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Theorem climcndslem2 14789
Description: Lemma for climcnds 14790: bound the condensed series by the original series. (Contributed by Mario Carneiro, 18-Jul-2014.) (Proof shortened by AV, 10-Jul-2022.)
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 6332 . . . . 5 (𝑥 = 1 → (seq1( + , 𝐺)‘𝑥) = (seq1( + , 𝐺)‘1))
2 oveq2 6801 . . . . . . . 8 (𝑥 = 1 → (2↑𝑥) = (2↑1))
3 2cn 11293 . . . . . . . . 9 2 ∈ ℂ
4 exp1 13073 . . . . . . . . 9 (2 ∈ ℂ → (2↑1) = 2)
53, 4ax-mp 5 . . . . . . . 8 (2↑1) = 2
62, 5syl6eq 2821 . . . . . . 7 (𝑥 = 1 → (2↑𝑥) = 2)
76fveq2d 6336 . . . . . 6 (𝑥 = 1 → (seq1( + , 𝐹)‘(2↑𝑥)) = (seq1( + , 𝐹)‘2))
87oveq2d 6809 . . . . 5 (𝑥 = 1 → (2 · (seq1( + , 𝐹)‘(2↑𝑥))) = (2 · (seq1( + , 𝐹)‘2)))
91, 8breq12d 4799 . . . 4 (𝑥 = 1 → ((seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥))) ↔ (seq1( + , 𝐺)‘1) ≤ (2 · (seq1( + , 𝐹)‘2))))
109imbi2d 329 . . 3 (𝑥 = 1 → ((𝜑 → (seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥)))) ↔ (𝜑 → (seq1( + , 𝐺)‘1) ≤ (2 · (seq1( + , 𝐹)‘2)))))
11 fveq2 6332 . . . . 5 (𝑥 = 𝑗 → (seq1( + , 𝐺)‘𝑥) = (seq1( + , 𝐺)‘𝑗))
12 oveq2 6801 . . . . . . 7 (𝑥 = 𝑗 → (2↑𝑥) = (2↑𝑗))
1312fveq2d 6336 . . . . . 6 (𝑥 = 𝑗 → (seq1( + , 𝐹)‘(2↑𝑥)) = (seq1( + , 𝐹)‘(2↑𝑗)))
1413oveq2d 6809 . . . . 5 (𝑥 = 𝑗 → (2 · (seq1( + , 𝐹)‘(2↑𝑥))) = (2 · (seq1( + , 𝐹)‘(2↑𝑗))))
1511, 14breq12d 4799 . . . 4 (𝑥 = 𝑗 → ((seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥))) ↔ (seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗)))))
1615imbi2d 329 . . 3 (𝑥 = 𝑗 → ((𝜑 → (seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥)))) ↔ (𝜑 → (seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗))))))
17 fveq2 6332 . . . . 5 (𝑥 = (𝑗 + 1) → (seq1( + , 𝐺)‘𝑥) = (seq1( + , 𝐺)‘(𝑗 + 1)))
18 oveq2 6801 . . . . . . 7 (𝑥 = (𝑗 + 1) → (2↑𝑥) = (2↑(𝑗 + 1)))
1918fveq2d 6336 . . . . . 6 (𝑥 = (𝑗 + 1) → (seq1( + , 𝐹)‘(2↑𝑥)) = (seq1( + , 𝐹)‘(2↑(𝑗 + 1))))
2019oveq2d 6809 . . . . 5 (𝑥 = (𝑗 + 1) → (2 · (seq1( + , 𝐹)‘(2↑𝑥))) = (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1)))))
2117, 20breq12d 4799 . . . 4 (𝑥 = (𝑗 + 1) → ((seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥))) ↔ (seq1( + , 𝐺)‘(𝑗 + 1)) ≤ (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1))))))
2221imbi2d 329 . . 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 6332 . . . . 5 (𝑥 = 𝑁 → (seq1( + , 𝐺)‘𝑥) = (seq1( + , 𝐺)‘𝑁))
24 oveq2 6801 . . . . . . 7 (𝑥 = 𝑁 → (2↑𝑥) = (2↑𝑁))
2524fveq2d 6336 . . . . . 6 (𝑥 = 𝑁 → (seq1( + , 𝐹)‘(2↑𝑥)) = (seq1( + , 𝐹)‘(2↑𝑁)))
2625oveq2d 6809 . . . . 5 (𝑥 = 𝑁 → (2 · (seq1( + , 𝐹)‘(2↑𝑥))) = (2 · (seq1( + , 𝐹)‘(2↑𝑁))))
2723, 26breq12d 4799 . . . 4 (𝑥 = 𝑁 → ((seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥))) ↔ (seq1( + , 𝐺)‘𝑁) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑁)))))
2827imbi2d 329 . . 3 (𝑥 = 𝑁 → ((𝜑 → (seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥)))) ↔ (𝜑 → (seq1( + , 𝐺)‘𝑁) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑁))))))
29 fveq2 6332 . . . . . . . 8 (𝑘 = 1 → (𝐹𝑘) = (𝐹‘1))
3029breq2d 4798 . . . . . . 7 (𝑘 = 1 → (0 ≤ (𝐹𝑘) ↔ 0 ≤ (𝐹‘1)))
