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Theorem climcndslem2 14507
 Description: Lemma for climcnds 14508: bound the condensed series by the original series. (Contributed by Mario Carneiro, 18-Jul-2014.)
Hypotheses
Ref Expression
climcnds.1 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℝ)
climcnds.2 ((𝜑𝑘 ∈ ℕ) → 0 ≤ (𝐹𝑘))
climcnds.3 ((𝜑𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
climcnds.4 ((𝜑𝑛 ∈ ℕ0) → (𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))
Assertion
Ref Expression
climcndslem2 ((𝜑𝑁 ∈ ℕ) → (seq1( + , 𝐺)‘𝑁) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑁))))
Distinct variable groups:   𝑘,𝑛,𝐹   𝑘,𝐺,𝑛   𝜑,𝑘,𝑛
Allowed substitution hints:   𝑁(𝑘,𝑛)

Proof of Theorem climcndslem2
Dummy variables 𝑗 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6148 . . . . 5 (𝑥 = 1 → (seq1( + , 𝐺)‘𝑥) = (seq1( + , 𝐺)‘1))
2 oveq2 6612 . . . . . . . 8 (𝑥 = 1 → (2↑𝑥) = (2↑1))
3 2cn 11035 . . . . . . . . 9 2 ∈ ℂ
4 exp1 12806 . . . . . . . . 9 (2 ∈ ℂ → (2↑1) = 2)
53, 4ax-mp 5 . . . . . . . 8 (2↑1) = 2
62, 5syl6eq 2671 . . . . . . 7 (𝑥 = 1 → (2↑𝑥) = 2)
76fveq2d 6152 . . . . . 6 (𝑥 = 1 → (seq1( + , 𝐹)‘(2↑𝑥)) = (seq1( + , 𝐹)‘2))
87oveq2d 6620 . . . . 5 (𝑥 = 1 → (2 · (seq1( + , 𝐹)‘(2↑𝑥))) = (2 · (seq1( + , 𝐹)‘2)))
91, 8breq12d 4626 . . . 4 (𝑥 = 1 → ((seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥))) ↔ (seq1( + , 𝐺)‘1) ≤ (2 · (seq1( + , 𝐹)‘2))))
109imbi2d 330 . . 3 (𝑥 = 1 → ((𝜑 → (seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥)))) ↔ (𝜑 → (seq1( + , 𝐺)‘1) ≤ (2 · (seq1( + , 𝐹)‘2)))))
11 fveq2 6148 . . . . 5 (𝑥 = 𝑗 → (seq1( + , 𝐺)‘𝑥) = (seq1( + , 𝐺)‘𝑗))
12 oveq2 6612 . . . . . . 7 (𝑥 = 𝑗 → (2↑𝑥) = (2↑𝑗))
1312fveq2d 6152 . . . . . 6 (𝑥 = 𝑗 → (seq1( + , 𝐹)‘(2↑𝑥)) = (seq1( + , 𝐹)‘(2↑𝑗)))
1413oveq2d 6620 . . . . 5 (𝑥 = 𝑗 → (2 · (seq1( + , 𝐹)‘(2↑𝑥))) = (2 · (seq1( + , 𝐹)‘(2↑𝑗))))
1511, 14breq12d 4626 . . . 4 (𝑥 = 𝑗 → ((seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥))) ↔ (seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗)))))
1615imbi2d 330 . . 3 (𝑥 = 𝑗 → ((𝜑 → (seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥)))) ↔ (𝜑 → (seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗))))))
17 fveq2 6148 . . . . 5 (𝑥 = (𝑗 + 1) → (seq1( + , 𝐺)‘𝑥) = (seq1( + , 𝐺)‘(𝑗 + 1)))
18 oveq2 6612 . . . . . . 7 (𝑥 = (𝑗 + 1) → (2↑𝑥) = (2↑(𝑗 + 1)))
1918fveq2d 6152 . . . . . 6 (𝑥 = (𝑗 + 1) → (seq1( + , 𝐹)‘(2↑𝑥)) = (seq1( + , 𝐹)‘(2↑(𝑗 + 1))))
2019oveq2d 6620 . . . . 5 (𝑥 = (𝑗 + 1) → (2 · (seq1( + , 𝐹)‘(2↑𝑥))) = (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1)))))
2117, 20breq12d 4626 . . . 4 (𝑥 = (𝑗 + 1) → ((seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥))) ↔ (seq1( + , 𝐺)‘(𝑗 + 1)) ≤ (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1))))))
2221imbi2d 330 . . 3 (𝑥 = (𝑗 + 1) → ((𝜑 → (seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥)))) ↔ (𝜑 → (seq1( + , 𝐺)‘(𝑗 + 1)) ≤ (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1)))))))
23 fveq2 6148 . . . . 5 (𝑥 = 𝑁 → (seq1( + , 𝐺)‘𝑥) = (seq1( + , 𝐺)‘𝑁))
24 oveq2 6612 . . . . . . 7 (𝑥 = 𝑁 → (2↑𝑥) = (2↑𝑁))
2524fveq2d 6152 . . . . . 6 (𝑥 = 𝑁 → (seq1( + , 𝐹)‘(2↑𝑥)) = (seq1( + , 𝐹)‘(2↑𝑁)))
2625oveq2d 6620 . . . . 5 (𝑥 = 𝑁 → (2 · (seq1( + , 𝐹)‘(2↑𝑥))) = (2 · (seq1( + , 𝐹)‘(2↑𝑁))))
2723, 26breq12d 4626 . . . 4 (𝑥 = 𝑁 → ((seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥))) ↔ (seq1( + , 𝐺)‘𝑁) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑁)))))
2827imbi2d 330 . . 3 (𝑥 = 𝑁 → ((𝜑 → (seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥)))) ↔ (𝜑 → (seq1( + , 𝐺)‘𝑁) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑁))))))
29 1nn 10975 . . . . . . 7 1 ∈ ℕ
