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Theorem cvgrat 14596
Description: Ratio test for convergence of a complex infinite series. If the ratio 𝐴 of the absolute values of successive terms in an infinite sequence 𝐹 is less than 1 for all terms beyond some index 𝐵, then the infinite sum of the terms of 𝐹 converges to a complex number. Equivalent to first part of Exercise 4 of [Gleason] p. 182. (Contributed by NM, 26-Apr-2005.) (Proof shortened by Mario Carneiro, 27-Apr-2014.)
Hypotheses
Ref Expression
cvgrat.1 𝑍 = (ℤ𝑀)
cvgrat.2 𝑊 = (ℤ𝑁)
cvgrat.3 (𝜑𝐴 ∈ ℝ)
cvgrat.4 (𝜑𝐴 < 1)
cvgrat.5 (𝜑𝑁𝑍)
cvgrat.6 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
cvgrat.7 ((𝜑𝑘𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹𝑘))))
Assertion
Ref Expression
cvgrat (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
Distinct variable groups:   𝐴,𝑘   𝑘,𝐹   𝑘,𝑀   𝑘,𝑁   𝜑,𝑘   𝑘,𝑊   𝑘,𝑍

Proof of Theorem cvgrat
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 cvgrat.2 . . 3 𝑊 = (ℤ𝑁)
2 cvgrat.5 . . . . . . 7 (𝜑𝑁𝑍)
3 cvgrat.1 . . . . . . 7 𝑍 = (ℤ𝑀)
42, 3syl6eleq 2709 . . . . . 6 (𝜑𝑁 ∈ (ℤ𝑀))
5 eluzelz 11682 . . . . . 6 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
64, 5syl 17 . . . . 5 (𝜑𝑁 ∈ ℤ)
7 uzid 11687 . . . . 5 (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ𝑁))
86, 7syl 17 . . . 4 (𝜑𝑁 ∈ (ℤ𝑁))
98, 1syl6eleqr 2710 . . 3 (𝜑𝑁𝑊)
10 oveq1 6642 . . . . . . 7 (𝑛 = 𝑘 → (𝑛𝑁) = (𝑘𝑁))
1110oveq2d 6651 . . . . . 6 (𝑛 = 𝑘 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))
12 eqid 2620 . . . . . 6 (𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁))) = (𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))
13 ovex 6663 . . . . . 6 (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)) ∈ V
1411, 12, 13fvmpt 6269 . . . . 5 (𝑘𝑊 → ((𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))
1514adantl 482 . . . 4 ((𝜑𝑘𝑊) → ((𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))
16 0re 10025 . . . . . . 7 0 ∈ ℝ
17 cvgrat.3 . . . . . . 7 (𝜑𝐴 ∈ ℝ)
18 ifcl 4121 . . . . . . 7 ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ)
1916, 17, 18sylancr 694 . . . . . 6 (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ)
2019adantr 481 . . . . 5 ((𝜑𝑘𝑊) → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ)
21 simpr 477 . . . . . . 7 ((𝜑𝑘𝑊) → 𝑘𝑊)
2221, 1syl6eleq 2709 . . . . . 6 ((𝜑𝑘𝑊) → 𝑘 ∈ (ℤ𝑁))
23 uznn0sub 11704 . . . . . 6 (𝑘 ∈ (ℤ𝑁) → (𝑘𝑁) ∈ ℕ0)
2422, 23syl 17 . . . . 5 ((𝜑𝑘𝑊) → (𝑘𝑁) ∈ ℕ0)
2520, 24reexpcld 13008 . . . 4 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)) ∈ ℝ)
2615, 25eqeltrd 2699 . . 3 ((𝜑𝑘𝑊) → ((𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))‘𝑘) ∈ ℝ)
27 uzss 11693 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → (ℤ𝑁) ⊆ (ℤ𝑀))
284, 27syl 17 . . . . . 6 (𝜑 → (ℤ𝑁) ⊆ (ℤ𝑀))
2928, 1, 33sstr4g 3638 . . . . 5 (𝜑𝑊𝑍)
3029sselda 3595 . . . 4 ((𝜑𝑘𝑊) → 𝑘𝑍)
31 cvgrat.6 . . . 4 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
3230, 31syldan 487 . . 3 ((𝜑𝑘𝑊) → (𝐹𝑘) ∈ ℂ)
3323adantl 482 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝑁)) → (𝑘𝑁) ∈ ℕ0)
34 oveq2 6643 . . . . . . . . 9 (𝑛 = (𝑘𝑁) → (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))
35 eqid 2620 . . . . . . . . 9 (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) = (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))
3634, 35, 13fvmpt 6269 . . . . . . . 8 ((𝑘𝑁) ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))‘(𝑘𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))
3733, 36syl 17 . . . . . . 7 ((𝜑𝑘 ∈ (ℤ𝑁)) → ((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))‘(𝑘𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))
