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Theorem cvgrat 15022
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 2869 . . . . . 6 (𝜑𝑁 ∈ (ℤ𝑀))
5 eluzelz 12006 . . . . . 6 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
64, 5syl 17 . . . . 5 (𝜑𝑁 ∈ ℤ)
7 uzid 12011 . . . . 5 (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ𝑁))
86, 7syl 17 . . . 4 (𝜑𝑁 ∈ (ℤ𝑁))
98, 1syl6eleqr 2870 . . 3 (𝜑𝑁𝑊)
10 oveq1 6931 . . . . . . 7 (𝑛 = 𝑘 → (𝑛𝑁) = (𝑘𝑁))
1110oveq2d 6940 . . . . . 6 (𝑛 = 𝑘 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))
12 eqid 2778 . . . . . 6 (𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁))) = (𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))
13 ovex 6956 . . . . . 6 (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)) ∈ V
1411, 12, 13fvmpt 6544 . . . . 5 (𝑘𝑊 → ((𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))
1514adantl 475 . . . 4 ((𝜑𝑘𝑊) → ((𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))
16 0re 10380 . . . . . . 7 0 ∈ ℝ
17 cvgrat.3 . . . . . . 7 (𝜑𝐴 ∈ ℝ)
18 ifcl 4351 . . . . . . 7 ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ)
1916, 17, 18sylancr 581 . . . . . 6 (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ)
2019adantr 474 . . . . 5 ((𝜑𝑘𝑊) → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ)
21 simpr 479 . . . . . . 7 ((𝜑𝑘𝑊) → 𝑘𝑊)
2221, 1syl6eleq 2869 . . . . . 6 ((𝜑𝑘𝑊) → 𝑘 ∈ (ℤ𝑁))
23 uznn0sub 12029 . . . . . 6 (𝑘 ∈ (ℤ𝑁) → (𝑘𝑁) ∈ ℕ0)
2422, 23syl 17 . . . . 5 ((𝜑𝑘𝑊) → (𝑘𝑁) ∈ ℕ0)
2520, 24reexpcld 13348 . . . 4 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)) ∈ ℝ)
2615, 25eqeltrd 2859 . . 3 ((𝜑𝑘𝑊) → ((𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))‘𝑘) ∈ ℝ)
27 uzss 12017 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → (ℤ𝑁) ⊆ (ℤ𝑀))
284, 27syl 17 . . . . . 6 (𝜑 → (ℤ𝑁) ⊆ (ℤ𝑀))
2928, 1, 33sstr4g 3865 . . . . 5 (𝜑𝑊𝑍)
3029sselda 3821 . . . 4 ((𝜑𝑘𝑊) → 𝑘𝑍)
31 cvgrat.6 . . . 4 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
3230, 31syldan 585 . . 3 ((𝜑𝑘𝑊) → (𝐹𝑘) ∈ ℂ)
3323adantl 475 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝑁)) → (𝑘𝑁) ∈ ℕ0)
34 oveq2 6932 . . . . . . . . 9 (𝑛 = (𝑘𝑁) → (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))
35 eqid 2778 . . . . . . . . 9 (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) = (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))
3634, 35, 13fvmpt 6544 . . . . . . . 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 11839 . . . . . . . 8 (𝜑𝑁 ∈ ℂ)
39 eluzelz 12006 . . . . . . . . 9 (𝑘 ∈ (ℤ𝑁) → 𝑘 ∈ ℤ)
4039zcnd 11839 . . . . . . . 8 (𝑘 ∈ (ℤ𝑁) → 𝑘 ∈ ℂ)
41 nn0ex 11653 . . . . . . . . . 10 0 ∈ V
4241mptex 6760 . . . . . . . . 9 (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) ∈ V
4342shftval 14225 . . . . . . . 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 589 . . . . . . 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 479 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝑁)) → 𝑘 ∈ (ℤ𝑁))
4645, 1syl6eleqr 2870 . . . . . . . 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 2825 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑁)) → ((𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))‘𝑘) = (((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) shift 𝑁)‘𝑘))
496, 48seqfeq 13148 . . . . 5 (𝜑 → seq𝑁( + , (𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))) = seq𝑁( + , ((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) shift 𝑁)))
5042seqshft 14236 . . . . . 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 579 . . . . 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 10724 . . . . . . 7 (𝜑 → (𝑁𝑁) = 0)
5352seqeq1d 13129 . . . . . 6 (𝜑 → seq(𝑁𝑁)( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) = seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))))
5453oveq1d 6939 . . . . 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 2818 . . . 4 (𝜑 → seq𝑁( + , (𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))) = (seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁))