30 climcnds.2 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 0 ≤ (𝐹𝑘))
3130ralrimiva 2960 . . . . . . 7 (𝜑 → ∀𝑘 ∈ ℕ 0 ≤ (𝐹𝑘))
32 fveq2 6148 . . . . . . . . 9 (𝑘 = 1 → (𝐹𝑘) = (𝐹‘1))
3332breq2d 4625 . . . . . . . 8 (𝑘 = 1 → (0 ≤ (𝐹𝑘) ↔ 0 ≤ (𝐹‘1)))
3433rspcv 3291 . . . . . . 7 (1 ∈ ℕ → (∀𝑘 ∈ ℕ 0 ≤ (𝐹𝑘) → 0 ≤ (𝐹‘1)))
3529, 31, 34mpsyl 68 . . . . . 6 (𝜑 → 0 ≤ (𝐹‘1))
36 2nn 11129 . . . . . . . 8 2 ∈ ℕ
37 climcnds.1 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℝ)
3837ralrimiva 2960 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ ℕ (𝐹𝑘) ∈ ℝ)
39 fveq2 6148 . . . . . . . . . 10 (𝑘 = 2 → (𝐹𝑘) = (𝐹‘2))
4039eleq1d 2683 . . . . . . . . 9 (𝑘 = 2 → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘2) ∈ ℝ))
4140rspcv 3291 . . . . . . . 8 (2 ∈ ℕ → (∀𝑘 ∈ ℕ (𝐹𝑘) ∈ ℝ → (𝐹‘2) ∈ ℝ))
4236, 38, 41mpsyl 68 . . . . . . 7 (𝜑 → (𝐹‘2) ∈ ℝ)
4332eleq1d 2683 . . . . . . . . 9 (𝑘 = 1 → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘1) ∈ ℝ))
4443rspcv 3291 . . . . . . . 8 (1 ∈ ℕ → (∀𝑘 ∈ ℕ (𝐹𝑘) ∈ ℝ → (𝐹‘1) ∈ ℝ))
4529, 38, 44mpsyl 68 . . . . . . 7 (𝜑 → (𝐹‘1) ∈ ℝ)
4642, 45addge02d 10560 . . . . . 6 (𝜑 → (0 ≤ (𝐹‘1) ↔ (𝐹‘2) ≤ ((𝐹‘1) + (𝐹‘2))))
4735, 46mpbid 222 . . . . 5 (𝜑 → (𝐹‘2) ≤ ((𝐹‘1) + (𝐹‘2)))
4845, 42readdcld 10013 . . . . . 6 (𝜑 → ((𝐹‘1) + (𝐹‘2)) ∈ ℝ)
49 2re 11034 . . . . . . . 8 2 ∈ ℝ
50 2pos 11056 . . . . . . . 8 0 < 2
5149, 50pm3.2i 471 . . . . . . 7 (2 ∈ ℝ ∧ 0 < 2)
5251a1i 11 . . . . . 6 (𝜑 → (2 ∈ ℝ ∧ 0 < 2))
53 lemul2 10820 . . . . . 6 (((𝐹‘2) ∈ ℝ ∧ ((𝐹‘1) + (𝐹‘2)) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((𝐹‘2) ≤ ((𝐹‘1) + (𝐹‘2)) ↔ (2 · (𝐹‘2)) ≤ (2 · ((𝐹‘1) + (𝐹‘2)))))
5442, 48, 52, 53syl3anc 1323 . . . . 5 (𝜑 → ((𝐹‘2) ≤ ((𝐹‘1) + (𝐹‘2)) ↔ (2 · (𝐹‘2)) ≤ (2 · ((𝐹‘1) + (𝐹‘2)))))
5547, 54mpbid 222 . . . 4 (𝜑 → (2 · (𝐹‘2)) ≤ (2 · ((𝐹‘1) + (𝐹‘2))))
56 1z 11351 . . . . 5 1 ∈ ℤ
57 1nn0 11252 . . . . . 6 1 ∈ ℕ0
58 climcnds.4 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → (𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))
5958ralrimiva 2960 . . . . . 6 (𝜑 → ∀𝑛 ∈ ℕ0 (𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))
60 fveq2 6148 . . . . . . . 8 (𝑛 = 1 → (𝐺𝑛) = (𝐺‘1))
61 oveq2 6612 . . . . . . . . . 10 (𝑛 = 1 → (2↑𝑛) = (2↑1))
6261, 5syl6eq 2671 . . . . . . . . 9 (𝑛 = 1 → (2↑𝑛) = 2)
6362fveq2d 6152 . . . . . . . . 9 (𝑛 = 1 → (𝐹‘(2↑𝑛)) = (𝐹‘2))
6462, 63oveq12d 6622 . . . . . . . 8 (𝑛 = 1 → ((2↑𝑛) · (𝐹‘(2↑𝑛))) = (2 · (𝐹‘2)))
6560, 64eqeq12d 2636 . . . . . . 7 (𝑛 = 1 → ((𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))) ↔ (𝐺‘1) = (2 · (𝐹‘2))))
6665rspcv 3291 . . . . . 6 (1 ∈ ℕ0 → (∀𝑛 ∈ ℕ0 (𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))) → (𝐺‘1) = (2 · (𝐹‘2))))
6757, 59, 66mpsyl 68 . . . . 5 (𝜑 → (𝐺‘1) = (2 · (𝐹‘2)))
6856, 67seq1i 12755 . . . 4 (𝜑 → (seq1( + , 𝐺)‘1) = (2 · (𝐹‘2)))
69 nnuz 11667 . . . . . 6 ℕ = (ℤ‘1)
70 df-2 11023 . . . . . 6 2 = (1 + 1)
71 eqidd 2622 . . . . . . 7 (𝜑 → (𝐹‘1) = (𝐹‘1))
7256, 71seq1i 12755 . . . . . 6 (𝜑 → (seq1( + , 𝐹)‘1) = (𝐹‘1))
73 eqidd 2622 . . . . . 6 (𝜑 → (𝐹‘2) = (𝐹‘2))
7469, 29, 70, 72, 73seqp1i 12757 . . . . 5 (𝜑 → (seq1( + , 𝐹)‘2) = ((𝐹‘1) + (𝐹‘2)))
7574oveq2d 6620 . . . 4 (𝜑 → (2 · (seq1( + , 𝐹)‘2)) = (2 · ((𝐹‘1) + (𝐹‘2))))
7655, 68, 753brtr4d 4645 . . 3 (𝜑 → (seq1( + , 𝐺)‘1) ≤ (2 · (seq1( + , 𝐹)‘2)))
77 peano2nn 10976 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
7877adantl 482 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ)
7978nnnn0d 11295 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ0)
8059adantr 481 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ∀𝑛 ∈ ℕ0 (𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))