386zcnd 11468 . . . . . . . 8 (𝜑𝑁 ∈ ℂ)
39 eluzelz 11682 . . . . . . . . 9 (𝑘 ∈ (ℤ𝑁) → 𝑘 ∈ ℤ)
4039zcnd 11468 . . . . . . . 8 (𝑘 ∈ (ℤ𝑁) → 𝑘 ∈ ℂ)
41 nn0ex 11283 . . . . . . . . . 10 0 ∈ V
4241mptex 6471 . . . . . . . . 9 (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) ∈ V
4342shftval 13795 . . . . . . . 8 ((𝑁 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) shift 𝑁)‘𝑘) = ((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))‘(𝑘𝑁)))
4438, 40, 43syl2an 494 . . . . . . 7 ((𝜑𝑘 ∈ (ℤ𝑁)) → (((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) shift 𝑁)‘𝑘) = ((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))‘(𝑘𝑁)))
45 simpr 477 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝑁)) → 𝑘 ∈ (ℤ𝑁))
4645, 1syl6eleqr 2710 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝑁)) → 𝑘𝑊)
4746, 14syl 17 . . . . . . 7 ((𝜑𝑘 ∈ (ℤ𝑁)) → ((𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))
4837, 44, 473eqtr4rd 2665 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑁)) → ((𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))‘𝑘) = (((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) shift 𝑁)‘𝑘))
496, 48seqfeq 12809 . . . . 5 (𝜑 → seq𝑁( + , (𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))) = seq𝑁( + , ((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) shift 𝑁)))
5042seqshft 13806 . . . . . 6 ((𝑁 ∈ ℤ ∧ 𝑁 ∈ ℤ) → seq𝑁( + , ((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) shift 𝑁)) = (seq(𝑁𝑁)( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁))
516, 6, 50syl2anc 692 . . . . 5 (𝜑 → seq𝑁( + , ((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) shift 𝑁)) = (seq(𝑁𝑁)( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁))
5238subidd 10365 . . . . . . 7 (𝜑 → (𝑁𝑁) = 0)
5352seqeq1d 12790 . . . . . 6 (𝜑 → seq(𝑁𝑁)( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) = seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))))
5453oveq1d 6650 . . . . 5 (𝜑 → (seq(𝑁𝑁)( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) = (seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁))
5549, 51, 543eqtrd 2658 . . . 4 (𝜑 → seq𝑁( + , (𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))) = (seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁))
5619recnd 10053 . . . . . . 7 (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℂ)
57 max2 12003 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → 0 ≤ if(𝐴 ≤ 0, 0, 𝐴))
5817, 16, 57sylancl 693 . . . . . . . . 9 (𝜑 → 0 ≤ if(𝐴 ≤ 0, 0, 𝐴))
5919, 58absidd 14142 . . . . . . . 8 (𝜑 → (abs‘if(𝐴 ≤ 0, 0, 𝐴)) = if(𝐴 ≤ 0, 0, 𝐴))
60 0lt1 10535 . . . . . . . . 9 0 < 1
61 cvgrat.4 . . . . . . . . 9 (𝜑𝐴 < 1)
62 breq1 4647 . . . . . . . . . 10 (0 = if(𝐴 ≤ 0, 0, 𝐴) → (0 < 1 ↔ if(𝐴 ≤ 0, 0, 𝐴) < 1))
63 breq1 4647 . . . . . . . . . 10 (𝐴 = if(𝐴 ≤ 0, 0, 𝐴) → (𝐴 < 1 ↔ if(𝐴 ≤ 0, 0, 𝐴) < 1))
6462, 63ifboth 4115 . . . . . . . . 9 ((0 < 1 ∧ 𝐴 < 1) → if(𝐴 ≤ 0, 0, 𝐴) < 1)
6560, 61, 64sylancr 694 . . . . . . . 8 (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) < 1)
6659, 65eqbrtrd 4666 . . . . . . 7 (𝜑 → (abs‘if(𝐴 ≤ 0, 0, 𝐴)) < 1)
67 oveq2 6643 . . . . . . . . 9 (𝑛 = 𝑘 → (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛) = (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘))
68 ovex 6663 . . . . . . . . 9 (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘) ∈ V
6967, 35, 68fvmpt 6269 . . . . . . . 8 (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘))
7069adantl 482 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘))
