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Theorem esumpcvgval 29921
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 11918 . . . 4 < Or ℝ*
21a1i 11 . . 3 (𝜑 → < Or ℝ*)
3 nnuz 11667 . . . . 5 ℕ = (ℤ‘1)
4 1zzd 11352 . . . . 5 (𝜑 → 1 ∈ ℤ)
5 esumpcvgval.1 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
6 esumpcvgval.2 . . . . . . . . . . . 12 (𝑘 = 𝑙𝐴 = 𝐵)
7 eqcom 2628 . . . . . . . . . . . 12 (𝑘 = 𝑙𝑙 = 𝑘)
8 eqcom 2628 . . . . . . . . . . . 12 (𝐴 = 𝐵𝐵 = 𝐴)
96, 7, 83imtr3i 280 . . . . . . . . . . 11 (𝑙 = 𝑘𝐵 = 𝐴)
109cbvmptv 4710 . . . . . . . . . 10 (𝑙 ∈ ℕ ↦ 𝐵) = (𝑘 ∈ ℕ ↦ 𝐴)
115, 10fmptd 6340 . . . . . . . . 9 (𝜑 → (𝑙 ∈ ℕ ↦ 𝐵):ℕ⟶(0[,)+∞))
1211ffvelrnda 6315 . . . . . . . 8 ((𝜑𝑥 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞))
13 elrege0 12220 . . . . . . . . 9 (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ (0[,)+∞) ↔ (((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑥)))
1413simplbi 476 . . . . . . . 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 12770 . . . . . 6 (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)):ℕ⟶ℝ)
1711adantr 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑙 ∈ ℕ ↦ 𝐵):ℕ⟶(0[,)+∞))
18 simpr 477 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
1918peano2nnd 10981 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
2017, 19ffvelrnd 6316 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞))
21 elrege0 12220 . . . . . . . . . 10 (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ (0[,)+∞) ↔ (((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1)) ∈ ℝ ∧ 0 ≤ ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
2221simprbi 480 . . . . . . . . 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 6315 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ∈ ℝ)
2521simplbi 476 . . . . . . . . . 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 10559 . . . . . . . 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 222 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) + ((𝑙 ∈ ℕ ↦ 𝐵)‘(𝑛 + 1))))
2918, 3syl6eleq 2708 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
30 seqp1 12756 . . . . . . . 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 4641 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘(𝑛 + 1)))
33 simpr 477 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
3410fvmpt2 6248 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ 𝐴 ∈ (0[,)+∞)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
3533, 5, 34syl2anc 692 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
36 rge0ssre 12222 . . . . . . . . 9 (0[,)+∞) ⊆ ℝ
3736, 5sseldi 3581 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
3816feqmptd 6206 . . . . . . . . . 10 (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)))
39 simpll 789 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
40 elfznn 12312 . . . . . . . . . . . . . . 15 (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
4140adantl 482 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
4239, 41, 35syl2anc 692 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
4337recnd 10012 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ ℂ)
4439, 41, 43syl2anc 692 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ ℂ)
4542, 29, 44fsumser 14394 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
4645eqcomd 2627 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) = Σ𝑘 ∈ (1...𝑛)𝐴)
4746mpteq2dva 4704 . . . . . . . . . 10 (𝜑 → (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴))