81 fveq2 6148 . . . . . . . . . . . 12 (𝑛 = (𝑗 + 1) → (𝐺𝑛) = (𝐺‘(𝑗 + 1)))
82 oveq2 6612 . . . . . . . . . . . . 13 (𝑛 = (𝑗 + 1) → (2↑𝑛) = (2↑(𝑗 + 1)))
8382fveq2d 6152 . . . . . . . . . . . . 13 (𝑛 = (𝑗 + 1) → (𝐹‘(2↑𝑛)) = (𝐹‘(2↑(𝑗 + 1))))
8482, 83oveq12d 6622 . . . . . . . . . . . 12 (𝑛 = (𝑗 + 1) → ((2↑𝑛) · (𝐹‘(2↑𝑛))) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))))
8581, 84eqeq12d 2636 . . . . . . . . . . 11 (𝑛 = (𝑗 + 1) → ((𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))) ↔ (𝐺‘(𝑗 + 1)) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1))))))
8685rspcv 3291 . . . . . . . . . 10 ((𝑗 + 1) ∈ ℕ0 → (∀𝑛 ∈ ℕ0 (𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))) → (𝐺‘(𝑗 + 1)) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1))))))
8779, 80, 86sylc 65 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐺‘(𝑗 + 1)) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))))
88 nnnn0 11243 . . . . . . . . . . . . 13 (𝑗 ∈ ℕ → 𝑗 ∈ ℕ0)
8988adantl 482 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ0)
90 expp1 12807 . . . . . . . . . . . 12 ((2 ∈ ℂ ∧ 𝑗 ∈ ℕ0) → (2↑(𝑗 + 1)) = ((2↑𝑗) · 2))
913, 89, 90sylancr 694 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) = ((2↑𝑗) · 2))
92 nnexpcl 12813 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑗 ∈ ℕ0) → (2↑𝑗) ∈ ℕ)
9336, 88, 92sylancr 694 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ → (2↑𝑗) ∈ ℕ)
9493adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℕ)
9594nncnd 10980 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℂ)
96 mulcom 9966 . . . . . . . . . . . 12 (((2↑𝑗) ∈ ℂ ∧ 2 ∈ ℂ) → ((2↑𝑗) · 2) = (2 · (2↑𝑗)))
9795, 3, 96sylancl 693 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · 2) = (2 · (2↑𝑗)))
9891, 97eqtrd 2655 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) = (2 · (2↑𝑗)))
9998oveq1d 6619 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))) = ((2 · (2↑𝑗)) · (𝐹‘(2↑(𝑗 + 1)))))
1003a1i 11 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → 2 ∈ ℂ)
101 nnexpcl 12813 . . . . . . . . . . . . 13 ((2 ∈ ℕ ∧ (𝑗 + 1) ∈ ℕ0) → (2↑(𝑗 + 1)) ∈ ℕ)
10236, 79, 101sylancr 694 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ ℕ)
10338adantr 481 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ∀𝑘 ∈ ℕ (𝐹𝑘) ∈ ℝ)
104 fveq2 6148 . . . . . . . . . . . . . 14 (𝑘 = (2↑(𝑗 + 1)) → (𝐹𝑘) = (𝐹‘(2↑(𝑗 + 1))))
105104eleq1d 2683 . . . . . . . . . . . . 13 (𝑘 = (2↑(𝑗 + 1)) → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ))
106105rspcv 3291 . . . . . . . . . . . 12 ((2↑(𝑗 + 1)) ∈ ℕ → (∀𝑘 ∈ ℕ (𝐹𝑘) ∈ ℝ → (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ))
107102, 103, 106sylc 65 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ)
108107recnd 10012 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℂ)
109100, 95, 108mulassd 10007 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ((2 · (2↑𝑗)) · (𝐹‘(2↑(𝑗 + 1)))) = (2 · ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1))))))
11087, 99, 1093eqtrd 2659 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐺‘(𝑗 + 1)) = (2 · ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1))))))
11194nnnn0d 11295 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℕ0)
112 hashfz1 13074 . . . . . . . . . . . . . . 15 ((2↑𝑗) ∈ ℕ0 → (#‘(1...(2↑𝑗))) = (2↑𝑗))
113111, 112syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (#‘(1...(2↑𝑗))) = (2↑𝑗))
114113, 95eqeltrd 2698 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (#‘(1...(2↑𝑗))) ∈ ℂ)
115 fzfid 12712 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (((2↑𝑗) + 1)...(2↑(𝑗 + 1))) ∈ Fin)
116 hashcl 13087 . . . . . . . . . . . . . . 15 ((((2↑𝑗) + 1)...(2↑(𝑗 + 1))) ∈ Fin → (#‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∈ ℕ0)