5619recnd 10407 . . . . . . 7 (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℂ)
57 max2 12334 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → 0 ≤ if(𝐴 ≤ 0, 0, 𝐴))
5817, 16, 57sylancl 580 . . . . . . . . 9 (𝜑 → 0 ≤ if(𝐴 ≤ 0, 0, 𝐴))
5919, 58absidd 14573 . . . . . . . 8 (𝜑 → (abs‘if(𝐴 ≤ 0, 0, 𝐴)) = if(𝐴 ≤ 0, 0, 𝐴))
60 0lt1 10899 . . . . . . . . 9 0 < 1
61 cvgrat.4 . . . . . . . . 9 (𝜑𝐴 < 1)
62 breq1 4891 . . . . . . . . . 10 (0 = if(𝐴 ≤ 0, 0, 𝐴) → (0 < 1 ↔ if(𝐴 ≤ 0, 0, 𝐴) < 1))
63 breq1 4891 . . . . . . . . . 10 (𝐴 = if(𝐴 ≤ 0, 0, 𝐴) → (𝐴 < 1 ↔ if(𝐴 ≤ 0, 0, 𝐴) < 1))
6462, 63ifboth 4345 . . . . . . . . 9 ((0 < 1 ∧ 𝐴 < 1) → if(𝐴 ≤ 0, 0, 𝐴) < 1)
6560, 61, 64sylancr 581 . . . . . . . 8 (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) < 1)
6659, 65eqbrtrd 4910 . . . . . . 7 (𝜑 → (abs‘if(𝐴 ≤ 0, 0, 𝐴)) < 1)
67 oveq2 6932 . . . . . . . . 9 (𝑛 = 𝑘 → (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛) = (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘))
68 ovex 6956 . . . . . . . . 9 (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘) ∈ V
6967, 35, 68fvmpt 6544 . . . . . . . 8 (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘))
7069adantl 475 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘))
7156, 66, 70geolim 15009 . . . . . 6 (𝜑 → seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))))
72 seqex 13125 . . . . . . 7 seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ∈ V
73 climshft 14719 . . . . . . 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 580 . . . . . 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 249 . . . . 5 (𝜑 → (seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))))
76 ovex 6956 . . . . . 6 (seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ∈ V
77 ovex 6956 . . . . . 6 (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))) ∈ V
7876, 77breldm 5576 . . . . 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 2859 . . 3 (𝜑 → seq𝑁( + , (𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))) ∈ dom ⇝ )
81 fveq2 6448 . . . . . 6 (𝑘 = 𝑁 → (𝐹𝑘) = (𝐹𝑁))
8281eleq1d 2844 . . . . 5 (𝑘 = 𝑁 → ((𝐹𝑘) ∈ ℂ ↔ (𝐹𝑁) ∈ ℂ))
8331ralrimiva 3148 . . . . 5 (𝜑 → ∀𝑘𝑍 (𝐹𝑘) ∈ ℂ)
8482, 83, 2rspcdva 3517 . . . 4 (𝜑 → (𝐹𝑁) ∈ ℂ)
8584abscld 14587 . . 3 (𝜑 → (abs‘(𝐹𝑁)) ∈ ℝ)
86 2fveq3 6453 . . . . . . . 8 (𝑛 = 𝑁 → (abs‘(𝐹𝑛)) = (abs‘(𝐹𝑁)))
87 oveq1 6931 . . . . . . . . . 10 (𝑛 = 𝑁 → (𝑛𝑁) = (𝑁𝑁))
8887oveq2d 6940 . . . . . . . . 9 (𝑛 = 𝑁 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁)))
8988oveq2d 6940 . . . . . . . 8 (𝑛 = 𝑁 → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁))) = ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁))))
9086, 89breq12d 4901 . . . . . . 7 (𝑛 = 𝑁 → ((abs‘(𝐹𝑛)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁))) ↔ (abs‘(𝐹𝑁)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁)))))
9190imbi2d 332 . . . . . 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 6453 . . . . . . . 8 (𝑛 = 𝑘 → (abs‘(𝐹𝑛)) = (abs‘(𝐹𝑘)))
9311oveq2d 6940 . . . . . . . 8 (𝑛 = 𝑘 → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁))) = ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))))
9492, 93breq12d 4901 . . . . . . 7 (𝑛 = 𝑘 → ((abs‘(𝐹𝑛)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁))) ↔ (abs‘(𝐹𝑘)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))))
9594imbi2d 332 . . . . . 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 6453 . . . . . . . 8 (𝑛 = (𝑘 + 1) → (abs‘(𝐹𝑛)) = (abs‘(𝐹‘(𝑘 + 1))))
97 oveq1 6931 . . . . . . . . . 10 (𝑛 = (𝑘 + 1) → (𝑛𝑁) = ((𝑘 + 1) − 𝑁))
9897oveq2d 6940 . . . . . . . . 9 (𝑛 = (𝑘 + 1) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))
9998oveq2d 6940 . . . . . . . 8 (𝑛 = (𝑘 + 1) → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁))) = ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))))
10096, 99breq12d 4901 . . . . . . 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 332 . . . . . 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 10943 . . . . . . 7 (𝜑 → (abs‘(𝐹𝑁)) ≤ (abs‘(𝐹𝑁)))