4838, 47eqtr2d 2656 . . . . . . . . 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 2699 . . . . . . . 8 (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ∈ dom ⇝ )
513, 4, 35, 37, 50isumrecl 14424 . . . . . . 7 (𝜑 → Σ𝑘 ∈ ℕ 𝐴 ∈ ℝ)
52 1zzd 11352 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 1 ∈ ℤ)
53 fzfid 12712 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
54 fzssuz 12324 . . . . . . . . . . . 12 (1...𝑛) ⊆ (ℤ‘1)
5554, 3sseqtr4i 3617 . . . . . . . . . . 11 (1...𝑛) ⊆ ℕ
5655a1i 11 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
5735adantlr 750 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
5837adantlr 750 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
595adantlr 750 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
60 elrege0 12220 . . . . . . . . . . . 12 (𝐴 ∈ (0[,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
6160simprbi 480 . . . . . . . . . . 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 14502 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → Σ𝑘 ∈ (1...𝑛)𝐴 ≤ Σ𝑘 ∈ ℕ 𝐴)
6545, 64eqbrtrrd 4637 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴)
6665ralrimiva 2960 . . . . . . 7 (𝜑 → ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴)
67 breq2 4617 . . . . . . . . 9 (𝑠 = Σ𝑘 ∈ ℕ 𝐴 → ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ↔ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴))
6867ralbidv 2980 . . . . . . . 8 (𝑠 = Σ𝑘 ∈ ℕ 𝐴 → (∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ↔ ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ Σ𝑘 ∈ ℕ 𝐴))
6968rspcev 3295 . . . . . . 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)
7051, 66, 69syl2anc 692 . . . . . 6 (𝜑 → ∃𝑠 ∈ ℝ ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠)
713, 4, 16, 32, 70climsup 14334 . . . . 5 (𝜑 → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⇝ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
723, 4, 71, 24climrecl 14248 . . . 4 (𝜑 → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ)
7372rexrd 10033 . . 3 (𝜑 → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ*)
74 eqid 2621 . . . . . . 7 (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴) = (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)
75 sumex 14352 . . . . . . 7 Σ𝑘𝑏 𝐴 ∈ V
7674, 75elrnmpti 5336 . . . . . 6 (𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘𝑏 𝐴)
77 ssnnssfz 29391 . . . . . . . . . 10 (𝑏 ∈ (𝒫 ℕ ∩ Fin) → ∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚))
78 fzfid 12712 . . . . . . . . . . . . . 14 ((𝜑𝑏 ⊆ (1...𝑚)) → (1...𝑚) ∈ Fin)
79 elfznn 12312 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (1...𝑚) → 𝑘 ∈ ℕ)
8079, 5sylan2 491 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (1...𝑚)) → 𝐴 ∈ (0[,)+∞))
8160simplbi 476 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (0[,)+∞) → 𝐴 ∈ ℝ)
8280, 81syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ)
8382adantlr 750 . . . . . . . . . . . . . 14 (((𝜑𝑏 ⊆ (1...𝑚)) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ)
8480, 61syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...𝑚)) → 0 ≤ 𝐴)
8584adantlr 750 . . . . . . . . . . . . . 14 (((𝜑𝑏 ⊆ (1...𝑚)) ∧ 𝑘 ∈ (1...𝑚)) → 0 ≤ 𝐴)
86 simpr 477 . . . . . . . . . . . . . 14 ((𝜑𝑏 ⊆ (1...𝑚)) → 𝑏 ⊆ (1...𝑚))
8778, 83, 85, 86fsumless 14455 . . . . . . . . . . . . 13 ((𝜑𝑏 ⊆ (1...𝑚)) → Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
8887ex 450 . . . . . . . . . . . 12 (𝜑 → (𝑏 ⊆ (1...𝑚) → Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
8988reximdv 3010 . . . . . . . . . . 11 (𝜑 → (∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚) → ∃𝑚 ∈ ℕ Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