117115, 116syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (#‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∈ ℕ0)
118117nn0cnd 11297 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (#‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∈ ℂ)
119 simpr 477 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
120119nnzd 11425 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℤ)
121 uzid 11646 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ𝑗))
122 peano2uz 11685 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (ℤ𝑗) → (𝑗 + 1) ∈ (ℤ𝑗))
123 1le2 11185 . . . . . . . . . . . . . . . . . . 19 1 ≤ 2
124 leexp2a 12856 . . . . . . . . . . . . . . . . . . 19 ((2 ∈ ℝ ∧ 1 ≤ 2 ∧ (𝑗 + 1) ∈ (ℤ𝑗)) → (2↑𝑗) ≤ (2↑(𝑗 + 1)))
12549, 123, 124mp3an12 1411 . . . . . . . . . . . . . . . . . 18 ((𝑗 + 1) ∈ (ℤ𝑗) → (2↑𝑗) ≤ (2↑(𝑗 + 1)))
126120, 121, 122, 1254syl 19 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ≤ (2↑(𝑗 + 1)))
12794, 69syl6eleq 2708 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ (ℤ‘1))
128102nnzd 11425 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ ℤ)
129 elfz5 12276 . . . . . . . . . . . . . . . . . 18 (((2↑𝑗) ∈ (ℤ‘1) ∧ (2↑(𝑗 + 1)) ∈ ℤ) → ((2↑𝑗) ∈ (1...(2↑(𝑗 + 1))) ↔ (2↑𝑗) ≤ (2↑(𝑗 + 1))))
130127, 128, 129syl2anc 692 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) ∈ (1...(2↑(𝑗 + 1))) ↔ (2↑𝑗) ≤ (2↑(𝑗 + 1))))
131126, 130mpbird 247 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ (1...(2↑(𝑗 + 1))))
132 fzsplit 12309 . . . . . . . . . . . . . . . 16 ((2↑𝑗) ∈ (1...(2↑(𝑗 + 1))) → (1...(2↑(𝑗 + 1))) = ((1...(2↑𝑗)) ∪ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))))
133131, 132syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (1...(2↑(𝑗 + 1))) = ((1...(2↑𝑗)) ∪ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))))
134133fveq2d 6152 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (#‘(1...(2↑(𝑗 + 1)))) = (#‘((1...(2↑𝑗)) ∪ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))))))
13595times2d 11220 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · 2) = ((2↑𝑗) + (2↑𝑗)))
13691, 135eqtrd 2655 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) = ((2↑𝑗) + (2↑𝑗)))
137102nnnn0d 11295 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ ℕ0)
138 hashfz1 13074 . . . . . . . . . . . . . . . 16 ((2↑(𝑗 + 1)) ∈ ℕ0 → (#‘(1...(2↑(𝑗 + 1)))) = (2↑(𝑗 + 1)))
139137, 138syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (#‘(1...(2↑(𝑗 + 1)))) = (2↑(𝑗 + 1)))
140113oveq1d 6619 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → ((#‘(1...(2↑𝑗))) + (2↑𝑗)) = ((2↑𝑗) + (2↑𝑗)))
141136, 139, 1403eqtr4d 2665 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (#‘(1...(2↑(𝑗 + 1)))) = ((#‘(1...(2↑𝑗))) + (2↑𝑗)))
142 fzfid 12712 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (1...(2↑𝑗)) ∈ Fin)
14394nnred 10979 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℝ)
144143ltp1d 10898 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) < ((2↑𝑗) + 1))
145 fzdisj 12310 . . . . . . . . . . . . . . . 16 ((2↑𝑗) < ((2↑𝑗) + 1) → ((1...(2↑𝑗)) ∩ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) = ∅)
146144, 145syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → ((1...(2↑𝑗)) ∩ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) = ∅)
147 hashun 13111 . . . . . . . . . . . . . . 15 (((1...(2↑𝑗)) ∈ Fin ∧ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))) ∈ Fin ∧ ((1...(2↑𝑗)) ∩ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) = ∅) → (#‘((1...(2↑𝑗)) ∪ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))))) = ((#‘(1...(2↑𝑗))) + (#‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1))))))