10352oveq2d 6940 . . . . . . . . . 10 (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑0))
10456exp0d 13325 . . . . . . . . . 10 (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑0) = 1)
105103, 104eqtrd 2814 . . . . . . . . 9 (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁)) = 1)
106105oveq2d 6940 . . . . . . . 8 (𝜑 → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁))) = ((abs‘(𝐹𝑁)) · 1))
10785recnd 10407 . . . . . . . . 9 (𝜑 → (abs‘(𝐹𝑁)) ∈ ℂ)
108107mulid1d 10396 . . . . . . . 8 (𝜑 → ((abs‘(𝐹𝑁)) · 1) = (abs‘(𝐹𝑁)))
109106, 108eqtrd 2814 . . . . . . 7 (𝜑 → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁))) = (abs‘(𝐹𝑁)))
110102, 109breqtrrd 4916 . . . . . 6 (𝜑 → (abs‘(𝐹𝑁)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁))))
11132abscld 14587 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (abs‘(𝐹𝑘)) ∈ ℝ)
11285adantr 474 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → (abs‘(𝐹𝑁)) ∈ ℝ)
113112, 25remulcld 10409 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))) ∈ ℝ)
11458adantr 474 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → 0 ≤ if(𝐴 ≤ 0, 0, 𝐴))
115 lemul2a 11234 . . . . . . . . . . . . 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 403 . . . . . . . . . . . 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 1442 . . . . . . . . . . 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 474 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℂ)
119107adantr 474 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → (abs‘(𝐹𝑁)) ∈ ℂ)
12025recnd 10407 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)) ∈ ℂ)
121118, 119, 120mul12d 10587 . . . . . . . . . . . . 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 13332 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘𝑁) + 1)) = ((if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)) · if(𝐴 ≤ 0, 0, 𝐴)))
12340, 1eleq2s 2877 . . . . . . . . . . . . . . . . 17 (𝑘𝑊𝑘 ∈ ℂ)
124 ax-1cn 10332 . . . . . . . . . . . . . . . . . 18 1 ∈ ℂ
125 addsub 10636 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑘 + 1) − 𝑁) = ((𝑘𝑁) + 1))
126124, 125mp3an2 1522 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑘 + 1) − 𝑁) = ((𝑘𝑁) + 1))
127123, 38, 126syl2anr 590 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑊) → ((𝑘 + 1) − 𝑁) = ((𝑘𝑁) + 1))
128127oveq2d 6940 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘𝑁) + 1)))
129118, 120mulcomd 10400 . . . . . . . . . . . . . . 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 2825 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))
131130oveq2d 6940 . . . . . . . . . . . . 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 2814 . . . . . . . . . . . 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 4900 . . . . . . . . . . 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 231 . . . . . . . . . 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 6448 . . . . . . . . . . . . . . 15 (𝑛 = (𝑘 + 1) → (𝐹𝑛) = (𝐹‘(𝑘 + 1)))
136135eleq1d 2844 . . . . . . . . . . . . . 14 (𝑛 = (𝑘 + 1) → ((𝐹𝑛) ∈ ℂ ↔ (𝐹‘(𝑘 + 1)) ∈ ℂ))
137 fveq2 6448 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
138137eleq1d 2844 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → ((𝐹𝑘) ∈ ℂ ↔ (𝐹𝑛) ∈ ℂ))
139138cbvralv 3367 . . . . . . . . . . . . . . . 16 (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ ↔ ∀𝑛𝑍 (𝐹𝑛) ∈ ℂ)
14083, 139sylib 210 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑛𝑍 (𝐹𝑛) ∈ ℂ)
141140adantr 474 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → ∀𝑛𝑍 (𝐹𝑛) ∈ ℂ)
1421peano2uzs 12052 . . . . . . . . . . . . . . 15 (𝑘𝑊 → (𝑘 + 1) ∈ 𝑊)
14329sselda 3821 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 + 1) ∈ 𝑊) → (𝑘 + 1) ∈ 𝑍)
144142, 143sylan2 586 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → (𝑘 + 1) ∈ 𝑍)
145136, 141, 144rspcdva 3517 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → (𝐹‘(𝑘 + 1)) ∈ ℂ)
146145abscld 14587 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ)
14717adantr 474 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → 𝐴 ∈ ℝ)