9089imp 445 . . . . . . . . . 10 ((𝜑 ∧ ∃𝑚 ∈ ℕ 𝑏 ⊆ (1...𝑚)) → ∃𝑚 ∈ ℕ Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
9177, 90sylan2 491 . . . . . . . . 9 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → ∃𝑚 ∈ ℕ Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
92 breq1 4616 . . . . . . . . . 10 (𝑥 = Σ𝑘𝑏 𝐴 → (𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴 ↔ Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
9392rexbidv 3045 . . . . . . . . 9 (𝑥 = Σ𝑘𝑏 𝐴 → (∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴 ↔ ∃𝑚 ∈ ℕ Σ𝑘𝑏 𝐴 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
9491, 93syl5ibrcom 237 . . . . . . . 8 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (𝑥 = Σ𝑘𝑏 𝐴 → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
9594rexlimdva 3024 . . . . . . 7 (𝜑 → (∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘𝑏 𝐴 → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴))
9695imp 445 . . . . . 6 ((𝜑 ∧ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘𝑏 𝐴) → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
9776, 96sylan2b 492 . . . . 5 ((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) → ∃𝑚 ∈ ℕ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
98 simpr 477 . . . . . . . . . 10 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘𝑏 𝐴) → 𝑥 = Σ𝑘𝑏 𝐴)
99 inss2 3812 . . . . . . . . . . . . 13 (𝒫 ℕ ∩ Fin) ⊆ Fin
100 simpr 477 . . . . . . . . . . . . 13 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ (𝒫 ℕ ∩ Fin))
10199, 100sseldi 3581 . . . . . . . . . . . 12 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ Fin)
102 simpll 789 . . . . . . . . . . . . . 14 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝜑)
103 inss1 3811 . . . . . . . . . . . . . . . . 17 (𝒫 ℕ ∩ Fin) ⊆ 𝒫 ℕ
104 simplr 791 . . . . . . . . . . . . . . . . 17 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝑏 ∈ (𝒫 ℕ ∩ Fin))
105103, 104sseldi 3581 . . . . . . . . . . . . . . . 16 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝑏 ∈ 𝒫 ℕ)
106105elpwid 4141 . . . . . . . . . . . . . . 15 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝑏 ⊆ ℕ)
107 simpr 477 . . . . . . . . . . . . . . 15 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝑘𝑏)
108106, 107sseldd 3584 . . . . . . . . . . . . . 14 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝑘 ∈ ℕ)
109102, 108, 5syl2anc 692 . . . . . . . . . . . . 13 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝐴 ∈ (0[,)+∞))
110109, 81syl 17 . . . . . . . . . . . 12 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝐴 ∈ ℝ)
111101, 110fsumrecl 14398 . . . . . . . . . . 11 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → Σ𝑘𝑏 𝐴 ∈ ℝ)
112111adantr 481 . . . . . . . . . 10 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘𝑏 𝐴) → Σ𝑘𝑏 𝐴 ∈ ℝ)
11398, 112eqeltrd 2698 . . . . . . . . 9 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑥 = Σ𝑘𝑏 𝐴) → 𝑥 ∈ ℝ)
114113r19.29an 3070 . . . . . . . 8 ((𝜑 ∧ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑥 = Σ𝑘𝑏 𝐴) → 𝑥 ∈ ℝ)
11576, 114sylan2b 492 . . . . . . 7 ((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) → 𝑥 ∈ ℝ)
116115adantr 481 . . . . . 6 (((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → 𝑥 ∈ ℝ)
117 fzfid 12712 . . . . . . . 8 (𝜑 → (1...𝑚) ∈ Fin)
118117, 82fsumrecl 14398 . . . . . . 7 (𝜑 → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ℝ)
119118ad2antrr 761 . . . . . 6 (((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ℝ)
12072ad2antrr 761 . . . . . 6 (((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ)