31 climcnds.2 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 0 ≤ (𝐹𝑘))
3231ralrimiva 3115 . . . . . . 7 (𝜑 → ∀𝑘 ∈ ℕ 0 ≤ (𝐹𝑘))
33 1nn 11233 . . . . . . . 8 1 ∈ ℕ
3433a1i 11 . . . . . . 7 (𝜑 → 1 ∈ ℕ)
3530, 32, 34rspcdva 3466 . . . . . 6 (𝜑 → 0 ≤ (𝐹‘1))
36 fveq2 6332 . . . . . . . . 9 (𝑘 = 2 → (𝐹𝑘) = (𝐹‘2))
3736eleq1d 2835 . . . . . . . 8 (𝑘 = 2 → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘2) ∈ ℝ))
38 climcnds.1 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℝ)
3938ralrimiva 3115 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ ℕ (𝐹𝑘) ∈ ℝ)
40 2nn 11387 . . . . . . . . 9 2 ∈ ℕ
4140a1i 11 . . . . . . . 8 (𝜑 → 2 ∈ ℕ)
4237, 39, 41rspcdva 3466 . . . . . . 7 (𝜑 → (𝐹‘2) ∈ ℝ)
4329eleq1d 2835 . . . . . . . 8 (𝑘 = 1 → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘1) ∈ ℝ))
4443, 39, 34rspcdva 3466 . . . . . . 7 (𝜑 → (𝐹‘1) ∈ ℝ)
4542, 44addge02d 10818 . . . . . 6 (𝜑 → (0 ≤ (𝐹‘1) ↔ (𝐹‘2) ≤ ((𝐹‘1) + (𝐹‘2))))
4635, 45mpbid 222 . . . . 5 (𝜑 → (𝐹‘2) ≤ ((𝐹‘1) + (𝐹‘2)))
4744, 42readdcld 10271 . . . . . 6 (𝜑 → ((𝐹‘1) + (𝐹‘2)) ∈ ℝ)
4841nnrpd 12073 . . . . . 6 (𝜑 → 2 ∈ ℝ+)
4942, 47, 48lemul2d 12119 . . . . 5 (𝜑 → ((𝐹‘2) ≤ ((𝐹‘1) + (𝐹‘2)) ↔ (2 · (𝐹‘2)) ≤ (2 · ((𝐹‘1) + (𝐹‘2)))))
5046, 49mpbid 222 . . . 4 (𝜑 → (2 · (𝐹‘2)) ≤ (2 · ((𝐹‘1) + (𝐹‘2))))
51 1z 11609 . . . . 5 1 ∈ ℤ
52 fveq2 6332 . . . . . . 7 (𝑛 = 1 → (𝐺𝑛) = (𝐺‘1))
53 oveq2 6801 . . . . . . . . 9 (𝑛 = 1 → (2↑𝑛) = (2↑1))
5453, 5syl6eq 2821 . . . . . . . 8 (𝑛 = 1 → (2↑𝑛) = 2)
5554fveq2d 6336 . . . . . . . 8 (𝑛 = 1 → (𝐹‘(2↑𝑛)) = (𝐹‘2))
5654, 55oveq12d 6811 . . . . . . 7 (𝑛 = 1 → ((2↑𝑛) · (𝐹‘(2↑𝑛))) = (2 · (𝐹‘2)))
5752, 56eqeq12d 2786 . . . . . 6 (𝑛 = 1 → ((𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))) ↔ (𝐺‘1) = (2 · (𝐹‘2))))
58 climcnds.4 . . . . . . 7 ((𝜑𝑛 ∈ ℕ0) → (𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))
5958ralrimiva 3115 . . . . . 6 (𝜑 → ∀𝑛 ∈ ℕ0 (𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))
60 1nn0 11510 . . . . . . 7 1 ∈ ℕ0
6160a1i 11 . . . . . 6 (𝜑 → 1 ∈ ℕ0)
6257, 59, 61rspcdva 3466 . . . . 5 (𝜑 → (𝐺‘1) = (2 · (𝐹‘2)))
6351, 62seq1i 13022 . . . 4 (𝜑 → (seq1( + , 𝐺)‘1) = (2 · (𝐹‘2)))
64 nnuz 11925 . . . . . 6 ℕ = (ℤ‘1)
65 df-2 11281 . . . . . 6 2 = (1 + 1)
66 eqidd 2772 . . . . . . 7 (𝜑 → (𝐹‘1) = (𝐹‘1))
6751, 66seq1i 13022 . . . . . 6 (𝜑 → (seq1( + , 𝐹)‘1) = (𝐹‘1))
68 eqidd 2772 . . . . . 6 (𝜑 → (𝐹‘2) = (𝐹‘2))
6964, 33, 65, 67, 68seqp1i 13024 . . . . 5 (𝜑 → (seq1( + , 𝐹)‘2) = ((𝐹‘1) + (𝐹‘2)))
7069oveq2d 6809 . . . 4 (𝜑 → (2 · (seq1( + , 𝐹)‘2)) = (2 · ((𝐹‘1) + (𝐹‘2))))
7150, 63, 703brtr4d 4818 . . 3 (𝜑 → (seq1( + , 𝐺)‘1) ≤ (2 · (seq1( + , 𝐹)‘2)))
72 fveq2 6332 . . . . . . . . . . 11 (𝑛 = (𝑗 + 1) → (𝐺𝑛) = (𝐺‘(𝑗 + 1)))
73 oveq2 6801 . . . . . . . . . . . 12 (𝑛 = (𝑗 + 1) → (2↑𝑛) = (2↑(𝑗 + 1)))
7473fveq2d 6336 . . . . . . . . . . . 12 (𝑛 = (𝑗 + 1) → (𝐹‘(2↑𝑛)) = (𝐹‘(2↑(𝑗 + 1))))
7573, 74oveq12d 6811 . . . . . . . . . . 11 (𝑛 = (𝑗 + 1) → ((2↑𝑛) · (𝐹‘(2↑𝑛))) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))))
7672, 75eqeq12d 2786 . . . . . . . . . 10 (𝑛 = (𝑗 + 1) → ((𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))) ↔ (𝐺‘(𝑗 + 1)) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1))))))
