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Theorem esumpcvgval 31341
 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 12537 . . . 4 < Or ℝ*
21a1i 11 . . 3 (𝜑 → < Or ℝ*)
3 nnuz 12284 . . . . 5 ℕ = (ℤ‘1)
4 1zzd 12016 . . . . 5 (𝜑 → 1 ∈ ℤ)
5 esumpcvgval.1 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
6 esumpcvgval.2 . . . . . . . . . . . 12 (𝑘 = 𝑙𝐴 = 𝐵)
7 eqcom 2831 . . . . . . . . . . . 12 (𝑘 = 𝑙𝑙 = 𝑘)
8 eqcom 2831 . . . . . . . . . . . 12 (𝐴 = 𝐵𝐵 = 𝐴)
96, 7, 83imtr3i 293 . . . . . . . . . . 11 (𝑙 = 𝑘𝐵 = 𝐴)
109cbvmptv 5172 . . . . . . . . . 10 (𝑙 ∈ ℕ ↦ 𝐵) = (𝑘 ∈ ℕ ↦ 𝐴)
115, 10fmptd 6881 . . . . . . . . 9 (𝜑 → (𝑙 ∈ ℕ ↦ 𝐵):ℕ⟶(0[,)+∞))
1211ffvelrnda 6854 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞))
13 elrege0 12845 . . . . . . . . 9 (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞) ↔ (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥)))
1413simplbi 500 . . . . . . . 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 13402 . . . . . 6 (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)):ℕ⟶ℝ)
1711adantr 483 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑙 ∈ ℕ ↦ 𝐵):ℕ⟶(0[,)+∞))
18 simpr 487 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
1918peano2nnd 11658 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
2017, 19ffvelrnd 6855 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞))
21 elrege0 12845 . . . . . . . . . 10 (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞) ↔ (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ ℝ ∧ 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
2221simprbi 499 . . . . . . . . 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)))
2416ffvelrnda 6854 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ∈ ℝ)
2521simplbi 500 . . . . . . . . . 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 11231 . . . . . . . 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 234 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) + ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
2918, 3eleqtrdi 2926 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
30 seqp1 13387 . . . . . . . 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 5097 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘(𝑛 + 1)))
33 simpr 487 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
3410fvmpt2 6782 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ 𝐴 ∈ (0[,)+∞)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
3533, 5, 34syl2anc 586 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
36 rge0ssre 12847 . . . . . . . . 9 (0[,)+∞) ⊆ ℝ
3736, 5sseldi 3968 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
3816feqmptd 6736 . . . . . . . . . 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 12939 . . . . . . . . . . . . . . 15 (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
4140adantl 484 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
4239, 41, 35syl2anc 586 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
4337recnd 10672 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ ℂ)
4439, 41, 43syl2anc 586 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ ℂ)
4542, 29, 44fsumser 15090 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
4645eqcomd 2830 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) = Σ𝑘 ∈ (1...𝑛)𝐴)
4746mpteq2dva 5164 . . . . . . . . . 10 (𝜑 → (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴))
4838, 47eqtr2d 2860 . . . . . . . . 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 2917 . . . . . . . 8 (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ∈ dom ⇝ )
513, 4, 35, 37, 50isumrecl 15123 . . . . . . 7 (𝜑 → Σ𝑘 ∈ ℕ 𝐴 ∈ ℝ)
52 1zzd 12016 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 1 ∈ ℤ)
53 fzfid 13344 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
54 fzssuz 12951 . . . . . . . . . . . 12 (1...𝑛) ⊆ (ℤ‘1)
5554, 3sseqtrri 4007 . . . . . . . . . . 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 12845 . . . . . . . . . . . 12 (𝐴 ∈ (0[,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