121 simprr 795 . . . . . 6 (((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)
122 frn 6010 . . . . . . . . 9 (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)):ℕ⟶ℝ → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ)
12316, 122syl 17 . . . . . . . 8 (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ)
124123ad2antrr 761 . . . . . . 7 (((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ)
125 1nn 10975 . . . . . . . . . 10 1 ∈ ℕ
126125ne0ii 3899 . . . . . . . . 9 ℕ ≠ ∅
127 dm0rn0 5302 . . . . . . . . . . 11 (dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅)
128 fdm 6008 . . . . . . . . . . . . 13 (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)):ℕ⟶ℝ → dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ℕ)
12916, 128syl 17 . . . . . . . . . . . 12 (𝜑 → dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ℕ)
130129eqeq1d 2623 . . . . . . . . . . 11 (𝜑 → (dom seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ℕ = ∅))
131127, 130syl5bbr 274 . . . . . . . . . 10 (𝜑 → (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = ∅ ↔ ℕ = ∅))
132131necon3bid 2834 . . . . . . . . 9 (𝜑 → (ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅ ↔ ℕ ≠ ∅))
133126, 132mpbiri 248 . . . . . . . 8 (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅)
134133ad2antrr 761 . . . . . . 7 (((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅)
135 1z 11351 . . . . . . . . . . . . . . . 16 1 ∈ ℤ
136 seqfn 12753 . . . . . . . . . . . . . . . 16 (1 ∈ ℤ → seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn (ℤ‘1))
137135, 136ax-mp 5 . . . . . . . . . . . . . . 15 seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn (ℤ‘1)
1383fneq2i 5944 . . . . . . . . . . . . . . 15 (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ ↔ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn (ℤ‘1))
139137, 138mpbir 221 . . . . . . . . . . . . . 14 seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ
140 dffn5 6198 . . . . . . . . . . . . . 14 (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) Fn ℕ ↔ seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)))
141139, 140mpbi 220 . . . . . . . . . . . . 13 seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) = (𝑛 ∈ ℕ ↦ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
142 fvex 6158 . . . . . . . . . . . . 13 (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ∈ V
143141, 142elrnmpti 5336 . . . . . . . . . . . 12 (𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
144 r19.29 3065 . . . . . . . . . . . . 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)))
145 breq1 4616 . . . . . . . . . . . . . . 15 (𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → (𝑧𝑠 ↔ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠))
146145biimparc 504 . . . . . . . . . . . . . 14 (((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑧𝑠)
147146rexlimivw 3022 . . . . . . . . . . . . 13 (∃𝑛 ∈ ℕ ((seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑧𝑠)
148144, 147syl 17 . . . . . . . . . . . 12 ((∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 ∧ ∃𝑛 ∈ ℕ 𝑧 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑧𝑠)
149143, 148sylan2b 492 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))) → 𝑧𝑠)
150149ralrimiva 2960 . . . . . . . . . 10 (∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 → ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧𝑠)
151150reximi 3005 . . . . . . . . 9 (∃𝑠 ∈ ℝ ∀𝑛 ∈ ℕ (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ≤ 𝑠 → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧𝑠)
15270, 151syl 17 . . . . . . . 8 (𝜑 → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧𝑠)
