MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cvgrat Structured version   Visualization version   GIF version

Theorem cvgrat 15768
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 2848 . . . . . 6 (𝜑𝑁 ∈ (ℤ𝑀))
5 eluzelz 12773 . . . . . 6 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
64, 5syl 17 . . . . 5 (𝜑𝑁 ∈ ℤ)
7 uzid 12778 . . . . 5 (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ𝑁))
86, 7syl 17 . . . 4 (𝜑𝑁 ∈ (ℤ𝑁))
98, 1eleqtrrdi 2849 . . 3 (𝜑𝑁𝑊)
10 oveq1 7364 . . . . . . 7 (𝑛 = 𝑘 → (𝑛𝑁) = (𝑘𝑁))
1110oveq2d 7373 . . . . . 6 (𝑛 = 𝑘 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))
12 eqid 2736 . . . . . 6 (𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁))) = (𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))
13 ovex 7390 . . . . . 6 (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)) ∈ V
1411, 12, 13fvmpt 6948 . . . . 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 11157 . . . . . . 7 0 ∈ ℝ
17 cvgrat.3 . . . . . . 7 (𝜑𝐴 ∈ ℝ)
18 ifcl 4531 . . . . . . 7 ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ)
1916, 17, 18sylancr 587 . . . . . 6 (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ)
2019adantr 481 . . . . 5 ((𝜑𝑘𝑊) → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ)
21 simpr 485 . . . . . . 7 ((𝜑𝑘𝑊) → 𝑘𝑊)
2221, 1eleqtrdi 2848 . . . . . 6 ((𝜑𝑘𝑊) → 𝑘 ∈ (ℤ𝑁))
23 uznn0sub 12802 . . . . . 6 (𝑘 ∈ (ℤ𝑁) → (𝑘𝑁) ∈ ℕ0)
2422, 23syl 17 . . . . 5 ((𝜑𝑘𝑊) → (𝑘𝑁) ∈ ℕ0)
2520, 24reexpcld 14068 . . . 4 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)) ∈ ℝ)
2615, 25eqeltrd 2838 . . 3 ((𝜑𝑘𝑊) → ((𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))‘𝑘) ∈ ℝ)
27 uzss 12786 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → (ℤ𝑁) ⊆ (ℤ𝑀))
284, 27syl 17 . . . . . 6 (𝜑 → (ℤ𝑁) ⊆ (ℤ𝑀))
2928, 1, 33sstr4g 3989 . . . . 5 (𝜑𝑊𝑍)
3029sselda 3944 . . . 4 ((𝜑𝑘𝑊) → 𝑘𝑍)
31 cvgrat.6 . . . 4 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
3230, 31syldan 591 . . 3 ((𝜑𝑘𝑊) → (𝐹𝑘) ∈ ℂ)
3323adantl 482 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝑁)) → (𝑘𝑁) ∈ ℕ0)
34 oveq2 7365 . . . . . . . . 9 (𝑛 = (𝑘𝑁) → (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))
35 eqid 2736 . . . . . . . . 9 (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) = (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))
3634, 35, 13fvmpt 6948 . . . . . . . 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 12608 . . . . . . . 8 (𝜑𝑁 ∈ ℂ)
39 eluzelz 12773 . . . . . . . . 9 (𝑘 ∈ (ℤ𝑁) → 𝑘 ∈ ℤ)
4039zcnd 12608 . . . . . . . 8 (𝑘 ∈ (ℤ𝑁) → 𝑘 ∈ ℂ)
41 nn0ex 12419 . . . . . . . . . 10 0 ∈ V
4241mptex 7173 . . . . . . . . 9 (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) ∈ V
4342shftval 14959 . . . . . . . 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 485 . . . . . . . . 9 ((𝜑𝑘 ∈ (ℤ𝑁)) → 𝑘 ∈ (ℤ𝑁))
4645, 1eleqtrrdi 2849 . . . . . . . 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 2787 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑁)) → ((𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))‘𝑘) = (((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) shift 𝑁)‘𝑘))
496, 48seqfeq 13933 . . . . 5 (𝜑 → seq𝑁( + , (𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))) = seq𝑁( + , ((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) shift 𝑁)))
5042seqshft 14970 . . . . . 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 11500 . . . . . . 7 (𝜑 → (𝑁𝑁) = 0)
5352seqeq1d 13912 . . . . . 6 (𝜑 → seq(𝑁𝑁)( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) = seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))))