7759adantr 466 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ∀𝑛 ∈ ℕ0 (𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))
78 peano2nn 11234 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
7978adantl 467 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ)
8079nnnn0d 11553 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ0)
8176, 77, 80rspcdva 3466 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐺‘(𝑗 + 1)) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))))
82 nnnn0 11501 . . . . . . . . . . . . 13 (𝑗 ∈ ℕ → 𝑗 ∈ ℕ0)
8382adantl 467 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ0)
84 expp1 13074 . . . . . . . . . . . 12 ((2 ∈ ℂ ∧ 𝑗 ∈ ℕ0) → (2↑(𝑗 + 1)) = ((2↑𝑗) · 2))
853, 83, 84sylancr 575 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) = ((2↑𝑗) · 2))
86 nnexpcl 13080 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑗 ∈ ℕ0) → (2↑𝑗) ∈ ℕ)
8740, 82, 86sylancr 575 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ → (2↑𝑗) ∈ ℕ)
8887adantl 467 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℕ)
8988nncnd 11238 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℂ)
90 mulcom 10224 . . . . . . . . . . . 12 (((2↑𝑗) ∈ ℂ ∧ 2 ∈ ℂ) → ((2↑𝑗) · 2) = (2 · (2↑𝑗)))
9189, 3, 90sylancl 574 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · 2) = (2 · (2↑𝑗)))
9285, 91eqtrd 2805 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) = (2 · (2↑𝑗)))
9392oveq1d 6808 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))) = ((2 · (2↑𝑗)) · (𝐹‘(2↑(𝑗 + 1)))))
943a1i 11 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → 2 ∈ ℂ)
95 fveq2 6332 . . . . . . . . . . . . 13 (𝑘 = (2↑(𝑗 + 1)) → (𝐹𝑘) = (𝐹‘(2↑(𝑗 + 1))))
9695eleq1d 2835 . . . . . . . . . . . 12 (𝑘 = (2↑(𝑗 + 1)) → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ))
9739adantr 466 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ∀𝑘 ∈ ℕ (𝐹𝑘) ∈ ℝ)
98 nnexpcl 13080 . . . . . . . . . . . . 13 ((2 ∈ ℕ ∧ (𝑗 + 1) ∈ ℕ0) → (2↑(𝑗 + 1)) ∈ ℕ)
9940, 80, 98sylancr 575 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ ℕ)
10096, 97, 99rspcdva 3466 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ)
101100recnd 10270 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℂ)
10294, 89, 101mulassd 10265 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ((2 · (2↑𝑗)) · (𝐹‘(2↑(𝑗 + 1)))) = (2 · ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1))))))
10381, 93, 1023eqtrd 2809 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐺‘(𝑗 + 1)) = (2 · ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1))))))
10488nnnn0d 11553 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℕ0)
105 hashfz1 13338 . . . . . . . . . . . . . . 15 ((2↑𝑗) ∈ ℕ0 → (♯‘(1...(2↑𝑗))) = (2↑𝑗))
106104, 105syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (♯‘(1...(2↑𝑗))) = (2↑𝑗))
107106, 89eqeltrd 2850 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (♯‘(1...(2↑𝑗))) ∈ ℂ)
108 fzfid 12980 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (((2↑𝑗) + 1)...(2↑(𝑗 + 1))) ∈ Fin)