153152ad2antrr 761 . . . . . . 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)
154 simpr 477 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
155 simpll 789 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝜑)
15679adantl 482 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝑘 ∈ ℕ)
157155, 156, 35syl2anc 692 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → ((𝑙 ∈ ℕ ↦ 𝐵)‘𝑘) = 𝐴)
158154, 3syl6eleq 2708 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ (ℤ‘1))
159155, 156, 5syl2anc 692 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ (0[,)+∞))
160159, 81syl 17 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℝ)
161160recnd 10012 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑚)) → 𝐴 ∈ ℂ)
162157, 158, 161fsumser 14394 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚))
163 fveq2 6148 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚))
164163eqeq2d 2631 . . . . . . . . . . 11 (𝑛 = 𝑚 → (Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) ↔ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚)))
165164rspcev 3295 . . . . . . . . . 10 ((𝑚 ∈ ℕ ∧ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑚)) → ∃𝑛 ∈ ℕ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
166154, 162, 165syl2anc 692 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → ∃𝑛 ∈ ℕ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
167141, 142elrnmpti 5336 . . . . . . . . 9 𝑘 ∈ (1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ Σ𝑘 ∈ (1...𝑚)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
168166, 167sylibr 224 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → Σ𝑘 ∈ (1...𝑚)𝐴 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)))
169168ad2ant2r 782 . . . . . . 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)))
170 suprub 10928 . . . . . . 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, < ))
171124, 134, 153, 169, 170syl31anc 1326 . . . . . 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, < ))
172116, 119, 120, 121, 171letrd 10138 . . . . 5 (((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) ∧ (𝑚 ∈ ℕ ∧ 𝑥 ≤ Σ𝑘 ∈ (1...𝑚)𝐴)) → 𝑥 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
17397, 172rexlimddv 3028 . . . 4 ((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) → 𝑥 ≤ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
17472adantr 481 . . . . 5 ((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ)
175115, 174lenltd 10127 . . . 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))
176173, 175mpbid 222 . . 3 ((𝜑𝑥 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)) → ¬ sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) < 𝑥)
177 simpr1r 1117 . . . . . . 7 ((𝜑 ∧ ((𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0 ≤ 𝑥𝑥 = +∞)) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
1781773anassrs 1287 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
17973ad3antrrr 765 . . . . . . . 8 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ*)
180 pnfnlt 11906 . . . . . . . 8 (sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ∈ ℝ* → ¬ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
181179, 180syl 17 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → ¬ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
182 breq1 4616 . . . . . . . . 9 (𝑥 = +∞ → (𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )))
183182notbid 308 . . . . . . . 8 (𝑥 = +∞ → (¬ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ) ↔ ¬ +∞ < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )))
