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Theorem cvgrat 15919
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, 3eleqtrdi 2851 . . . . . 6 (𝜑𝑁 ∈ (ℤ𝑀))
5 eluzelz 12888 . . . . . 6 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
64, 5syl 17 . . . . 5 (𝜑𝑁 ∈ ℤ)
7 uzid 12893 . . . . 5 (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ𝑁))
86, 7syl 17 . . . 4 (𝜑𝑁 ∈ (ℤ𝑁))
98, 1eleqtrrdi 2852 . . 3 (𝜑𝑁𝑊)
10 oveq1 7438 . . . . . . 7 (𝑛 = 𝑘 → (𝑛𝑁) = (𝑘𝑁))
1110oveq2d 7447 . . . . . 6 (𝑛 = 𝑘 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))
12 eqid 2737 . . . . . 6 (𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁))) = (𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))
13 ovex 7464 . . . . . 6 (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)) ∈ V
1411, 12, 13fvmpt 7016 . . . . 5 (𝑘𝑊 → ((𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))
1514adantl 481 . . . 4 ((𝜑𝑘𝑊) → ((𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))
16 0re 11263 . . . . . . 7 0 ∈ ℝ
17 cvgrat.3 . . . . . . 7 (𝜑𝐴 ∈ ℝ)
18 ifcl 4571 . . . . . . 7 ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ)
1916, 17, 18sylancr 587 . . . . . 6 (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ)
2019adantr 480 . . . . 5 ((𝜑𝑘𝑊) → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ)
21 simpr 484 . . . . . . 7 ((𝜑𝑘𝑊) → 𝑘𝑊)
2221, 1eleqtrdi 2851 . . . . . 6 ((𝜑𝑘𝑊) → 𝑘 ∈ (ℤ𝑁))
23 uznn0sub 12917 . . . . . 6 (𝑘 ∈ (ℤ𝑁) → (𝑘𝑁) ∈ ℕ0)
2422, 23syl 17 . . . . 5 ((𝜑𝑘𝑊) → (𝑘𝑁) ∈ ℕ0)
2520, 24reexpcld 14203 . . . 4 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)) ∈ ℝ)
2615, 25eqeltrd 2841 . . 3 ((𝜑𝑘𝑊) → ((𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))‘𝑘) ∈ ℝ)
27 uzss 12901 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → (ℤ𝑁) ⊆ (ℤ𝑀))
284, 27syl 17 . . . . . 6 (𝜑 → (ℤ𝑁) ⊆ (ℤ𝑀))
2928, 1, 33sstr4g 4037 . . . . 5 (𝜑𝑊𝑍)
3029sselda 3983 . . . 4 ((𝜑𝑘𝑊) → 𝑘𝑍)
31 cvgrat.6 . . . 4 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
3230, 31syldan 591 . . 3 ((𝜑𝑘𝑊) → (𝐹𝑘) ∈ ℂ)
3323adantl 481 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝑁)) → (𝑘𝑁) ∈ ℕ0)
34 oveq2 7439 . . . . . . . . 9 (𝑛 = (𝑘𝑁) → (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))
35 eqid 2737 . . . . . . . . 9 (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) = (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))
3634, 35, 13fvmpt 7016 . . . . . . . 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 12723 . . . . . . . 8 (𝜑𝑁 ∈ ℂ)
39 eluzelz 12888 . . . . . . . . 9 (𝑘 ∈ (ℤ𝑁) → 𝑘 ∈ ℤ)
4039zcnd 12723 . . . . . . . 8 (𝑘 ∈ (ℤ𝑁) → 𝑘 ∈ ℂ)
41 nn0ex 12532 . . . . . . . . . 10 0 ∈ V
4241mptex 7243 . . . . . . . . 9 (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) ∈ V
4342shftval 15113 . . . . . . . 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 596 . . . . . . 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 484 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝑁)) → 𝑘 ∈ (ℤ𝑁))
4645, 1eleqtrrdi 2852 . . . . . . . 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 2788 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑁)) → ((𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))‘𝑘) = (((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) shift 𝑁)‘𝑘))
496, 48seqfeq 14068 . . . . 5 (𝜑 → seq𝑁( + , (𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))) = seq𝑁( + , ((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) shift 𝑁)))
5042seqshft 15124 . . . . . 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 584 . . . . 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 11608 . . . . . . 7 (𝜑 → (𝑁𝑁) = 0)
5352seqeq1d 14048 . . . . . 6 (𝜑 → seq(𝑁𝑁)( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) = seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))))
5453oveq1d 7446 . . . . 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 2781 . . . 4 (𝜑 → seq𝑁( + , (𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))) = (seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁))
5619recnd 11289 . . . . . . 7 (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℂ)