109 hashcl 13349 . . . . . . . . . . . . . . 15 ((((2↑𝑗) + 1)...(2↑(𝑗 + 1))) ∈ Fin → (♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∈ ℕ0)
110108, 109syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∈ ℕ0)
111110nn0cnd 11555 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∈ ℂ)
112 simpr 471 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
113112nnzd 11683 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℤ)
114 uzid 11903 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ𝑗))
115 peano2uz 11943 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (ℤ𝑗) → (𝑗 + 1) ∈ (ℤ𝑗))
116 2re 11292 . . . . . . . . . . . . . . . . . . 19 2 ∈ ℝ
117 1le2 11443 . . . . . . . . . . . . . . . . . . 19 1 ≤ 2
118 leexp2a 13123 . . . . . . . . . . . . . . . . . . 19 ((2 ∈ ℝ ∧ 1 ≤ 2 ∧ (𝑗 + 1) ∈ (ℤ𝑗)) → (2↑𝑗) ≤ (2↑(𝑗 + 1)))
119116, 117, 118mp3an12 1562 . . . . . . . . . . . . . . . . . 18 ((𝑗 + 1) ∈ (ℤ𝑗) → (2↑𝑗) ≤ (2↑(𝑗 + 1)))
120113, 114, 115, 1194syl 19 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ≤ (2↑(𝑗 + 1)))
12188, 64syl6eleq 2860 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ (ℤ‘1))
12299nnzd 11683 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ ℤ)
123 elfz5 12541 . . . . . . . . . . . . . . . . . 18 (((2↑𝑗) ∈ (ℤ‘1) ∧ (2↑(𝑗 + 1)) ∈ ℤ) → ((2↑𝑗) ∈ (1...(2↑(𝑗 + 1))) ↔ (2↑𝑗) ≤ (2↑(𝑗 + 1))))
124121, 122, 123syl2anc 573 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) ∈ (1...(2↑(𝑗 + 1))) ↔ (2↑𝑗) ≤ (2↑(𝑗 + 1))))
125120, 124mpbird 247 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ (1...(2↑(𝑗 + 1))))
126 fzsplit 12574 . . . . . . . . . . . . . . . 16 ((2↑𝑗) ∈ (1...(2↑(𝑗 + 1))) → (1...(2↑(𝑗 + 1))) = ((1...(2↑𝑗)) ∪ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))))
127125, 126syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (1...(2↑(𝑗 + 1))) = ((1...(2↑𝑗)) ∪ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))))
128127fveq2d 6336 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (♯‘(1...(2↑(𝑗 + 1)))) = (♯‘((1...(2↑𝑗)) ∪ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))))))
12989times2d 11478 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · 2) = ((2↑𝑗) + (2↑𝑗)))
13085, 129eqtrd 2805 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) = ((2↑𝑗) + (2↑𝑗)))
13199nnnn0d 11553 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ ℕ0)
132 hashfz1 13338 . . . . . . . . . . . . . . . 16 ((2↑(𝑗 + 1)) ∈ ℕ0 → (♯‘(1...(2↑(𝑗 + 1)))) = (2↑(𝑗 + 1)))
133131, 132syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (♯‘(1...(2↑(𝑗 + 1)))) = (2↑(𝑗 + 1)))
134106oveq1d 6808 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → ((♯‘(1...(2↑𝑗))) + (2↑𝑗)) = ((2↑𝑗) + (2↑𝑗)))
135130, 133, 1343eqtr4d 2815 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (♯‘(1...(2↑(𝑗 + 1)))) = ((♯‘(1...(2↑𝑗))) + (2↑𝑗)))
136 fzfid 12980 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (1...(2↑𝑗)) ∈ Fin)
13788nnred 11237 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℝ)
138137ltp1d 11156 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) < ((2↑𝑗) + 1))
139 fzdisj 12575 . . . . . . . . . . . . . . . 16 ((2↑𝑗) < ((2↑𝑗) + 1) → ((1...(2↑𝑗)) ∩ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) = ∅)