184183adantl 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, < )))
185181, 184mpbird 247 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 = +∞) → ¬ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
186178, 185pm2.21dd 186 . . . . 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)
187 simplll 797 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝜑)
188 simpr1l 1116 . . . . . . . 8 ((𝜑 ∧ ((𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0 ≤ 𝑥𝑥 < +∞)) → 𝑥 ∈ ℝ*)
1891883anassrs 1287 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 ∈ ℝ*)
190 simplr 791 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 0 ≤ 𝑥)
191 simpr 477 . . . . . . 7 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 < +∞)
192 0xr 10030 . . . . . . . 8 0 ∈ ℝ*
193 pnfxr 10036 . . . . . . . 8 +∞ ∈ ℝ*
194 elico1 12160 . . . . . . . 8 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥𝑥 < +∞)))
195192, 193, 194mp2an 707 . . . . . . 7 (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥𝑥 < +∞))
196189, 190, 191, 195syl3anbrc 1244 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 ∈ (0[,)+∞))
197 simpr1r 1117 . . . . . . 7 ((𝜑 ∧ ((𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < )) ∧ 0 ≤ 𝑥𝑥 < +∞)) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
1981973anassrs 1287 . . . . . 6 ((((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) ∧ 𝑥 < +∞) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
199123adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ℝ)
200133adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ≠ ∅)
201152adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑠 ∈ ℝ ∀𝑧 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑧𝑠)
202199, 200, 2013jca 1240 . . . . . . . 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))
203 simprl 793 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → 𝑥 ∈ (0[,)+∞))
20436, 203sseldi 3581 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → 𝑥 ∈ ℝ)
205 simprr 795 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
206 suprlub 10931 . . . . . . . . 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))
207206biimpa 501 . . . . . . . 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)
208202, 204, 205, 207syl21anc 1322 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦)
20940ssriv 3587 . . . . . . . . . . . . . . . . 17 (1...𝑛) ⊆ ℕ
210 ovex 6632 . . . . . . . . . . . . . . . . . 18 (1...𝑛) ∈ V
211210elpw 4136 . . . . . . . . . . . . . . . . 17 ((1...𝑛) ∈ 𝒫 ℕ ↔ (1...𝑛) ⊆ ℕ)
212209, 211mpbir 221 . . . . . . . . . . . . . . . 16 (1...𝑛) ∈ 𝒫 ℕ
213 fzfi 12711 . . . . . . . . . . . . . . . 16 (1...𝑛) ∈ Fin
214 elin 3774 . . . . . . . . . . . . . . . 16 ((1...𝑛) ∈ (𝒫 ℕ ∩ Fin) ↔ ((1...𝑛) ∈ 𝒫 ℕ ∧ (1...𝑛) ∈ Fin))
215212, 213, 214mpbir2an 954 . . . . . . . . . . . . . . 15 (1...𝑛) ∈ (𝒫 ℕ ∩ Fin)
216215a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → (1...𝑛) ∈ (𝒫 ℕ ∩ Fin))
217 simpr 477 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
21845adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → Σ𝑘 ∈ (1...𝑛)𝐴 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
219217, 218eqtr4d 2658 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → 𝑦 = Σ𝑘 ∈ (1...𝑛)𝐴)
220 sumeq1 14353 . . . . . . . . . . . . . . . 16 (𝑏 = (1...𝑛) → Σ𝑘𝑏 𝐴 = Σ𝑘 ∈ (1...𝑛)𝐴)
221220eqeq2d 2631 . . . . . . . . . . . . . . 15 (𝑏 = (1...𝑛) → (𝑦 = Σ𝑘𝑏 𝐴𝑦 = Σ𝑘 ∈ (1...𝑛)𝐴))
222221rspcev 3295 . . . . . . . . . . . . . 14 (((1...𝑛) ∈ (𝒫 ℕ ∩ Fin) ∧ 𝑦 = Σ𝑘 ∈ (1...𝑛)𝐴) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘𝑏 𝐴)
