  df-mrc 17630  df-acs 17632  df-ps 18611  df-tsr 18612  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-grp 18954  df-minusg 18955  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-ur 20179  df-ring 20232  df-cring 20233  df-fbas 21361  df-fg 21362  df-cnfld 21365  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-ntr 23028  df-nei 23106  df-cn 23235  df-haus 23323  df-fil 23854  df-fm 23946  df-flim 23947  df-flf 23948  df-tsms 24135  df-esum 34029
This theorem is referenced by:  esumcvg  34087  esumcvgsum  34089
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