140138, 139syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → ((1...(2↑𝑗)) ∩ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) = ∅)
141 hashun 13373 . . . . . . . . . . . . . . 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))))))
142136, 108, 140, 141syl3anc 1476 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (♯‘((1...(2↑𝑗)) ∪ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))))) = ((♯‘(1...(2↑𝑗))) + (♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1))))))
143128, 135, 1423eqtr3d 2813 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ((♯‘(1...(2↑𝑗))) + (2↑𝑗)) = ((♯‘(1...(2↑𝑗))) + (♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1))))))
144107, 89, 111, 143addcanad 10443 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) = (♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))))
145144oveq1d 6808 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) = ((♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) · (𝐹‘(2↑(𝑗 + 1)))))
146 fsumconst 14729 . . . . . . . . . . . 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)))))
147108, 101, 146syl2anc 573 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘(2↑(𝑗 + 1))) = ((♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) · (𝐹‘(2↑(𝑗 + 1)))))
148145, 147eqtr4d 2808 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) = Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘(2↑(𝑗 + 1))))
149100adantr 466 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ)
150 simpl 468 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → 𝜑)
151 peano2nn 11234 . . . . . . . . . . . . . 14 ((2↑𝑗) ∈ ℕ → ((2↑𝑗) + 1) ∈ ℕ)
15288, 151syl 17 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) + 1) ∈ ℕ)
153 elfzuz 12545 . . . . . . . . . . . . 13 (𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))) → 𝑘 ∈ (ℤ‘((2↑𝑗) + 1)))
154 eluznn 11961 . . . . . . . . . . . . 13 ((((2↑𝑗) + 1) ∈ ℕ ∧ 𝑘 ∈ (ℤ‘((2↑𝑗) + 1))) → 𝑘 ∈ ℕ)
155152, 153, 154syl2an 583 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → 𝑘 ∈ ℕ)
156150, 155, 38syl2an2r 664 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (𝐹𝑘) ∈ ℝ)
157 elfzuz3 12546 . . . . . . . . . . . . . . 15 (𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))) → (2↑(𝑗 + 1)) ∈ (ℤ𝑛))
158157adantl 467 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (2↑(𝑗 + 1)) ∈ (ℤ𝑛))
159 simplll 758 . . . . . . . . . . . . . . 15 ((((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...(2↑(𝑗 + 1)))) → 𝜑)
160 elfzuz 12545 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))) → 𝑛 ∈ (ℤ‘((2↑𝑗) + 1)))
161 eluznn 11961 . . . . . . . . . . . . . . . . 17 ((((2↑𝑗) + 1) ∈ ℕ ∧ 𝑛 ∈ (ℤ‘((2↑𝑗) + 1))) → 𝑛 ∈ ℕ)
162152, 160, 161syl2an 583 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → 𝑛 ∈ ℕ)
163 elfzuz 12545 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (𝑛...(2↑(𝑗 + 1))) → 𝑘 ∈ (ℤ𝑛))
164 eluznn 11961 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑘 ∈ ℕ)
165162, 163, 164syl2an 583 . . . . . . . . . . . . . . 15 ((((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...(2↑(𝑗 + 1)))) → 𝑘 ∈ ℕ)