6160simprbi 499 . . . . . . . . . . 11 (𝐴 ∈ (0[,)+∞) → 0 ≤ 𝐴)
6259, 61syl 17 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 0 ≤ 𝐴)
6350adantr 483 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ∈ dom ⇝ )
643, 52, 53, 56, 57, 58, 62, 63isumless 15203 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 ≤ Σ𝑘 ∈ ℕ 𝐴)
6545, 64eqbrtrrd 5093 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴)
6665ralrimiva 3185 . . . . . . 7 (𝜑 → ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴)
67 brralrspcev 5129 . . . . . . 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 586 . . . . . 6 (𝜑 → ∃𝑠 ∈ ℝ ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠)
693, 4, 16, 32, 68climsup 15029 . . . . 5 (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⇝ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
703, 4, 69, 24climrecl 14943 . . . 4 (𝜑 → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ)
7170rexrd 10694 . . 3 (𝜑 → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ*)
72 eqid 2824 . . . . . . 7 (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴) = (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)
73 sumex 15047 . . . . . . 7 Σ𝑘𝑏 𝐴 ∈ V
7472, 73elrnmpti 5835 . . . . . 6 (𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘𝑏 𝐴)
75 ssnnssfz 30513 . . . . . . . . . 10 (𝑏 ∈ (𝒫 ℕ ∩ Fin) → ∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚))
76 fzfid 13344 . . . . . . . . . . . . . 14 ((𝜑𝑏 ⊆ (1...𝑚)) → (1...𝑚) ∈ Fin)
77 elfznn 12939 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ)
7877, 5sylan2 594 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (1...𝑚)) → 𝐴 ∈ (0[,)+∞))
7960simplbi 500 . . . . . . . . . . . . . . . 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 487 . . . . . . . . . . . . . 14 ((𝜑𝑏 ⊆ (1...𝑚)) → 𝑏 ⊆ (1...𝑚))
8576, 81, 83, 84fsumless 15154 . . . . . . . . . . . . 13 ((𝜑𝑏 ⊆ (1...𝑚)) → Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
8685ex 415 . . . . . . . . . . . 12 (𝜑 → (𝑏 ⊆ (1...𝑚) → Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
8786reximdv 3276 . . . . . . . . . . 11 (𝜑 → (∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚) → ∃𝑚 ∈ ℕ Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
8887imp 409 . . . . . . . . . 10 ((𝜑 ∧ ∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚)) → ∃𝑚 ∈ ℕ Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
8975, 88sylan2 594 . . . . . . . . 9 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → ∃𝑚 ∈ ℕ Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
90 breq1 5072 . . . . . . . . . 10 (𝑥 = Σ𝑘𝑏 𝐴 → (𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴 ↔ Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
9190rexbidv 3300 . . . . . . . . 9 (𝑥 = Σ𝑘𝑏 𝐴 → (∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴 ↔ ∃𝑚 ∈ ℕ Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
9289, 91syl5ibrcom 249 . . . . . . . 8 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (𝑥 = Σ𝑘𝑏 𝐴 → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
9392rexlimdva 3287 . . . . . . 7 (𝜑 → (∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘𝑏 𝐴 → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
9493imp 409 . . . . . 6 ((𝜑 ∧ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘𝑏 𝐴) → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
9574, 94sylan2b 595 . . . . 5 ((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
96 simpr 487 . . . . . . . . . 10 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘𝑏 𝐴) → 𝑥 = Σ𝑘𝑏 𝐴)
97 inss2 4209 . . . . . . . . . . . . 13 (𝒫 ℕ ∩ Fin) ⊆ Fin
98 simpr 487 . . . . . . . . . . . . 13 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ (𝒫 ℕ ∩ Fin))
9997, 98sseldi 3968 . . . . . . . . . . . 12 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ Fin)
100 simpll 765 . . . . . . . . . . . . . 14 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝜑)