148147, 111remulcld 10409 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (𝐴 · (abs‘(𝐹𝑘))) ∈ ℝ)
14920, 111remulcld 10409 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹𝑘))) ∈ ℝ)
150 cvgrat.7 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹𝑘))))
15132absge0d 14595 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → 0 ≤ (abs‘(𝐹𝑘)))
152 max1 12332 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → 𝐴 ≤ if(𝐴 ≤ 0, 0, 𝐴))
15317, 16, 152sylancl 580 . . . . . . . . . . . . . 14 (𝜑𝐴 ≤ if(𝐴 ≤ 0, 0, 𝐴))
154153adantr 474 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → 𝐴 ≤ if(𝐴 ≤ 0, 0, 𝐴))
155147, 20, 111, 151, 154lemul1ad 11319 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (𝐴 · (abs‘(𝐹𝑘))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹𝑘))))
156146, 148, 149, 150, 155letrd 10535 . . . . . . . . . . 11 ((𝜑𝑘𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹𝑘))))
157 peano2uz 12051 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (ℤ𝑁) → (𝑘 + 1) ∈ (ℤ𝑁))
15822, 157syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑊) → (𝑘 + 1) ∈ (ℤ𝑁))
159 uznn0sub 12029 . . . . . . . . . . . . . . 15 ((𝑘 + 1) ∈ (ℤ𝑁) → ((𝑘 + 1) − 𝑁) ∈ ℕ0)
160158, 159syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → ((𝑘 + 1) − 𝑁) ∈ ℕ0)
16120, 160reexpcld 13348 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)) ∈ ℝ)
162112, 161remulcld 10409 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))) ∈ ℝ)
163 letr 10472 . . . . . . . . . . . 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 1439 . . . . . . . . . . 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 685 . . . . . . . . . 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 585 . . . . . . . 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 404 . . . . . . 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 12060 . . . . 5 (𝑘 ∈ (ℤ𝑁) → (𝜑 → (abs‘(𝐹𝑘)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))))
171170impcom 398 . . . 4 ((𝜑𝑘 ∈ (ℤ𝑁)) → (abs‘(𝐹𝑘)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))))
17247oveq2d 6940 . . . 4 ((𝜑𝑘 ∈ (ℤ𝑁)) → ((abs‘(𝐹𝑁)) · ((𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))‘𝑘)) = ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))))
173171, 172breqtrrd 4916 . . 3 ((𝜑𝑘 ∈ (ℤ𝑁)) → (abs‘(𝐹𝑘)) ≤ ((abs‘(𝐹𝑁)) · ((𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))‘𝑘)))
1741, 9, 26, 32, 80, 85, 173cvgcmpce 14958 . 2 (𝜑 → seq𝑁( + , 𝐹) ∈ dom ⇝ )
1753, 2, 31iserex 14799 . 2 (𝜑 → (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑁( + , 𝐹) ∈ dom ⇝ ))
176174, 175mpbird 249 1 (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1071   = wceq 1601  wcel 2107  wral 3090  Vcvv 3398  wss 3792  ifcif 4307   class class class wbr 4888  cmpt 4967  dom cdm 5357  cfv 6137  (class class class)co 6924  cc 10272  cr 10273  0cc0 10274  1c1 10275   + caddc 10277   · cmul 10279   < clt 10413  cle 10414  cmin 10608   / cdiv 11034  0cn0 11646  cz 11732  cuz 11996  seqcseq 13123  cexp 13182   shift cshi 14217  abscabs 14385  cli 14627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-inf2 8837  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351  ax-pre-sup 10352  ax-addf 10353  ax-mulf 10354
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-se 5317  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-isom 6146  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-oadd 7849  df-er 8028  df-pm 8145  df-en 8244  df-dom 8245  df-sdom 8246  df-fin 8247  df-sup 8638  df-inf 8639  df-oi 8706  df-card 9100  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-div 11035  df-nn 11379  df-2 11442  df-3 11443  df-n0 11647  df-z 11733  df-uz 11997  df-rp 12142  df-ico 12497  df-fz 12648  df-fzo 12789  df-fl 12916  df-seq 13124  df-exp 13183  df-hash 13440  df-shft 14218  df-cj 14250  df-re 14251  df-im 14252  df-sqrt 14386  df-abs 14387  df-limsup 14614  df-clim 14631  df-rlim 14632  df-sum 14829
This theorem is referenced by:  efcllem  15214  cvgdvgrat  39478
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