148142, 115, 146, 147syl3anc 1323 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (#‘((1...(2↑𝑗)) ∪ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))))) = ((#‘(1...(2↑𝑗))) + (#‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1))))))
149134, 141, 1483eqtr3d 2663 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ((#‘(1...(2↑𝑗))) + (2↑𝑗)) = ((#‘(1...(2↑𝑗))) + (#‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1))))))
150114, 95, 118, 149addcanad 10185 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) = (#‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))))
151150oveq1d 6619 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) = ((#‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) · (𝐹‘(2↑(𝑗 + 1)))))
152 fsumconst 14450 . . . . . . . . . . . 12 (((((2↑𝑗) + 1)...(2↑(𝑗 + 1))) ∈ Fin ∧ (𝐹‘(2↑(𝑗 + 1))) ∈ ℂ) → Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘(2↑(𝑗 + 1))) = ((#‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) · (𝐹‘(2↑(𝑗 + 1)))))
153115, 108, 152syl2anc 692 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘(2↑(𝑗 + 1))) = ((#‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) · (𝐹‘(2↑(𝑗 + 1)))))
154151, 153eqtr4d 2658 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) = Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘(2↑(𝑗 + 1))))
155107adantr 481 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ)
156 simpl 473 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → 𝜑)
157156adantr 481 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → 𝜑)
158 peano2nn 10976 . . . . . . . . . . . . . 14 ((2↑𝑗) ∈ ℕ → ((2↑𝑗) + 1) ∈ ℕ)
15994, 158syl 17 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) + 1) ∈ ℕ)
160 elfzuz 12280 . . . . . . . . . . . . 13 (𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))) → 𝑘 ∈ (ℤ‘((2↑𝑗) + 1)))
161 eluznn 11702 . . . . . . . . . . . . 13 ((((2↑𝑗) + 1) ∈ ℕ ∧ 𝑘 ∈ (ℤ‘((2↑𝑗) + 1))) → 𝑘 ∈ ℕ)
162159, 160, 161syl2an 494 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → 𝑘 ∈ ℕ)
163157, 162, 37syl2anc 692 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (𝐹𝑘) ∈ ℝ)
164 elfzuz3 12281 . . . . . . . . . . . . . . 15 (𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))) → (2↑(𝑗 + 1)) ∈ (ℤ𝑛))
165164adantl 482 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (2↑(𝑗 + 1)) ∈ (ℤ𝑛))
166 simplll 797 . . . . . . . . . . . . . . 15 ((((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...(2↑(𝑗 + 1)))) → 𝜑)
167 elfzuz 12280 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))) → 𝑛 ∈ (ℤ‘((2↑𝑗) + 1)))
168 eluznn 11702 . . . . . . . . . . . . . . . . 17 ((((2↑𝑗) + 1) ∈ ℕ ∧ 𝑛 ∈ (ℤ‘((2↑𝑗) + 1))) → 𝑛 ∈ ℕ)
169159, 167, 168syl2an 494 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → 𝑛 ∈ ℕ)
170 elfzuz 12280 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (𝑛...(2↑(𝑗 + 1))) → 𝑘 ∈ (ℤ𝑛))
171 eluznn 11702 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑘 ∈ ℕ)
172169, 170, 171syl2an 494 . . . . . . . . . . . . . . 15 ((((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...(2↑(𝑗 + 1)))) → 𝑘 ∈ ℕ)
173166, 172, 37syl2anc 692 . . . . . . . . . . . . . 14 ((((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...(2↑(𝑗 + 1)))) → (𝐹𝑘) ∈ ℝ)
174 simplll 797 . . . . . . . . . . . . . . 15 ((((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...((2↑(𝑗 + 1)) − 1))) → 𝜑)
175 elfzuz 12280 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (𝑛...((2↑(𝑗 + 1)) − 1)) → 𝑘 ∈ (ℤ𝑛))
176169, 175, 171syl2an 494 . . . . . . . . . . . . . . 15 ((((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...((2↑(𝑗 + 1)) − 1))) → 𝑘 ∈ ℕ)
177 climcnds.3 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