57 max2 13229 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → 0 ≤ if(𝐴 ≤ 0, 0, 𝐴))
5817, 16, 57sylancl 586 . . . . . . . . 9 (𝜑 → 0 ≤ if(𝐴 ≤ 0, 0, 𝐴))
5919, 58absidd 15461 . . . . . . . 8 (𝜑 → (abs‘if(𝐴 ≤ 0, 0, 𝐴)) = if(𝐴 ≤ 0, 0, 𝐴))
60 0lt1 11785 . . . . . . . . 9 0 < 1
61 cvgrat.4 . . . . . . . . 9 (𝜑𝐴 < 1)
62 breq1 5146 . . . . . . . . . 10 (0 = if(𝐴 ≤ 0, 0, 𝐴) → (0 < 1 ↔ if(𝐴 ≤ 0, 0, 𝐴) < 1))
63 breq1 5146 . . . . . . . . . 10 (𝐴 = if(𝐴 ≤ 0, 0, 𝐴) → (𝐴 < 1 ↔ if(𝐴 ≤ 0, 0, 𝐴) < 1))
6462, 63ifboth 4565 . . . . . . . . 9 ((0 < 1 ∧ 𝐴 < 1) → if(𝐴 ≤ 0, 0, 𝐴) < 1)
6560, 61, 64sylancr 587 . . . . . . . 8 (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) < 1)
6659, 65eqbrtrd 5165 . . . . . . 7 (𝜑 → (abs‘if(𝐴 ≤ 0, 0, 𝐴)) < 1)
67 oveq2 7439 . . . . . . . . 9 (𝑛 = 𝑘 → (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛) = (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘))
68 ovex 7464 . . . . . . . . 9 (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘) ∈ V
6967, 35, 68fvmpt 7016 . . . . . . . 8 (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘))
7069adantl 481 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘))
7156, 66, 70geolim 15906 . . . . . 6 (𝜑 → seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))))
72 seqex 14044 . . . . . . 7 seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ∈ V
73 climshft 15612 . . . . . . 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 586 . . . . . 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 257 . . . . 5 (𝜑 → (seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))))
76 ovex 7464 . . . . . 6 (seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ∈ V
77 ovex 7464 . . . . . 6 (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))) ∈ V
7876, 77breldm 5919 . . . . 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 2841 . . 3 (𝜑 → seq𝑁( + , (𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))) ∈ dom ⇝ )
81 fveq2 6906 . . . . . 6 (𝑘 = 𝑁 → (𝐹𝑘) = (𝐹𝑁))
8281eleq1d 2826 . . . . 5 (𝑘 = 𝑁 → ((𝐹𝑘) ∈ ℂ ↔ (𝐹𝑁) ∈ ℂ))
8331ralrimiva 3146 . . . . 5 (𝜑 → ∀𝑘𝑍 (𝐹𝑘) ∈ ℂ)
8482, 83, 2rspcdva 3623 . . . 4 (𝜑 → (𝐹𝑁) ∈ ℂ)
8584abscld 15475 . . 3 (𝜑 → (abs‘(𝐹𝑁)) ∈ ℝ)
86 2fveq3 6911 . . . . . . . 8 (𝑛 = 𝑁 → (abs‘(𝐹𝑛)) = (abs‘(𝐹𝑁)))
87 oveq1 7438 . . . . . . . . . 10 (𝑛 = 𝑁 → (𝑛𝑁) = (𝑁𝑁))
8887oveq2d 7447 . . . . . . . . 9 (𝑛 = 𝑁 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁)))
8988oveq2d 7447 . . . . . . . 8 (𝑛 = 𝑁 → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁))) = ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁))))
9086, 89breq12d 5156 . . . . . . 7 (𝑛 = 𝑁 → ((abs‘(𝐹𝑛)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁))) ↔ (abs‘(𝐹𝑁)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁)))))
9190imbi2d 340 . . . . . 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))))))
92 2fveq3 6911 . . . . . . . 8 (𝑛 = 𝑘 → (abs‘(𝐹𝑛)) = (abs‘(𝐹𝑘)))
9311oveq2d 7447 . . . . . . . 8 (𝑛 = 𝑘 → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁))) = ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))))
9492, 93breq12d 5156 . . . . . . 7 (𝑛 = 𝑘 → ((abs‘(𝐹𝑛)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁))) ↔ (abs‘(𝐹𝑘)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))))
9594imbi2d 340 . . . . . 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))))))
96 2fveq3 6911 . . . . . . . 8 (𝑛 = (𝑘 + 1) → (abs‘(𝐹𝑛)) = (abs‘(𝐹‘(𝑘 + 1))))
97 oveq1 7438 . . . . . . . . . 10 (𝑛 = (𝑘 + 1) → (𝑛𝑁) = ((𝑘 + 1) − 𝑁))
9897oveq2d 7447 . . . . . . . . 9 (𝑛 = (𝑘 + 1) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))
9998oveq2d 7447 . . . . . . . 8 (𝑛 = (𝑘 + 1) → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁))) = ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))))
10096, 99breq12d 5156 . . . . . . 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)))))
101100imbi2d 340 . . . . . 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))))))
10285leidd 11829 . . . . . . 7 (𝜑 → (abs‘(𝐹𝑁)) ≤ (abs‘(𝐹𝑁)))