7156, 66, 70geolim 14582 . . . . . 6 (𝜑 → seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))))
72 seqex 12786 . . . . . . 7 seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ∈ V
73 climshft 14288 . . . . . . 7 ((𝑁 ∈ ℤ ∧ seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ∈ V) → ((seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))) ↔ seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴)))))
746, 72, 73sylancl 693 . . . . . 6 (𝜑 → ((seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))) ↔ seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴)))))
7571, 74mpbird 247 . . . . 5 (𝜑 → (seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))))
76 ovex 6663 . . . . . 6 (seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ∈ V
77 ovex 6663 . . . . . 6 (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))) ∈ V
7876, 77breldm 5318 . . . . 5 ((seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))) → (seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ∈ dom ⇝ )
7975, 78syl 17 . . . 4 (𝜑 → (seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ∈ dom ⇝ )
8055, 79eqeltrd 2699 . . 3 (𝜑 → seq𝑁( + , (𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))) ∈ dom ⇝ )
8131ralrimiva 2963 . . . . 5 (𝜑 → ∀𝑘𝑍 (𝐹𝑘) ∈ ℂ)
82 fveq2 6178 . . . . . . 7 (𝑘 = 𝑁 → (𝐹𝑘) = (𝐹𝑁))
8382eleq1d 2684 . . . . . 6 (𝑘 = 𝑁 → ((𝐹𝑘) ∈ ℂ ↔ (𝐹𝑁) ∈ ℂ))
8483rspcv 3300 . . . . 5 (𝑁𝑍 → (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ → (𝐹𝑁) ∈ ℂ))
852, 81, 84sylc 65 . . . 4 (𝜑 → (𝐹𝑁) ∈ ℂ)
8685abscld 14156 . . 3 (𝜑 → (abs‘(𝐹𝑁)) ∈ ℝ)
87 fveq2 6178 . . . . . . . . 9 (𝑛 = 𝑁 → (𝐹𝑛) = (𝐹𝑁))
8887fveq2d 6182 . . . . . . . 8 (𝑛 = 𝑁 → (abs‘(𝐹𝑛)) = (abs‘(𝐹𝑁)))
89 oveq1 6642 . . . . . . . . . 10 (𝑛 = 𝑁 → (𝑛𝑁) = (𝑁𝑁))
9089oveq2d 6651 . . . . . . . . 9 (𝑛 = 𝑁 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁)))
9190oveq2d 6651 . . . . . . . 8 (𝑛 = 𝑁 → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁))) = ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁))))
9288, 91breq12d 4657 . . . . . . 7 (𝑛 = 𝑁 → ((abs‘(𝐹𝑛)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁))) ↔ (abs‘(𝐹𝑁)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁)))))
9392imbi2d 330 . . . . . 6 (𝑛 = 𝑁 → ((𝜑 → (abs‘(𝐹𝑛)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))) ↔ (𝜑 → (abs‘(𝐹𝑁)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁))))))
94 fveq2 6178 . . . . . . . . 9 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
9594fveq2d 6182 . . . . . . . 8 (𝑛 = 𝑘 → (abs‘(𝐹𝑛)) = (abs‘(𝐹𝑘)))
9611oveq2d 6651 . . . . . . . 8 (𝑛 = 𝑘 → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁))) = ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))))
9795, 96breq12d 4657 . . . . . . 7 (𝑛 = 𝑘 → ((abs‘(𝐹𝑛)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁))) ↔ (abs‘(𝐹𝑘)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))))
9897imbi2d 330 . . . . . 6 (𝑛 = 𝑘 → ((𝜑 → (abs‘(𝐹𝑛)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))) ↔ (𝜑 → (abs‘(𝐹𝑘)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))))))
99 fveq2 6178 . . . . . . . . 9 (𝑛 = (𝑘 + 1) → (𝐹𝑛) = (𝐹‘(𝑘 + 1)))
10099fveq2d 6182 . . . . . . . 8 (𝑛 = (𝑘 + 1) → (abs‘(𝐹𝑛)) = (abs‘(𝐹‘(𝑘 + 1))))
101 oveq1 6642 . . . . . . . . . 10 (𝑛 = (𝑘 + 1) → (𝑛𝑁) = ((𝑘 + 1) − 𝑁))
102101oveq2d 6651 . . . . . . . . 9 (𝑛 = (𝑘 + 1) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))
103102oveq2d 6651 . . . . . . . 8 (𝑛 = (𝑘 + 1) → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁))) = ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))))