178174, 176, 177syl2anc 692 . . . . . . . . . . . . . 14 ((((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...((2↑(𝑗 + 1)) − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
179165, 173, 178monoord2 12772 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑛))
180179ralrimiva 2960 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ∀𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑛))
181 fveq2 6148 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
182181breq2d 4625 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → ((𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑛) ↔ (𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑘)))
183182rspccva 3294 . . . . . . . . . . . 12 ((∀𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑛) ∧ 𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑘))
184180, 183sylan 488 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑘))
185115, 155, 163, 184fsumle 14458 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘(2↑(𝑗 + 1))) ≤ Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘))
186154, 185eqbrtrd 4635 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) ≤ Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘))
187143, 107remulcld 10014 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) ∈ ℝ)
188115, 163fsumrecl 14398 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘) ∈ ℝ)
18951a1i 11 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (2 ∈ ℝ ∧ 0 < 2))
190 lemul2 10820 . . . . . . . . . 10 ((((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) ∈ ℝ ∧ Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) ≤ Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘) ↔ (2 · ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1))))) ≤ (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘))))
191187, 188, 189, 190syl3anc 1323 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) ≤ Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘) ↔ (2 · ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1))))) ≤ (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘))))
192186, 191mpbid 222 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (2 · ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1))))) ≤ (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)))
193110, 192eqbrtrd 4635 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (𝐺‘(𝑗 + 1)) ≤ (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)))
194 1zzd 11352 . . . . . . . . . 10 (𝜑 → 1 ∈ ℤ)
195 nnnn0 11243 . . . . . . . . . . 11 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
196 simpr 477 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
197 nnexpcl 12813 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
19836, 196, 197sylancr 694 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
199198nnred 10979 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℝ)
20038adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ0) → ∀𝑘 ∈ ℕ (𝐹𝑘) ∈ ℝ)
201 fveq2 6148 . . . . . . . . . . . . . . . 16 (𝑘 = (2↑𝑛) → (𝐹𝑘) = (𝐹‘(2↑𝑛)))
202201eleq1d 2683 . . . . . . . . . . . . . . 15 (𝑘 = (2↑𝑛) → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘(2↑𝑛)) ∈ ℝ))
203202rspcv 3291 . . . . . . . . . . . . . 14 ((2↑𝑛) ∈ ℕ → (∀𝑘 ∈ ℕ (𝐹𝑘) ∈ ℝ → (𝐹‘(2↑𝑛)) ∈ ℝ))
204198, 200, 203sylc 65 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ0) → (𝐹‘(2↑𝑛)) ∈ ℝ)
205199, 204remulcld 10014 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ0) → ((2↑𝑛) · (𝐹‘(2↑𝑛))) ∈ ℝ)
20658, 205eqeltrd 2698 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → (𝐺𝑛) ∈ ℝ)
207195, 206sylan2 491 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ ℝ)