166159, 165, 38syl2anc 573 . . . . . . . . . . . . . 14 ((((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...(2↑(𝑗 + 1)))) → (𝐹𝑘) ∈ ℝ)
167 simplll 758 . . . . . . . . . . . . . . 15 ((((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...((2↑(𝑗 + 1)) − 1))) → 𝜑)
168 elfzuz 12545 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (𝑛...((2↑(𝑗 + 1)) − 1)) → 𝑘 ∈ (ℤ𝑛))
169162, 168, 164syl2an 583 . . . . . . . . . . . . . . 15 ((((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...((2↑(𝑗 + 1)) − 1))) → 𝑘 ∈ ℕ)
170 climcnds.3 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
171167, 169, 170syl2anc 573 . . . . . . . . . . . . . 14 ((((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...((2↑(𝑗 + 1)) − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
172158, 166, 171monoord2 13039 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑛))
173172ralrimiva 3115 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ∀𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑛))
174 fveq2 6332 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
175174breq2d 4798 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → ((𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑛) ↔ (𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑘)))
176175rspccva 3459 . . . . . . . . . . . 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))
177173, 176sylan 569 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑘))
178108, 149, 156, 177fsumle 14738 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘(2↑(𝑗 + 1))) ≤ Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘))
179148, 178eqbrtrd 4808 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) ≤ Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘))
180137, 100remulcld 10272 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) ∈ ℝ)
181108, 156fsumrecl 14673 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘) ∈ ℝ)
182 2rp 12040 . . . . . . . . . . 11 2 ∈ ℝ+
183182a1i 11 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → 2 ∈ ℝ+)
184180, 181, 183lemul2d 12119 . . . . . . . . 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))))
185179, 184mpbid 222 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (2 · ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1))))) ≤ (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)))
186103, 185eqbrtrd 4808 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (𝐺‘(𝑗 + 1)) ≤ (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)))
187 1zzd 11610 . . . . . . . . . 10 (𝜑 → 1 ∈ ℤ)
188 nnnn0 11501 . . . . . . . . . . 11 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
189 simpr 471 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
190 nnexpcl 13080 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
19140, 189, 190sylancr 575 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
192191nnred 11237 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℝ)
193 fveq2 6332 . . . . . . . . . . . . . . 15 (𝑘 = (2↑𝑛) → (𝐹𝑘) = (𝐹‘(2↑𝑛)))
194193eleq1d 2835 . . . . . . . . . . . . . 14 (𝑘 = (2↑𝑛) → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘(2↑𝑛)) ∈ ℝ))