5453oveq1d 7372 . . . . 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 2780 . . . 4 (𝜑 → seq𝑁( + , (𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))) = (seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁))
5619recnd 11183 . . . . . . 7 (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℂ)
57 max2 13106 . . . . . . . . . 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 15307 . . . . . . . 8 (𝜑 → (abs‘if(𝐴 ≤ 0, 0, 𝐴)) = if(𝐴 ≤ 0, 0, 𝐴))
60 0lt1 11677 . . . . . . . . 9 0 < 1
61 cvgrat.4 . . . . . . . . 9 (𝜑𝐴 < 1)
62 breq1 5108 . . . . . . . . . 10 (0 = if(𝐴 ≤ 0, 0, 𝐴) → (0 < 1 ↔ if(𝐴 ≤ 0, 0, 𝐴) < 1))
63 breq1 5108 . . . . . . . . . 10 (𝐴 = if(𝐴 ≤ 0, 0, 𝐴) → (𝐴 < 1 ↔ if(𝐴 ≤ 0, 0, 𝐴) < 1))
6462, 63ifboth 4525 . . . . . . . . 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 5127 . . . . . . 7 (𝜑 → (abs‘if(𝐴 ≤ 0, 0, 𝐴)) < 1)
67 oveq2 7365 . . . . . . . . 9 (𝑛 = 𝑘 → (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛) = (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘))
68 ovex 7390 . . . . . . . . 9 (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘) ∈ V
6967, 35, 68fvmpt 6948 . . . . . . . 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 15755 . . . . . 6 (𝜑 → seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))))
72 seqex 13908 . . . . . . 7 seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ∈ V
73 climshft 15458 . . . . . . 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 256 . . . . 5 (𝜑 → (seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))))
76 ovex 7390 . . . . . 6 (seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ∈ V
77 ovex 7390 . . . . . 6 (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))) ∈ V
7876, 77breldm 5864 . . . . 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 2838 . . 3 (𝜑 → seq𝑁( + , (𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))) ∈ dom ⇝ )
81 fveq2 6842 . . . . . 6 (𝑘 = 𝑁 → (𝐹𝑘) = (𝐹𝑁))
8281eleq1d 2822 . . . . 5 (𝑘 = 𝑁 → ((𝐹𝑘) ∈ ℂ ↔ (𝐹𝑁) ∈ ℂ))
8331ralrimiva 3143 . . . . 5 (𝜑 → ∀𝑘𝑍 (𝐹𝑘) ∈ ℂ)
8482, 83, 2rspcdva 3582 . . . 4 (𝜑 → (𝐹𝑁) ∈ ℂ)
8584abscld 15321 . . 3 (𝜑 → (abs‘(𝐹𝑁)) ∈ ℝ)
86 2fveq3 6847 . . . . . . . 8 (𝑛 = 𝑁 → (abs‘(𝐹𝑛)) = (abs‘(𝐹𝑁)))
87 oveq1 7364 . . . . . . . . . 10 (𝑛 = 𝑁 → (𝑛𝑁) = (𝑁𝑁))
8887oveq2d 7373 . . . . . . . . 9 (𝑛 = 𝑁 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁)))
8988oveq2d 7373 . . . . . . . 8 (𝑛 = 𝑁 → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁))) = ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁))))
9086, 89breq12d 5118 . . . . . . 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 6847 . . . . . . . 8 (𝑛 = 𝑘 → (abs‘(𝐹𝑛)) = (abs‘(𝐹𝑘)))
9311oveq2d 7373 . . . . . . . 8 (𝑛 = 𝑘 → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁))) = ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))))
9492, 93breq12d 5118 . . . . . . 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 6847 . . . . . . . 8 (𝑛 = (𝑘 + 1) → (abs‘(𝐹𝑛)) = (abs‘(𝐹‘(𝑘 + 1))))
97 oveq1 7364 . . . . . . . . . 10 (𝑛 = (𝑘 + 1) → (𝑛𝑁) = ((𝑘 + 1) − 𝑁))
9897oveq2d 7373 . . . . . . . . 9 (𝑛 = (𝑘 + 1) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))
9998oveq2d 7373 . . . . . . . 8 (𝑛 = (𝑘 + 1) → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁))) = ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))))
10096, 99breq12d 5118 . . . . . . 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 11721 . . . . . . 7 (𝜑 → (abs‘(𝐹𝑁)) ≤ (abs‘(𝐹𝑁)))
10352oveq2d 7373 . . . . . . . . . 10 (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑0))