101 inss1 4208 . . . . . . . . . . . . . . . . 17 (𝒫 ℕ ∩ Fin) ⊆ 𝒫 ℕ
102 simplr 767 . . . . . . . . . . . . . . . . 17 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝑏 ∈ (𝒫 ℕ ∩ Fin))
103101, 102sseldi 3968 . . . . . . . . . . . . . . . 16 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝑏 ∈ 𝒫 ℕ)
104103elpwid 4553 . . . . . . . . . . . . . . 15 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝑏 ⊆ ℕ)
105 simpr 487 . . . . . . . . . . . . . . 15 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝑘𝑏)
106104, 105sseldd 3971 . . . . . . . . . . . . . 14 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝑘 ∈ ℕ)
107100, 106, 5syl2anc 586 . . . . . . . . . . . . 13 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝐴 ∈ (0[,)+∞))
108107, 79syl 17 . . . . . . . . . . . 12 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝐴 ∈ ℝ)
10999, 108fsumrecl 15094 . . . . . . . . . . 11 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → Σ𝑘𝑏 𝐴 ∈ ℝ)
110109adantr 483 . . . . . . . . . 10 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘𝑏 𝐴) → Σ𝑘𝑏 𝐴 ∈ ℝ)
11196, 110eqeltrd 2916 . . . . . . . . 9 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘𝑏 𝐴) → 𝑥 ∈ ℝ)
112111r19.29an 3291 . . . . . . . 8 ((𝜑 ∧ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘𝑏 𝐴) → 𝑥 ∈ ℝ)
11374, 112sylan2b 595 . . . . . . 7 ((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) → 𝑥 ∈ ℝ)
114113adantr 483 . . . . . 6 (((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → 𝑥 ∈ ℝ)
115 fzfid 13344 . . . . . . . 8 (𝜑 → (1...𝑚) ∈ Fin)
116115, 80fsumrecl 15094 . . . . . . 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 6524 . . . . . . . 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 11652 . . . . . . . . . 10 1 ∈ ℕ
123122ne0ii 4306 . . . . . . . . 9 ℕ ≠ ∅
124 dm0rn0 5798 . . . . . . . . . . 11 (dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅)
12516fdmd 6526 . . . . . . . . . . . 12 (𝜑 → dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ℕ)
126125eqeq1d 2826 . . . . . . . . . . 11 (𝜑 → (dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ℕ = ∅))
127124, 126syl5bbr 287 . . . . . . . . . 10 (𝜑 → (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ℕ = ∅))
128127necon3bid 3063 . . . . . . . . 9 (𝜑 → (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅ ↔ ℕ ≠ ∅))
129123, 128mpbiri 260 . . . . . . . 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 12015 . . . . . . . . . . . . . . . 16 1 ∈ ℤ
132 seqfn 13384 . . . . . . . . . . . . . . . 16 (1 ∈ ℤ → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn (ℤ‘1))
133131, 132ax-mp 5 . . . . . . . . . . . . . . 15 seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn (ℤ‘1)
1343fneq2i 6454 . . . . . . . . . . . . . . 15 (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ ↔ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn (ℤ‘1))
135133, 134mpbir 233 . . . . . . . . . . . . . 14 seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ
136 dffn5 6727 . . . . . . . . . . . . . 14 (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ ↔ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)))
137135, 136mpbi 232 . . . . . . . . . . . . 13 seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
138 fvex 6686 . . . . . . . . . . . . 13 (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ∈ V
139137, 138elrnmpti 5835 . . . . . . . . . . . 12 (𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
140 r19.29 3257 . . . . . . . . . . . . 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 5072 . . . . . . . . . . . . . . 15 (𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → (𝑧𝑠 ↔ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠))
142141biimparc 482 . . . . . . . . . . . . . 14 (((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑧𝑠)
143142rexlimivw 3285 . . . . . . . . . . . . 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 595 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))) → 𝑧𝑠)