19539adantr 466 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ0) → ∀𝑘 ∈ ℕ (𝐹𝑘) ∈ ℝ)
196194, 195, 191rspcdva 3466 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ0) → (𝐹‘(2↑𝑛)) ∈ ℝ)
197192, 196remulcld 10272 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ0) → ((2↑𝑛) · (𝐹‘(2↑𝑛))) ∈ ℝ)
19858, 197eqeltrd 2850 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → (𝐺𝑛) ∈ ℝ)
199188, 198sylan2 580 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ ℝ)
20064, 187, 199serfre 13037 . . . . . . . . 9 (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
201200ffvelrnda 6502 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (seq1( + , 𝐺)‘𝑗) ∈ ℝ)
20272eleq1d 2835 . . . . . . . . 9 (𝑛 = (𝑗 + 1) → ((𝐺𝑛) ∈ ℝ ↔ (𝐺‘(𝑗 + 1)) ∈ ℝ))
203199ralrimiva 3115 . . . . . . . . . 10 (𝜑 → ∀𝑛 ∈ ℕ (𝐺𝑛) ∈ ℝ)
204203adantr 466 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ∀𝑛 ∈ ℕ (𝐺𝑛) ∈ ℝ)
205202, 204, 79rspcdva 3466 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐺‘(𝑗 + 1)) ∈ ℝ)
20664, 187, 38serfre 13037 . . . . . . . . . 10 (𝜑 → seq1( + , 𝐹):ℕ⟶ℝ)
207 ffvelrn 6500 . . . . . . . . . 10 ((seq1( + , 𝐹):ℕ⟶ℝ ∧ (2↑𝑗) ∈ ℕ) → (seq1( + , 𝐹)‘(2↑𝑗)) ∈ ℝ)
208206, 87, 207syl2an 583 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘(2↑𝑗)) ∈ ℝ)
209 remulcl 10223 . . . . . . . . 9 ((2 ∈ ℝ ∧ (seq1( + , 𝐹)‘(2↑𝑗)) ∈ ℝ) → (2 · (seq1( + , 𝐹)‘(2↑𝑗))) ∈ ℝ)
210116, 208, 209sylancr 575 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (2 · (seq1( + , 𝐹)‘(2↑𝑗))) ∈ ℝ)
211 remulcl 10223 . . . . . . . . 9 ((2 ∈ ℝ ∧ Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘) ∈ ℝ) → (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)) ∈ ℝ)
212116, 181, 211sylancr 575 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)) ∈ ℝ)
213 le2add 10712 . . . . . . . 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)))))
214201, 205, 210, 212, 213syl22anc 1477 . . . . . . 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)))))
215186, 214mpan2d 674 . . . . . 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)))))
216112, 64syl6eleq 2860 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ‘1))
217 seqp1 13023 . . . . . . . 8 (𝑗 ∈ (ℤ‘1) → (seq1( + , 𝐺)‘(𝑗 + 1)) = ((seq1( + , 𝐺)‘𝑗) + (𝐺‘(𝑗 + 1))))
218216, 217syl 17 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (seq1( + , 𝐺)‘(𝑗 + 1)) = ((seq1( + , 𝐺)‘𝑗) + (𝐺‘(𝑗 + 1))))
219 fzfid 12980 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (1...(2↑(𝑗 + 1))) ∈ Fin)
220 elfznn 12577 . . . . . . . . . . . 12 (𝑘 ∈ (1...(2↑(𝑗 + 1))) → 𝑘 ∈ ℕ)
22138recnd 10270 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℂ)
222150, 220, 221syl2an 583 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑(𝑗 + 1)))) → (𝐹𝑘) ∈ ℂ)
223140, 127, 219, 222fsumsplit 14679 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...(2↑(𝑗 + 1)))(𝐹𝑘) = (Σ𝑘 ∈ (1...(2↑𝑗))(𝐹𝑘) + Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)))
224 eqidd 2772 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑(𝑗 + 1)))) → (𝐹𝑘) = (𝐹𝑘))
22599, 64syl6eleq 2860 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ (ℤ‘1))