104100, 103breq12d 4657 . . . . . . 7 (𝑛 = (𝑘 + 1) → ((abs‘(𝐹𝑛)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁))) ↔ (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))))
105104imbi2d 330 . . . . . 6 (𝑛 = (𝑘 + 1) → ((𝜑 → (abs‘(𝐹𝑛)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))) ↔ (𝜑 → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))))))
10686leidd 10579 . . . . . . . 8 (𝜑 → (abs‘(𝐹𝑁)) ≤ (abs‘(𝐹𝑁)))
10752oveq2d 6651 . . . . . . . . . . 11 (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑0))
10856exp0d 12985 . . . . . . . . . . 11 (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑0) = 1)
109107, 108eqtrd 2654 . . . . . . . . . 10 (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁)) = 1)
110109oveq2d 6651 . . . . . . . . 9 (𝜑 → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁))) = ((abs‘(𝐹𝑁)) · 1))
11186recnd 10053 . . . . . . . . . 10 (𝜑 → (abs‘(𝐹𝑁)) ∈ ℂ)
112111mulid1d 10042 . . . . . . . . 9 (𝜑 → ((abs‘(𝐹𝑁)) · 1) = (abs‘(𝐹𝑁)))
113110, 112eqtrd 2654 . . . . . . . 8 (𝜑 → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁))) = (abs‘(𝐹𝑁)))
114106, 113breqtrrd 4672 . . . . . . 7 (𝜑 → (abs‘(𝐹𝑁)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁))))
115114a1i 11 . . . . . 6 (𝑁 ∈ ℤ → (𝜑 → (abs‘(𝐹𝑁)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁)))))
11632abscld 14156 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (abs‘(𝐹𝑘)) ∈ ℝ)
11786adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → (abs‘(𝐹𝑁)) ∈ ℝ)
118117, 25remulcld 10055 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))) ∈ ℝ)
11958adantr 481 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → 0 ≤ if(𝐴 ≤ 0, 0, 𝐴))
120 lemul2a 10863 . . . . . . . . . . . . 13 ((((abs‘(𝐹𝑘)) ∈ ℝ ∧ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))) ∈ ℝ ∧ (if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ ∧ 0 ≤ if(𝐴 ≤ 0, 0, 𝐴))) ∧ (abs‘(𝐹𝑘)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))) → (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹𝑘))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))))
121120ex 450 . . . . . . . . . . . 12 (((abs‘(𝐹𝑘)) ∈ ℝ ∧ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))) ∈ ℝ ∧ (if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ ∧ 0 ≤ if(𝐴 ≤ 0, 0, 𝐴))) → ((abs‘(𝐹𝑘)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))) → (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹𝑘))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))))))
122116, 118, 20, 119, 121syl112anc 1328 . . . . . . . . . . 11 ((𝜑𝑘𝑊) → ((abs‘(𝐹𝑘)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))) → (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹𝑘))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))))))
12356adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℂ)
124111adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → (abs‘(𝐹𝑁)) ∈ ℂ)
12525recnd 10053 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)) ∈ ℂ)
126123, 124, 125mul12d 10230 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴) · ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))) = ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))))
127123, 24expp1d 12992 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘𝑁) + 1)) = ((if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)) · if(𝐴 ≤ 0, 0, 𝐴)))
12840, 1eleq2s 2717 . . . . . . . . . . . . . . . . 17 (𝑘𝑊𝑘 ∈ ℂ)
129 ax-1cn 9979 . . . . . . . . . . . . . . . . . 18 1 ∈ ℂ
130 addsub 10277 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑘 + 1) − 𝑁) = ((𝑘𝑁) + 1))
131129, 130mp3an2 1410 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑘 + 1) − 𝑁) = ((𝑘𝑁) + 1))