223216, 219, 222syl2anc 692 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛)) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘𝑏 𝐴)
224223ex 450 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘𝑏 𝐴))
225224rexlimdva 3024 . . . . . . . . . . 11 (𝜑 → (∃𝑛 ∈ ℕ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘𝑏 𝐴))
226141, 142elrnmpti 5336 . . . . . . . . . . 11 (𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ↔ ∃𝑛 ∈ ℕ 𝑦 = (seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))‘𝑛))
22774, 75elrnmpti 5336 . . . . . . . . . . 11 (𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)𝑦 = Σ𝑘𝑏 𝐴)
228225, 226, 2273imtr4g 285 . . . . . . . . . 10 (𝜑 → (𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) → 𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)))
229228ssrdv 3589 . . . . . . . . 9 (𝜑 → ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)) ⊆ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴))
230 ssrexv 3646 . . . . . . . . 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))
231229, 230syl 17 . . . . . . . 8 (𝜑 → (∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦 → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)𝑥 < 𝑦))
232231imp 445 . . . . . . 7 ((𝜑 ∧ ∃𝑦 ∈ ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵))𝑥 < 𝑦) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)𝑥 < 𝑦)
233208, 232syldan 487 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)𝑥 < 𝑦)
234187, 196, 198, 233syl12anc 1321 . . . . 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)
235 simplrl 799 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) → 𝑥 ∈ ℝ*)
236 xrlelttric 29360 . . . . . . . 8 ((+∞ ∈ ℝ*𝑥 ∈ ℝ*) → (+∞ ≤ 𝑥𝑥 < +∞))
237193, 236mpan 705 . . . . . . 7 (𝑥 ∈ ℝ* → (+∞ ≤ 𝑥𝑥 < +∞))
238 xgepnf 11940 . . . . . . . 8 (𝑥 ∈ ℝ* → (+∞ ≤ 𝑥𝑥 = +∞))
239238orbi1d 738 . . . . . . 7 (𝑥 ∈ ℝ* → ((+∞ ≤ 𝑥𝑥 < +∞) ↔ (𝑥 = +∞ ∨ 𝑥 < +∞)))
240237, 239mpbid 222 . . . . . 6 (𝑥 ∈ ℝ* → (𝑥 = +∞ ∨ 𝑥 < +∞))
241235, 240syl 17 . . . . 5 (((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) → (𝑥 = +∞ ∨ 𝑥 < +∞))
242186, 234, 241mpjaodan 826 . . . 4 (((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) ∧ 0 ≤ 𝑥) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)𝑥 < 𝑦)
243 0elpw 4794 . . . . . . . . 9 ∅ ∈ 𝒫 ℕ
244 0fin 8132 . . . . . . . . 9 ∅ ∈ Fin
245 elin 3774 . . . . . . . . 9 (∅ ∈ (𝒫 ℕ ∩ Fin) ↔ (∅ ∈ 𝒫 ℕ ∧ ∅ ∈ Fin))
246243, 244, 245mpbir2an 954 . . . . . . . 8 ∅ ∈ (𝒫 ℕ ∩ Fin)
247 sum0 14385 . . . . . . . . 9 Σ𝑘 ∈ ∅ 𝐴 = 0
248247eqcomi 2630 . . . . . . . 8 0 = Σ𝑘 ∈ ∅ 𝐴
249 sumeq1 14353 . . . . . . . . . 10 (𝑏 = ∅ → Σ𝑘𝑏 𝐴 = Σ𝑘 ∈ ∅ 𝐴)
250249eqeq2d 2631 . . . . . . . . 9 (𝑏 = ∅ → (0 = Σ𝑘𝑏 𝐴 ↔ 0 = Σ𝑘 ∈ ∅ 𝐴))
251250rspcev 3295 . . . . . . . 8 ((∅ ∈ (𝒫 ℕ ∩ Fin) ∧ 0 = Σ𝑘 ∈ ∅ 𝐴) → ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘𝑏 𝐴)
252246, 248, 251mp2an 707 . . . . . . 7 𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘𝑏 𝐴
25374, 75elrnmpti 5336 . . . . . . 7 (0 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴) ↔ ∃𝑏 ∈ (𝒫 ℕ ∩ Fin)0 = Σ𝑘𝑏 𝐴)
254252, 253mpbir 221 . . . . . 6 0 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)
255 breq2 4617 . . . . . . 7 (𝑦 = 0 → (𝑥 < 𝑦𝑥 < 0))
256255rspcev 3295 . . . . . 6 ((0 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴) ∧ 𝑥 < 0) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)𝑥 < 𝑦)
257254, 256mpan 705 . . . . 5 (𝑥 < 0 → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)𝑥 < 𝑦)
258257adantl 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)
259 xrlelttric 29360 . . . . . 6 ((0 ∈ ℝ*𝑥 ∈ ℝ*) → (0 ≤ 𝑥𝑥 < 0))
260192, 259mpan 705 . . . . 5 (𝑥 ∈ ℝ* → (0 ≤ 𝑥𝑥 < 0))