10352oveq2d 7447 . . . . . . . . . 10 (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑0))
10456exp0d 14180 . . . . . . . . . 10 (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑0) = 1)
105103, 104eqtrd 2777 . . . . . . . . 9 (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁)) = 1)
106105oveq2d 7447 . . . . . . . 8 (𝜑 → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁))) = ((abs‘(𝐹𝑁)) · 1))
10785recnd 11289 . . . . . . . . 9 (𝜑 → (abs‘(𝐹𝑁)) ∈ ℂ)
108107mulridd 11278 . . . . . . . 8 (𝜑 → ((abs‘(𝐹𝑁)) · 1) = (abs‘(𝐹𝑁)))
109106, 108eqtrd 2777 . . . . . . 7 (𝜑 → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁))) = (abs‘(𝐹𝑁)))
110102, 109breqtrrd 5171 . . . . . 6 (𝜑 → (abs‘(𝐹𝑁)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁))))
11132abscld 15475 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (abs‘(𝐹𝑘)) ∈ ℝ)
11285adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → (abs‘(𝐹𝑁)) ∈ ℝ)
113112, 25remulcld 11291 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))) ∈ ℝ)
11458adantr 480 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → 0 ≤ if(𝐴 ≤ 0, 0, 𝐴))
115 lemul2a 12122 . . . . . . . . . . . . 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)))))
116115ex 412 . . . . . . . . . . . 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))))))
117111, 113, 20, 114, 116syl112anc 1376 . . . . . . . . . . 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))))))
11856adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℂ)
119107adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → (abs‘(𝐹𝑁)) ∈ ℂ)
12025recnd 11289 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)) ∈ ℂ)
121118, 119, 120mul12d 11470 . . . . . . . . . . . . 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)))))
122118, 24expp1d 14187 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘𝑁) + 1)) = ((if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)) · if(𝐴 ≤ 0, 0, 𝐴)))
12340, 1eleq2s 2859 . . . . . . . . . . . . . . . . 17 (𝑘𝑊𝑘 ∈ ℂ)
124 ax-1cn 11213 . . . . . . . . . . . . . . . . . 18 1 ∈ ℂ
125 addsub 11519 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑘 + 1) − 𝑁) = ((𝑘𝑁) + 1))
126124, 125mp3an2 1451 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑘 + 1) − 𝑁) = ((𝑘𝑁) + 1))
127123, 38, 126syl2anr 597 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑊) → ((𝑘 + 1) − 𝑁) = ((𝑘𝑁) + 1))
128127oveq2d 7447 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘𝑁) + 1)))
129118, 120mulcomd 11282 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))) = ((if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)) · if(𝐴 ≤ 0, 0, 𝐴)))
130122, 128, 1293eqtr4rd 2788 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))
131130oveq2d 7447 . . . . . . . . . . . . 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))))
132121, 131eqtrd 2777 . . . . . . . . . . . 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))))
133132breq2d 5155 . . . . . . . . . . 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)))))
134117, 133sylibd 239 . . . . . . . . . 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)))))
135 fveq2 6906 . . . . . . . . . . . . . . 15 (𝑛 = (𝑘 + 1) → (𝐹𝑛) = (𝐹‘(𝑘 + 1)))
136135eleq1d 2826 . . . . . . . . . . . . . 14 (𝑛 = (𝑘 + 1) → ((𝐹𝑛) ∈ ℂ ↔ (𝐹‘(𝑘 + 1)) ∈ ℂ))
137 fveq2 6906 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
138137eleq1d 2826 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → ((𝐹𝑘) ∈ ℂ ↔ (𝐹𝑛) ∈ ℂ))
139138cbvralvw 3237 . . . . . . . . . . . . . . . 16 (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ ↔ ∀𝑛𝑍 (𝐹𝑛) ∈ ℂ)
14083, 139sylib 218 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑛𝑍 (𝐹𝑛) ∈ ℂ)
141140adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → ∀𝑛𝑍 (𝐹𝑛) ∈ ℂ)
1421peano2uzs 12944 . . . . . . . . . . . . . . 15 (𝑘𝑊 → (𝑘 + 1) ∈ 𝑊)
14329sselda 3983 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 + 1) ∈ 𝑊) → (𝑘 + 1) ∈ 𝑍)
144142, 143sylan2 593 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → (𝑘 + 1) ∈ 𝑍)
145136, 141, 144rspcdva 3623 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → (𝐹‘(𝑘 + 1)) ∈ ℂ)