146145ralrimiva 3185 . . . . . . . . . 10 (∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 → ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧𝑠)
147146reximi 3246 . . . . . . . . 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 487 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
151 simpll 765 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝜑)
15277adantl 484 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ ℕ)
153151, 152, 35syl2anc 586 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
154150, 3eleqtrdi 2926 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ (ℤ‘1))
155151, 152, 5syl2anc 586 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ (0[,)+∞))
156155, 79syl 17 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ)
157156recnd 10672 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℂ)
158153, 154, 157fsumser 15090 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚))
159 fveq2 6673 . . . . . . . . . . 11 (𝑛 = 𝑚 → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚))
160159rspceeqv 3641 . . . . . . . . . 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 586 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → ∃𝑛 ∈ ℕ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
162137, 138elrnmpti 5835 . . . . . . . . 9 𝑘 ∈ (1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
163161, 162sylibr 236 . . . . . . . 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 11605 . . . . . . 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 1369 . . . . . 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 10800 . . . . 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 3294 . . . 4 ((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) → 𝑥 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
16970adantr 483 . . . . 5 ((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ)
170113, 169lenltd 10789 . . . 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 234 . . 3 ((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) → ¬ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) < 𝑥)
172 simpr1r 1227 . . . . . . 7 ((𝜑 ∧ ((𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0 ≤ 𝑥𝑥 = +∞)) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
1731723anassrs 1356 . . . . . 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 12526 . . . . . . . 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 5072 . . . . . . . . 9 (𝑥 = +∞ → (𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )))
178177notbid 320 . . . . . . . 8 (𝑥 = +∞ → (¬ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ ¬ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )))
179178adantl 484 . . . . . . 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 259 . . . . . 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 197 . . . . 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 1226 . . . . . . . 8 ((𝜑 ∧ ((𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0 ≤ 𝑥𝑥 < +∞)) → 𝑥 ∈ ℝ*)
1841833anassrs 1356 . . . . . . 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 487 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 < +∞)
187 0xr 10691 . . . . . . . 8 0 ∈ ℝ*
188 pnfxr 10698 . . . . . . . 8 +∞ ∈ ℝ*
189 elico1 12784 . . . . . . . 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 1339 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 ∈ (0[,)+∞))
192 simpr1r 1227 . . . . . . 7 ((𝜑 ∧ ((𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0 ≤ 𝑥𝑥 < +∞)) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
1931923anassrs 1356 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
194120adantr 483 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ)
195129adantr 483 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅)
196148adantr 483 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧𝑠)