261260ad2antrl 763 . . . 4 ((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → (0 ≤ 𝑥𝑥 < 0))
262242, 258, 261mpjaodan 826 . . 3 ((𝜑 ∧ (𝑥 ∈ ℝ*𝑥 < sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))) → ∃𝑦 ∈ ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴)𝑥 < 𝑦)
2632, 73, 176, 262eqsupd 8307 . 2 (𝜑 → sup(ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴), ℝ*, < ) = sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
264 nfv 1840 . . 3 𝑘𝜑
265 nfcv 2761 . . 3 𝑘
266 nnex 10970 . . . 4 ℕ ∈ V
267266a1i 11 . . 3 (𝜑 → ℕ ∈ V)
268 icossicc 12202 . . . 4 (0[,)+∞) ⊆ (0[,]+∞)
269268, 5sseldi 3581 . . 3 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
270 elex 3198 . . . . . 6 (𝑏 ∈ (𝒫 ℕ ∩ Fin) → 𝑏 ∈ V)
271270adantl 482 . . . . 5 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → 𝑏 ∈ V)
272 eqid 2621 . . . . . 6 (𝑘𝑏𝐴) = (𝑘𝑏𝐴)
273109, 272fmptd 6340 . . . . 5 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (𝑘𝑏𝐴):𝑏⟶(0[,)+∞))
274 esumpfinvallem 29917 . . . . 5 ((𝑏 ∈ V ∧ (𝑘𝑏𝐴):𝑏⟶(0[,)+∞)) → (ℂfld Σg (𝑘𝑏𝐴)) = ((ℝ*𝑠s (0[,]+∞)) Σg (𝑘𝑏𝐴)))
275271, 273, 274syl2anc 692 . . . 4 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (ℂfld Σg (𝑘𝑏𝐴)) = ((ℝ*𝑠s (0[,]+∞)) Σg (𝑘𝑏𝐴)))
276110recnd 10012 . . . . 5 (((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) ∧ 𝑘𝑏) → 𝐴 ∈ ℂ)
277101, 276gsumfsum 19732 . . . 4 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → (ℂfld Σg (𝑘𝑏𝐴)) = Σ𝑘𝑏 𝐴)
278275, 277eqtr3d 2657 . . 3 ((𝜑𝑏 ∈ (𝒫 ℕ ∩ Fin)) → ((ℝ*𝑠s (0[,]+∞)) Σg (𝑘𝑏𝐴)) = Σ𝑘𝑏 𝐴)
279264, 265, 267, 269, 278esumval 29889 . 2 (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = sup(ran (𝑏 ∈ (𝒫 ℕ ∩ Fin) ↦ Σ𝑘𝑏 𝐴), ℝ*, < ))
2803, 4, 35, 43, 71isumclim 14416 . 2 (𝜑 → Σ𝑘 ∈ ℕ 𝐴 = sup(ran seq1( + , (𝑙 ∈ ℕ ↦ 𝐵)), ℝ, < ))
281263, 279, 2803eqtr4d 2665 1 (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3186  cin 3554  wss 3555  c0 3891  𝒫 cpw 4130   class class class wbr 4613  cmpt 4673   Or wor 4994  dom cdm 5074  ran crn 5075   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  Fincfn 7899  supcsup 8290  cc 9878  cr 9879  0cc0 9880  1c1 9881   + caddc 9883  +∞cpnf 10015  *cxr 10017   < clt 10018  cle 10019  cn 10964  cz 11321  cuz 11631  [,)cico 12119  [,]cicc 12120  ...cfz 12268  seqcseq 12741  cli 14149  Σcsu 14350  s cress 15782   Σg cgsu 16022  *𝑠cxrs 16081  fldccnfld 19665  Σ*cesum 29870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958  ax-addf 9959  ax-mulf 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-fi 8261  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-q 11733  df-rp 11777  df-xadd 11891  df-ioo 12121  df-ioc 12122  df-ico 12123  df-icc 12124  df-fz 12269  df-fzo 12407  df-fl 12533  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-rlim 14154  df-sum 14351  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-starv 15877  df-tset 15881  df-ple 15882  df-ds 15885  df-unif 15886  df-rest 16004  df-topn 16005  df-0g 16023  df-gsum 16024  df-topgen 16025  df-ordt 16082  df-xrs 16083  df-mre 16167  df-mrc 16168  df-acs 16170  df-ps 17121  df-tsr 17122  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-grp 17346  df-minusg 17347  df-cntz 17671  df-cmn 18116  df-abl 18117  df-mgp 18411  df-ur 18423  df-ring 18470  df-cring 18471  df-fbas 19662  df-fg 19663  df-cnfld 19666  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-ntr 20734  df-nei 20812  df-cn 20941  df-haus 21029  df-fil 21560  df-fm 21652  df-flim 21653  df-flf 21654  df-tsms 21840  df-esum 29871
This theorem is referenced by:  esumcvg  29929  esumcvgsum  29931
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