146145abscld 15475 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ)
14717adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → 𝐴 ∈ ℝ)
148147, 111remulcld 11291 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (𝐴 · (abs‘(𝐹𝑘))) ∈ ℝ)
14920, 111remulcld 11291 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹𝑘))) ∈ ℝ)
150 cvgrat.7 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹𝑘))))
15132absge0d 15483 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → 0 ≤ (abs‘(𝐹𝑘)))
152 max1 13227 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → 𝐴 ≤ if(𝐴 ≤ 0, 0, 𝐴))
15317, 16, 152sylancl 586 . . . . . . . . . . . . . 14 (𝜑𝐴 ≤ if(𝐴 ≤ 0, 0, 𝐴))
154153adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → 𝐴 ≤ if(𝐴 ≤ 0, 0, 𝐴))
155147, 20, 111, 151, 154lemul1ad 12207 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (𝐴 · (abs‘(𝐹𝑘))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹𝑘))))
156146, 148, 149, 150, 155letrd 11418 . . . . . . . . . . 11 ((𝜑𝑘𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹𝑘))))
157 peano2uz 12943 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (ℤ𝑁) → (𝑘 + 1) ∈ (ℤ𝑁))
15822, 157syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑊) → (𝑘 + 1) ∈ (ℤ𝑁))
159 uznn0sub 12917 . . . . . . . . . . . . . . 15 ((𝑘 + 1) ∈ (ℤ𝑁) → ((𝑘 + 1) − 𝑁) ∈ ℕ0)
160158, 159syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → ((𝑘 + 1) − 𝑁) ∈ ℕ0)
16120, 160reexpcld 14203 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)) ∈ ℝ)
162112, 161remulcld 11291 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))) ∈ ℝ)
163 letr 11355 . . . . . . . . . . . 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)))))
164146, 149, 162, 163syl3anc 1373 . . . . . . . . . . 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)))))
165156, 164mpand 695 . . . . . . . . . 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)))))
166134, 165syld 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)))))
16746, 166syldan 591 . . . . . . . 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)))))
168167expcom 413 . . . . . . 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))))))
169168a2d 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))))))
17091, 95, 101, 95, 110, 169uzind4i 12952 . . . . 5 (𝑘 ∈ (ℤ𝑁) → (𝜑 → (abs‘(𝐹𝑘)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))))
171170impcom 407 . . . 4 ((𝜑𝑘 ∈ (ℤ𝑁)) → (abs‘(𝐹𝑘)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))))
17247oveq2d 7447 . . . 4 ((𝜑𝑘 ∈ (ℤ𝑁)) → ((abs‘(𝐹𝑁)) · ((𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))‘𝑘)) = ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))))
173171, 172breqtrrd 5171 . . 3 ((𝜑𝑘 ∈ (ℤ𝑁)) → (abs‘(𝐹𝑘)) ≤ ((abs‘(𝐹𝑁)) · ((𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))‘𝑘)))
1741, 9, 26, 32, 80, 85, 173cvgcmpce 15854 . 2 (𝜑 → seq𝑁( + , 𝐹) ∈ dom ⇝ )
1753, 2, 31iserex 15693 . 2 (𝜑 → (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑁( + , 𝐹) ∈ dom ⇝ ))
176174, 175mpbird 257 1 (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  wss 3951  ifcif 4525   class class class wbr 5143  cmpt 5225  dom cdm 5685  cfv 6561  (class class class)co 7431  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160   < clt 11295  cle 11296  cmin 11492   / cdiv 11920  0cn0 12526  cz 12613  cuz 12878  seqcseq 14042  cexp 14102   shift cshi 15105  abscabs 15273  cli 15520
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
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-op 4633  df-uni 4908  df-int 4947  df-iun 4993  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-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  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-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-ico 13393  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-shft 15106  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-limsup 15507  df-clim 15524  df-rlim 15525  df-sum 15723
This theorem is referenced by:  efcllem  16113  cvgdvgrat  44332
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