197194, 195, 1963jca 1124 . . . . . . . 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, 198sseldi 3968 . . . . . . . 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 11608 . . . . . . . . 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 479 . . . . . . . 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 835 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦)
20440ssriv 3974 . . . . . . . . . . . . . . . . 17 (1...𝑛) ⊆ ℕ
205 ovex 7192 . . . . . . . . . . . . . . . . . 18 (1...𝑛) ∈ V
206205elpw 4546 . . . . . . . . . . . . . . . . 17 ((1...𝑛) ∈ 𝒫 ℕ ↔ (1...𝑛) ⊆ ℕ)
207204, 206mpbir 233 . . . . . . . . . . . . . . . 16 (1...𝑛) ∈ 𝒫 ℕ
208 fzfi 13343 . . . . . . . . . . . . . . . 16 (1...𝑛) ∈ Fin
209 elin 4172 . . . . . . . . . . . . . . . 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 487 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
21345adantr 483 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → Σ𝑘 ∈ (1...𝑛)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
214212, 213eqtr4d 2862 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑦 = Σ𝑘 ∈ (1...𝑛)𝐴)
215 sumeq1 15048 . . . . . . . . . . . . . . 15 (𝑏 = (1...𝑛) → Σ𝑘𝑏 𝐴 = Σ𝑘 ∈ (1...𝑛)𝐴)
216215rspceeqv 3641 . . . . . . . . . . . . . 14 (((1...𝑛) ∈ (𝒫 ℕ ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ (1...𝑛)𝐴) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘𝑏 𝐴)
217211, 214, 216syl2anc 586 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘𝑏 𝐴)
218217ex 415 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘𝑏 𝐴))
219218rexlimdva 3287 . . . . . . . . . . 11 (𝜑 → (∃𝑛 ∈ ℕ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘𝑏 𝐴))
220137, 138elrnmpti 5835 . . . . . . . . . . 11 (𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
22172, 73elrnmpti 5835 . . . . . . . . . . 11 (𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘𝑏 𝐴)
222219, 220, 2213imtr4g 298 . . . . . . . . . 10 (𝜑 → (𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) → 𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)))
223222ssrdv 3976 . . . . . . . . 9 (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴))
224 ssrexv 4037 . . . . . . . . 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 409 . . . . . . 7 ((𝜑 ∧ ∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)𝑥 < 𝑦)
227203, 226syldan 593 . . . . . 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 834 . . . . 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 30479 . . . . . . . 8 ((+∞ ∈ ℝ*𝑥 ∈ ℝ*) → (+∞ ≤ 𝑥𝑥 < +∞))
231188, 230mpan 688 . . . . . . 7 (𝑥 ∈ ℝ* → (+∞ ≤ 𝑥𝑥 < +∞))
232 xgepnf 12561 . . . . . . . 8 (𝑥 ∈ ℝ* → (+∞ ≤ 𝑥𝑥 = +∞))
233232orbi1d 913 . . . . . . 7 (𝑥 ∈ ℝ* → ((+∞ ≤ 𝑥𝑥 < +∞) ↔ (𝑥 = +∞ ∨ 𝑥 < +∞)))
234231, 233mpbid 234 . . . . . 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 955 . . . 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 5259 . . . . . . . . 9 ∅ ∈ 𝒫 ℕ
238 0fin 8749 . . . . . . . . 9 ∅ ∈ Fin
239 elin 4172 . . . . . . . . 9 (∅ ∈ (𝒫 ℕ ∩ Fin) ↔ (∅ ∈ 𝒫 ℕ ∧ ∅ ∈ Fin))
240237, 238, 239mpbir2an 709 . . . . . . . 8 ∅ ∈ (𝒫 ℕ ∩ Fin)
241 sum0 15081 . . . . . . . . 9 Σ𝑘 ∈ ∅ 𝐴 = 0
242241eqcomi 2833 . . . . . . . 8 0 = Σ𝑘 ∈ ∅ 𝐴
243 sumeq1 15048 . . . . . . . . 9 (𝑏 = ∅ → Σ𝑘𝑏 𝐴 = Σ𝑘 ∈ ∅ 𝐴)
244243rspceeqv 3641 . . . . . . . 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 5835 . . . . . . 7 (0 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘𝑏 𝐴)
247245, 246mpbir 233 . . . . . 6 0 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)
248 breq2 5073 . . . . . . 7 (𝑦 = 0 → (𝑥 < 𝑦𝑥 < 0))
249248rspcev 3626 . . . . . 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 484 . . . 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 30479 . . . . . 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 955 . . 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 8924 . 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 2980 . . 3 𝑘
259 nnex 11647 . . . 4 ℕ ∈ V
260259a1i 11 . . 3 (𝜑 → ℕ ∈ V)
261 icossicc 12827 . . . 4 (0[,)+∞) ⊆ (0[,]+∞)
262261, 5sseldi 3968 . . 3 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