10456exp0d 14045 . . . . . . . . . 10 (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑0) = 1)
105103, 104eqtrd 2776 . . . . . . . . 9 (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁)) = 1)
106105oveq2d 7373 . . . . . . . 8 (𝜑 → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁))) = ((abs‘(𝐹𝑁)) · 1))
10785recnd 11183 . . . . . . . . 9 (𝜑 → (abs‘(𝐹𝑁)) ∈ ℂ)
108107mulid1d 11172 . . . . . . . 8 (𝜑 → ((abs‘(𝐹𝑁)) · 1) = (abs‘(𝐹𝑁)))
109106, 108eqtrd 2776 . . . . . . 7 (𝜑 → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁))) = (abs‘(𝐹𝑁)))
110102, 109breqtrrd 5133 . . . . . 6 (𝜑 → (abs‘(𝐹𝑁)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁𝑁))))
11132abscld 15321 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (abs‘(𝐹𝑘)) ∈ ℝ)
11285adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → (abs‘(𝐹𝑁)) ∈ ℝ)
113112, 25remulcld 11185 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))) ∈ ℝ)
11458adantr 481 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → 0 ≤ if(𝐴 ≤ 0, 0, 𝐴))
115 lemul2a 12010 . . . . . . . . . . . . 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 413 . . . . . . . . . . . 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 1374 . . . . . . . . . . 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 481 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℂ)
119107adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → (abs‘(𝐹𝑁)) ∈ ℂ)
12025recnd 11183 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)) ∈ ℂ)
121118, 119, 120mul12d 11364 . . . . . . . . . . . . 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 14052 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘𝑁) + 1)) = ((if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)) · if(𝐴 ≤ 0, 0, 𝐴)))
12340, 1eleq2s 2856 . . . . . . . . . . . . . . . . 17 (𝑘𝑊𝑘 ∈ ℂ)
124 ax-1cn 11109 . . . . . . . . . . . . . . . . . 18 1 ∈ ℂ
125 addsub 11412 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑘 + 1) − 𝑁) = ((𝑘𝑁) + 1))
126124, 125mp3an2 1449 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑘 + 1) − 𝑁) = ((𝑘𝑁) + 1))
127123, 38, 126syl2anr 597 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑊) → ((𝑘 + 1) − 𝑁) = ((𝑘𝑁) + 1))
128127oveq2d 7373 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘𝑁) + 1)))
129118, 120mulcomd 11176 . . . . . . . . . . . . . . 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 2787 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))
131130oveq2d 7373 . . . . . . . . . . . . 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 2776 . . . . . . . . . . . 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 5117 . . . . . . . . . . 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 238 . . . . . . . . . 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 6842 . . . . . . . . . . . . . . 15 (𝑛 = (𝑘 + 1) → (𝐹𝑛) = (𝐹‘(𝑘 + 1)))
136135eleq1d 2822 . . . . . . . . . . . . . 14 (𝑛 = (𝑘 + 1) → ((𝐹𝑛) ∈ ℂ ↔ (𝐹‘(𝑘 + 1)) ∈ ℂ))
137 fveq2 6842 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
138137eleq1d 2822 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → ((𝐹𝑘) ∈ ℂ ↔ (𝐹𝑛) ∈ ℂ))
139138cbvralvw 3225 . . . . . . . . . . . . . . . 16 (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ ↔ ∀𝑛𝑍 (𝐹𝑛) ∈ ℂ)
14083, 139sylib 217 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑛𝑍 (𝐹𝑛) ∈ ℂ)
141140adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → ∀𝑛𝑍 (𝐹𝑛) ∈ ℂ)
1421peano2uzs 12827 . . . . . . . . . . . . . . 15 (𝑘𝑊 → (𝑘 + 1) ∈ 𝑊)
14329sselda 3944 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 + 1) ∈ 𝑊) → (𝑘 + 1) ∈ 𝑍)