132128, 38, 131syl2anr 495 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑊) → ((𝑘 + 1) − 𝑁) = ((𝑘𝑁) + 1))
133132oveq2d 6651 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘𝑁) + 1)))
134123, 125mulcomd 10046 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))) = ((if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)) · if(𝐴 ≤ 0, 0, 𝐴)))
135127, 133, 1343eqtr4rd 2665 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))
136135oveq2d 6651 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))) = ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))))
137126, 136eqtrd 2654 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴) · ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))) = ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))))
138137breq2d 4656 . . . . . . . . . . 11 ((𝜑𝑘𝑊) → ((if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹𝑘))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))) ↔ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹𝑘))) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))))
139122, 138sylibd 229 . . . . . . . . . 10 ((𝜑𝑘𝑊) → ((abs‘(𝐹𝑘)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))) → (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹𝑘))) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))))
1401peano2uzs 11727 . . . . . . . . . . . . . . 15 (𝑘𝑊 → (𝑘 + 1) ∈ 𝑊)
14129sselda 3595 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 + 1) ∈ 𝑊) → (𝑘 + 1) ∈ 𝑍)
142140, 141sylan2 491 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → (𝑘 + 1) ∈ 𝑍)
143 fveq2 6178 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
144143eleq1d 2684 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → ((𝐹𝑘) ∈ ℂ ↔ (𝐹𝑛) ∈ ℂ))
145144cbvralv 3166 . . . . . . . . . . . . . . . 16 (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ ↔ ∀𝑛𝑍 (𝐹𝑛) ∈ ℂ)
14681, 145sylib 208 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑛𝑍 (𝐹𝑛) ∈ ℂ)
147146adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → ∀𝑛𝑍 (𝐹𝑛) ∈ ℂ)
14899eleq1d 2684 . . . . . . . . . . . . . . 15 (𝑛 = (𝑘 + 1) → ((𝐹𝑛) ∈ ℂ ↔ (𝐹‘(𝑘 + 1)) ∈ ℂ))
149148rspcv 3300 . . . . . . . . . . . . . 14 ((𝑘 + 1) ∈ 𝑍 → (∀𝑛𝑍 (𝐹𝑛) ∈ ℂ → (𝐹‘(𝑘 + 1)) ∈ ℂ))
150142, 147, 149sylc 65 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → (𝐹‘(𝑘 + 1)) ∈ ℂ)
151150abscld 14156 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ)
15217adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → 𝐴 ∈ ℝ)
153152, 116remulcld 10055 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (𝐴 · (abs‘(𝐹𝑘))) ∈ ℝ)
15420, 116remulcld 10055 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹𝑘))) ∈ ℝ)
155 cvgrat.7 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹𝑘))))
15632absge0d 14164 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → 0 ≤ (abs‘(𝐹𝑘)))
157 max1 12001 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → 𝐴 ≤ if(𝐴 ≤ 0, 0, 𝐴))
15817, 16, 157sylancl 693 . . . . . . . . . . . . . 14 (𝜑𝐴 ≤ if(𝐴 ≤ 0, 0, 𝐴))
159158adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → 𝐴 ≤ if(𝐴 ≤ 0, 0, 𝐴))
160152, 20, 116, 156, 159lemul1ad 10948 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (𝐴 · (abs‘(𝐹𝑘))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹𝑘))))
161151, 153, 154, 155, 160letrd 10179 . . . . . . . . . . 11 ((𝜑𝑘𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹𝑘))))
162 peano2uz 11726 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (ℤ𝑁) → (𝑘 + 1) ∈ (ℤ𝑁))
16322, 162syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑊) → (𝑘 + 1) ∈ (ℤ𝑁))
164 uznn0sub 11704 . . . . . . . . . . . . . . 15 ((𝑘 + 1) ∈ (ℤ𝑁) → ((𝑘 + 1) − 𝑁) ∈ ℕ0)