263 elex 3515 . . . . . 6 (𝑏 ∈ (𝒫 ℕ ∩ Fin) → 𝑏 ∈ V)
264263adantl 484 . . . . 5 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ V)
265107fmpttd 6882 . . . . 5 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (𝑘𝑏𝐴):𝑏⟶(0[,)+∞))
266 esumpfinvallem 31337 . . . . 5 ((𝑏 ∈ V ∧ (𝑘𝑏𝐴):𝑏⟶(0[,)+∞)) → (ℂfld Σg (𝑘𝑏𝐴)) = ((ℝ*𝑠s (0[,]+∞)) Σg (𝑘𝑏𝐴)))
267264, 265, 266syl2anc 586 . . . 4 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (ℂfld Σg (𝑘𝑏𝐴)) = ((ℝ*𝑠s (0[,]+∞)) Σg (𝑘𝑏𝐴)))
268108recnd 10672 . . . . 5 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝐴 ∈ ℂ)
26999, 268gsumfsum 20615 . . . 4 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (ℂfld Σg (𝑘𝑏𝐴)) = Σ𝑘𝑏 𝐴)
270267, 269eqtr3d 2861 . . 3 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → ((ℝ*𝑠s (0[,]+∞)) Σg (𝑘𝑏𝐴)) = Σ𝑘𝑏 𝐴)
271257, 258, 260, 262, 270esumval 31309 . 2 (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = sup(ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴), ℝ*, < ))
2723, 4, 35, 43, 69isumclim 15115 . 2 (𝜑 → Σ𝑘 ∈ ℕ 𝐴 = sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
273256, 271, 2723eqtr4d 2869 1 (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113   ≠ wne 3019  ∀wral 3141  ∃wrex 3142  Vcvv 3497   ∩ cin 3938   ⊆ wss 3939  ∅c0 4294  𝒫 cpw 4542   class class class wbr 5069   ↦ cmpt 5149   Or wor 5476  dom cdm 5558  ran crn 5559   Fn wfn 6353  ⟶wf 6354  ‘cfv 6358  (class class class)co 7159  Fincfn 8512  supcsup 8907  ℂcc 10538  ℝcr 10539  0cc0 10540  1c1 10541   + caddc 10543  +∞cpnf 10675  ℝ*cxr 10677   < clt 10678   ≤ cle 10679  ℕcn 11641  ℤcz 11984  ℤ≥cuz 12246  [,)cico 12743  [,]cicc 12744  ...cfz 12895  seqcseq 13372   ⇝ cli 14844  Σcsu 15045   ↾s cress 16487   Σg cgsu 16717  ℝ*𝑠cxrs 16776  ℂfldccnfld 20548  Σ*cesum 31290 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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-inf2 9107  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617  ax-pre-sup 10618  ax-addf 10619  ax-mulf 10620 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-fal 1549  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-iin 4925  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-se 5518  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-isom 6367  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-of 7412  df-om 7584  df-1st 7692  df-2nd 7693  df-supp 7834  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-oadd 8109  df-er 8292  df-map 8411  df-pm 8412  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-fsupp 8837  df-fi 8878  df-sup 8909  df-inf 8910  df-oi 8977  df-card 9371  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-div 11301  df-nn 11642  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-uz 12247  df-q 12352  df-rp 12393  df-xadd 12511  df-ioo 12745  df-ioc 12746  df-ico 12747  df-icc 12748  df-fz 12896  df-fzo 13037  df-fl 13165  df-seq 13373  df-exp 13433  df-hash 13694  df-cj 14461  df-re 14462  df-im 14463  df-sqrt 14597  df-abs 14598  df-clim 14848  df-rlim 14849  df-sum 15046  df-struct 16488  df-ndx 16489  df-slot 16490  df-base 16492  df-sets 16493  df-ress 16494  df-plusg 16581  df-mulr 16582  df-starv 16583  df-tset 16587  df-ple 16588  df-ds 16590  df-unif 16591  df-rest 16699  df-topn 16700  df-0g 16718  df-gsum 16719  df-topgen 16720  df-ordt 16777  df-xrs 16778  df-mre 16860  df-mrc 16861  df-acs 16863  df-ps 17813  df-tsr 17814  df-mgm 17855  df-sgrp 17904  df-mnd 17915  df-submnd 17960  df-grp 18109  df-minusg 18110  df-cntz 18450  df-cmn 18911  df-abl 18912  df-mgp 19243  df-ur 19255  df-ring 19302  df-cring 19303  df-fbas 20545  df-fg 20546  df-cnfld 20549  df-top 21505  df-topon 21522  df-topsp 21544  df-bases 21557  df-ntr 21631  df-nei 21709  df-cn 21838  df-haus 21926  df-fil 22457  df-fm 22549  df-flim 22550  df-flf 22551  df-tsms 22738  df-esum 31291 This theorem is referenced by:  esumcvg  31349  esumcvgsum  31351
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