144142, 143sylan2 593 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → (𝑘 + 1) ∈ 𝑍)
145136, 141, 144rspcdva 3582 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → (𝐹‘(𝑘 + 1)) ∈ ℂ)
146145abscld 15321 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ)
14717adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → 𝐴 ∈ ℝ)
148147, 111remulcld 11185 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (𝐴 · (abs‘(𝐹𝑘))) ∈ ℝ)
14920, 111remulcld 11185 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹𝑘))) ∈ ℝ)
150 cvgrat.7 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹𝑘))))
15132absge0d 15329 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → 0 ≤ (abs‘(𝐹𝑘)))
152 max1 13104 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → 𝐴 ≤ if(𝐴 ≤ 0, 0, 𝐴))
15317, 16, 152sylancl 586 . . . . . . . . . . . . . 14 (𝜑𝐴 ≤ if(𝐴 ≤ 0, 0, 𝐴))
154153adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → 𝐴 ≤ if(𝐴 ≤ 0, 0, 𝐴))
155147, 20, 111, 151, 154lemul1ad 12094 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → (𝐴 · (abs‘(𝐹𝑘))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹𝑘))))
156146, 148, 149, 150, 155letrd 11312 . . . . . . . . . . 11 ((𝜑𝑘𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹𝑘))))
157 peano2uz 12826 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (ℤ𝑁) → (𝑘 + 1) ∈ (ℤ𝑁))
15822, 157syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑊) → (𝑘 + 1) ∈ (ℤ𝑁))
159 uznn0sub 12802 . . . . . . . . . . . . . . 15 ((𝑘 + 1) ∈ (ℤ𝑁) → ((𝑘 + 1) − 𝑁) ∈ ℕ0)
160158, 159syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑘𝑊) → ((𝑘 + 1) − 𝑁) ∈ ℕ0)
16120, 160reexpcld 14068 . . . . . . . . . . . . 13 ((𝜑𝑘𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)) ∈ ℝ)
162112, 161remulcld 11185 . . . . . . . . . . . 12 ((𝜑𝑘𝑊) → ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))) ∈ ℝ)
163 letr 11249 . . . . . . . . . . . 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 1371 . . . . . . . . . . 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 693 . . . . . . . . . 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 414 . . . . . . 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 12835 . . . . 5 (𝑘 ∈ (ℤ𝑁) → (𝜑 → (abs‘(𝐹𝑘)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁)))))
171170impcom 408 . . . 4 ((𝜑𝑘 ∈ (ℤ𝑁)) → (abs‘(𝐹𝑘)) ≤ ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))))
17247oveq2d 7373 . . . 4 ((𝜑𝑘 ∈ (ℤ𝑁)) → ((abs‘(𝐹𝑁)) · ((𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))‘𝑘)) = ((abs‘(𝐹𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘𝑁))))
173171, 172breqtrrd 5133 . . 3 ((𝜑𝑘 ∈ (ℤ𝑁)) → (abs‘(𝐹𝑘)) ≤ ((abs‘(𝐹𝑁)) · ((𝑛𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛𝑁)))‘𝑘)))
1741, 9, 26, 32, 80, 85, 173cvgcmpce 15703 . 2 (𝜑 → seq𝑁( + , 𝐹) ∈ dom ⇝ )
1753, 2, 31iserex 15541 . 2 (𝜑 → (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑁( + , 𝐹) ∈ dom ⇝ ))
176174, 175mpbird 256 1 (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  wss 3910  ifcif 4486   class class class wbr 5105  cmpt 5188  dom cdm 5633  cfv 6496  (class class class)co 7357  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056   < clt 11189  cle 11190  cmin 11385   / cdiv 11812  0cn0 12413  cz 12499  cuz 12763  seqcseq 13906  cexp 13967   shift cshi 14951  abscabs 15119  cli 15366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-ico 13270  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-shft 14952  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371  df-sum 15571
This theorem is referenced by:  efcllem  15960  cvgdvgrat  42583
  Copyright terms: Public domain W3C validator