165163, 164syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → ((𝑘 + 1) − 𝑁) ∈ ℕ0)
16620, 165reexpcld 13008 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)) ∈ ℝ)
167117, 166remulcld 10055 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))) ∈ ℝ)
168 letr 10116 . . . . . . . . . . . 12 (((abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ ∧ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹𝑘))) ∈ ℝ ∧ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))) ∈ ℝ) → (((abs‘(𝐹‘(𝑘 + 1))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹𝑘))) ∧ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹𝑘))) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))))
169151, 154, 167, 168syl3anc 1324 . . . . . . . . . . 11 ((𝜑𝑘𝑊) → (((abs‘(𝐹‘(𝑘 + 1))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹𝑘))) ∧ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹𝑘))) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))))
170161, 169mpand 710 . . . . . . . . . 10 ((𝜑𝑘𝑊) → ((if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹𝑘))) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))))
171139, 170syld 47 . . . . . . . . 9 ((𝜑𝑘𝑊) → ((abs‘(𝐹𝑘)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))))
17246, 171syldan 487 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝑁)) → ((abs‘(𝐹𝑘)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))))
173172expcom 451 . . . . . . 7 (𝑘 ∈ (ℤ𝑁) → (𝜑 → ((abs‘(𝐹𝑘)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))))))
174173a2d 29 . . . . . 6 (𝑘 ∈ (ℤ𝑁) → ((𝜑 → (abs‘(𝐹𝑘)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))) → (𝜑 → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))))))
17593, 98, 105, 98, 115, 174uzind4 11731 . . . . 5 (𝑘 ∈ (ℤ𝑁) → (𝜑 → (abs‘(𝐹𝑘)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))))
176175impcom 446 . . . 4 ((𝜑𝑘 ∈ (ℤ𝑁)) → (abs‘(𝐹𝑘)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))))
17747oveq2d 6651 . . . 4 ((𝜑𝑘 ∈ (ℤ𝑁)) → ((abs‘(𝐹𝑁)) · ((𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))‘𝑘)) = ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))))
178176, 177breqtrrd 4672 . . 3 ((𝜑𝑘 ∈ (ℤ𝑁)) → (abs‘(𝐹𝑘)) ≤ ((abs‘(𝐹𝑁)) · ((𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))‘𝑘)))
1791, 9, 26, 32, 80, 86, 178cvgcmpce 14531 . 2 (𝜑 → seq𝑁( + , 𝐹) ∈ dom ⇝ )
1803, 2, 31iserex 14368 . 2 (𝜑 → (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑁( + , 𝐹) ∈ dom ⇝ ))
181179, 180mpbird 247 1 (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  Vcvv 3195  wss 3567  ifcif 4077   class class class wbr 4644  cmpt 4720  dom cdm 5104  cfv 5876  (class class class)co 6635  cc 9919  cr 9920  0cc0 9921  1c1 9922   + caddc 9924   · cmul 9926   < clt 10059  cle 10060  cmin 10251   / cdiv 10669  0cn0 11277  cz 11362  cuz 11672  seqcseq 12784  cexp 12843   shift cshi 13787  abscabs 13955  cli 14196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999  ax-addf 10000  ax-mulf 10001
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-pm 7845  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-sup 8333  df-inf 8334  df-oi 8400  df-card 8750  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-n0 11278  df-z 11363  df-uz 11673  df-rp 11818  df-ico 12166  df-fz 12312  df-fzo 12450  df-fl 12576  df-seq 12785  df-exp 12844  df-hash 13101  df-shft 13788  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-abs 13957  df-limsup 14183  df-clim 14200  df-rlim 14201  df-sum 14398
This theorem is referenced by:  efcllem  14789  cvgdvgrat  38332
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