20869, 194, 207serfre 12770 . . . . . . . . 9 (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
209208ffvelrnda 6315 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (seq1( + , 𝐺)‘𝑗) ∈ ℝ)
210207ralrimiva 2960 . . . . . . . . . 10 (𝜑 → ∀𝑛 ∈ ℕ (𝐺𝑛) ∈ ℝ)
211210adantr 481 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ∀𝑛 ∈ ℕ (𝐺𝑛) ∈ ℝ)
21281eleq1d 2683 . . . . . . . . . 10 (𝑛 = (𝑗 + 1) → ((𝐺𝑛) ∈ ℝ ↔ (𝐺‘(𝑗 + 1)) ∈ ℝ))
213212rspcv 3291 . . . . . . . . 9 ((𝑗 + 1) ∈ ℕ → (∀𝑛 ∈ ℕ (𝐺𝑛) ∈ ℝ → (𝐺‘(𝑗 + 1)) ∈ ℝ))
21478, 211, 213sylc 65 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐺‘(𝑗 + 1)) ∈ ℝ)
21569, 194, 37serfre 12770 . . . . . . . . . 10 (𝜑 → seq1( + , 𝐹):ℕ⟶ℝ)
216 ffvelrn 6313 . . . . . . . . . 10 ((seq1( + , 𝐹):ℕ⟶ℝ ∧ (2↑𝑗) ∈ ℕ) → (seq1( + , 𝐹)‘(2↑𝑗)) ∈ ℝ)
217215, 93, 216syl2an 494 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘(2↑𝑗)) ∈ ℝ)
218 remulcl 9965 . . . . . . . . 9 ((2 ∈ ℝ ∧ (seq1( + , 𝐹)‘(2↑𝑗)) ∈ ℝ) → (2 · (seq1( + , 𝐹)‘(2↑𝑗))) ∈ ℝ)
21949, 217, 218sylancr 694 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (2 · (seq1( + , 𝐹)‘(2↑𝑗))) ∈ ℝ)
220 remulcl 9965 . . . . . . . . 9 ((2 ∈ ℝ ∧ Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘) ∈ ℝ) → (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)) ∈ ℝ)
22149, 188, 220sylancr 694 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)) ∈ ℝ)
222 le2add 10454 . . . . . . . 8 ((((seq1( + , 𝐺)‘𝑗) ∈ ℝ ∧ (𝐺‘(𝑗 + 1)) ∈ ℝ) ∧ ((2 · (seq1( + , 𝐹)‘(2↑𝑗))) ∈ ℝ ∧ (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)) ∈ ℝ)) → (((seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗))) ∧ (𝐺‘(𝑗 + 1)) ≤ (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘))) → ((seq1( + , 𝐺)‘𝑗) + (𝐺‘(𝑗 + 1))) ≤ ((2 · (seq1( + , 𝐹)‘(2↑𝑗))) + (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)))))
223209, 214, 219, 221, 222syl22anc 1324 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (((seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗))) ∧ (𝐺‘(𝑗 + 1)) ≤ (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘))) → ((seq1( + , 𝐺)‘𝑗) + (𝐺‘(𝑗 + 1))) ≤ ((2 · (seq1( + , 𝐹)‘(2↑𝑗))) + (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)))))
224193, 223mpan2d 709 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ((seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗))) → ((seq1( + , 𝐺)‘𝑗) + (𝐺‘(𝑗 + 1))) ≤ ((2 · (seq1( + , 𝐹)‘(2↑𝑗))) + (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)))))
225119, 69syl6eleq 2708 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ‘1))
226 seqp1 12756 . . . . . . . 8 (𝑗 ∈ (ℤ‘1) → (seq1( + , 𝐺)‘(𝑗 + 1)) = ((seq1( + , 𝐺)‘𝑗) + (𝐺‘(𝑗 + 1))))
227225, 226syl 17 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (seq1( + , 𝐺)‘(𝑗 + 1)) = ((seq1( + , 𝐺)‘𝑗) + (𝐺‘(𝑗 + 1))))
228 fzfid 12712 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (1...(2↑(𝑗 + 1))) ∈ Fin)
229 elfznn 12312 . . . . . . . . . . . 12 (𝑘 ∈ (1...(2↑(𝑗 + 1))) → 𝑘 ∈ ℕ)
23037recnd 10012 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℂ)
231156, 229, 230syl2an 494 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑(𝑗 + 1)))) → (𝐹𝑘) ∈ ℂ)
232146, 133, 228, 231fsumsplit 14404 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...(2↑(𝑗 + 1)))(𝐹𝑘) = (Σ𝑘 ∈ (1...(2↑𝑗))(𝐹𝑘) + Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)))
233 eqidd 2622 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑(𝑗 + 1)))) → (𝐹𝑘) = (𝐹𝑘))
234102, 69syl6eleq 2708 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ (ℤ‘1))
235233, 234, 231fsumser 14394 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...(2↑(𝑗 + 1)))(𝐹𝑘) = (seq1( + , 𝐹)‘(2↑(𝑗 + 1))))