226224, 225, 222fsumser 14669 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...(2↑(𝑗 + 1)))(𝐹𝑘) = (seq1( + , 𝐹)‘(2↑(𝑗 + 1))))
227 eqidd 2772 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑𝑗))) → (𝐹𝑘) = (𝐹𝑘))
228 elfznn 12577 . . . . . . . . . . . . 13 (𝑘 ∈ (1...(2↑𝑗)) → 𝑘 ∈ ℕ)
229150, 228, 221syl2an 583 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑𝑗))) → (𝐹𝑘) ∈ ℂ)
230227, 121, 229fsumser 14669 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...(2↑𝑗))(𝐹𝑘) = (seq1( + , 𝐹)‘(2↑𝑗)))
231230oveq1d 6808 . . . . . . . . . 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)))
232223, 226, 2313eqtr3d 2813 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘(2↑(𝑗 + 1))) = ((seq1( + , 𝐹)‘(2↑𝑗)) + Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)))
233232oveq2d 6809 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1)))) = (2 · ((seq1( + , 𝐹)‘(2↑𝑗)) + Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘))))
234208recnd 10270 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘(2↑𝑗)) ∈ ℂ)
235181recnd 10270 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘) ∈ ℂ)
23694, 234, 235adddid 10266 . . . . . . . 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))))
237233, 236eqtrd 2805 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1)))) = ((2 · (seq1( + , 𝐹)‘(2↑𝑗))) + (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘))))
238218, 237breq12d 4799 . . . . . 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)))))
239215, 238sylibrd 249 . . . . 5 ((𝜑𝑗 ∈ ℕ) → ((seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗))) → (seq1( + , 𝐺)‘(𝑗 + 1)) ≤ (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1))))))
240239expcom 398 . . . 4 (𝑗 ∈ ℕ → (𝜑 → ((seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗))) → (seq1( + , 𝐺)‘(𝑗 + 1)) ≤ (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1)))))))
241240a2d 29 . . 3 (𝑗 ∈ ℕ → ((𝜑 → (seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗)))) → (𝜑 → (seq1( + , 𝐺)‘(𝑗 + 1)) ≤ (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1)))))))
24210, 16, 22, 28, 71, 241nnind 11240 . 2 (𝑁 ∈ ℕ → (𝜑 → (seq1( + , 𝐺)‘𝑁) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑁)))))
243242impcom 394 1 ((𝜑𝑁 ∈ ℕ) → (seq1( + , 𝐺)‘𝑁) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑁))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  cun 3721  cin 3722  c0 4063   class class class wbr 4786  wf 6027  cfv 6031  (class class class)co 6793  Fincfn 8109  cc 10136  cr 10137  0cc0 10138  1c1 10139   + caddc 10141   · cmul 10143   < clt 10276  cle 10277  cmin 10468  cn 11222  2c2 11272  0cn0 11494  cz 11579  cuz 11888  +crp 12035  ...cfz 12533  seqcseq 13008  cexp 13067  chash 13321  Σcsu 14624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-sup 8504  df-oi 8571  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-n0 11495  df-z 11580  df-uz 11889  df-rp 12036  df-ico 12386  df-fz 12534  df-fzo 12674  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-clim 14427  df-sum 14625
This theorem is referenced by:  climcnds  14790
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