236 eqidd 2622 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑𝑗))) → (𝐹𝑘) = (𝐹𝑘))
237 elfznn 12312 . . . . . . . . . . . . 13 (𝑘 ∈ (1...(2↑𝑗)) → 𝑘 ∈ ℕ)
238156, 237, 230syl2an 494 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑𝑗))) → (𝐹𝑘) ∈ ℂ)
239236, 127, 238fsumser 14394 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...(2↑𝑗))(𝐹𝑘) = (seq1( + , 𝐹)‘(2↑𝑗)))
240239oveq1d 6619 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (Σ𝑘 ∈ (1...(2↑𝑗))(𝐹𝑘) + Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)) = ((seq1( + , 𝐹)‘(2↑𝑗)) + Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)))
241232, 235, 2403eqtr3d 2663 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘(2↑(𝑗 + 1))) = ((seq1( + , 𝐹)‘(2↑𝑗)) + Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)))
242241oveq2d 6620 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1)))) = (2 · ((seq1( + , 𝐹)‘(2↑𝑗)) + Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘))))
243217recnd 10012 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘(2↑𝑗)) ∈ ℂ)
244188recnd 10012 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘) ∈ ℂ)
245100, 243, 244adddid 10008 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (2 · ((seq1( + , 𝐹)‘(2↑𝑗)) + Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘))) = ((2 · (seq1( + , 𝐹)‘(2↑𝑗))) + (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘))))
246242, 245eqtrd 2655 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1)))) = ((2 · (seq1( + , 𝐹)‘(2↑𝑗))) + (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘))))
247227, 246breq12d 4626 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ((seq1( + , 𝐺)‘(𝑗 + 1)) ≤ (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1)))) ↔ ((seq1( + , 𝐺)‘𝑗) + (𝐺‘(𝑗 + 1))) ≤ ((2 · (seq1( + , 𝐹)‘(2↑𝑗))) + (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)))))
248224, 247sylibrd 249 . . . . 5 ((𝜑𝑗 ∈ ℕ) → ((seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗))) → (seq1( + , 𝐺)‘(𝑗 + 1)) ≤ (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1))))))
249248expcom 451 . . . 4 (𝑗 ∈ ℕ → (𝜑 → ((seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗))) → (seq1( + , 𝐺)‘(𝑗 + 1)) ≤ (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1)))))))
250249a2d 29 . . 3 (𝑗 ∈ ℕ → ((𝜑 → (seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗)))) → (𝜑 → (seq1( + , 𝐺)‘(𝑗 + 1)) ≤ (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1)))))))
25110, 16, 22, 28, 76, 250nnind 10982 . 2 (𝑁 ∈ ℕ → (𝜑 → (seq1( + , 𝐺)‘𝑁) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑁)))))
252251impcom 446 1 ((𝜑𝑁 ∈ ℕ) → (seq1( + , 𝐺)‘𝑁) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑁))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2907   ∪ cun 3553   ∩ cin 3554  ∅c0 3891   class class class wbr 4613  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604  Fincfn 7899  ℂcc 9878  ℝcr 9879  0cc0 9880  1c1 9881   + caddc 9883   · cmul 9885   < clt 10018   ≤ cle 10019   − cmin 10210  ℕcn 10964  2c2 11014  ℕ0cn0 11236  ℤcz 11321  ℤ≥cuz 11631  ...cfz 12268  seqcseq 12741  ↑cexp 12800  #chash 13057  Σcsu 14350 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 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-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-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-oi 8359  df-card 8709  df-cda 8934  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-n0 11237  df-z 11322  df-uz 11632  df-rp 11777  df-ico 12123  df-fz 12269  df-fzo 12407  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-sum 14351 This theorem is referenced by:  climcnds  14508
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