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Theorem climcndslem2 15882
Description: Lemma for climcnds 15883: 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 6906 . . . . 5 (𝑥 = 1 → (seq1( + , 𝐺)‘𝑥) = (seq1( + , 𝐺)‘1))
2 oveq2 7438 . . . . . . . 8 (𝑥 = 1 → (2↑𝑥) = (2↑1))
3 2cn 12338 . . . . . . . . 9 2 ∈ ℂ
4 exp1 14104 . . . . . . . . 9 (2 ∈ ℂ → (2↑1) = 2)
53, 4ax-mp 5 . . . . . . . 8 (2↑1) = 2
62, 5eqtrdi 2790 . . . . . . 7 (𝑥 = 1 → (2↑𝑥) = 2)
76fveq2d 6910 . . . . . 6 (𝑥 = 1 → (seq1( + , 𝐹)‘(2↑𝑥)) = (seq1( + , 𝐹)‘2))
87oveq2d 7446 . . . . 5 (𝑥 = 1 → (2 · (seq1( + , 𝐹)‘(2↑𝑥))) = (2 · (seq1( + , 𝐹)‘2)))
91, 8breq12d 5160 . . . 4 (𝑥 = 1 → ((seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥))) ↔ (seq1( + , 𝐺)‘1) ≤ (2 · (seq1( + , 𝐹)‘2))))
109imbi2d 340 . . 3 (𝑥 = 1 → ((𝜑 → (seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥)))) ↔ (𝜑 → (seq1( + , 𝐺)‘1) ≤ (2 · (seq1( + , 𝐹)‘2)))))
11 fveq2 6906 . . . . 5 (𝑥 = 𝑗 → (seq1( + , 𝐺)‘𝑥) = (seq1( + , 𝐺)‘𝑗))
12 oveq2 7438 . . . . . . 7 (𝑥 = 𝑗 → (2↑𝑥) = (2↑𝑗))
1312fveq2d 6910 . . . . . 6 (𝑥 = 𝑗 → (seq1( + , 𝐹)‘(2↑𝑥)) = (seq1( + , 𝐹)‘(2↑𝑗)))
1413oveq2d 7446 . . . . 5 (𝑥 = 𝑗 → (2 · (seq1( + , 𝐹)‘(2↑𝑥))) = (2 · (seq1( + , 𝐹)‘(2↑𝑗))))
1511, 14breq12d 5160 . . . 4 (𝑥 = 𝑗 → ((seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥))) ↔ (seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗)))))
1615imbi2d 340 . . 3 (𝑥 = 𝑗 → ((𝜑 → (seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥)))) ↔ (𝜑 → (seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗))))))
17 fveq2 6906 . . . . 5 (𝑥 = (𝑗 + 1) → (seq1( + , 𝐺)‘𝑥) = (seq1( + , 𝐺)‘(𝑗 + 1)))
18 oveq2 7438 . . . . . . 7 (𝑥 = (𝑗 + 1) → (2↑𝑥) = (2↑(𝑗 + 1)))
1918fveq2d 6910 . . . . . 6 (𝑥 = (𝑗 + 1) → (seq1( + , 𝐹)‘(2↑𝑥)) = (seq1( + , 𝐹)‘(2↑(𝑗 + 1))))
2019oveq2d 7446 . . . . 5 (𝑥 = (𝑗 + 1) → (2 · (seq1( + , 𝐹)‘(2↑𝑥))) = (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1)))))
2117, 20breq12d 5160 . . . 4 (𝑥 = (𝑗 + 1) → ((seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥))) ↔ (seq1( + , 𝐺)‘(𝑗 + 1)) ≤ (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1))))))
2221imbi2d 340 . . 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 6906 . . . . 5 (𝑥 = 𝑁 → (seq1( + , 𝐺)‘𝑥) = (seq1( + , 𝐺)‘𝑁))
24 oveq2 7438 . . . . . . 7 (𝑥 = 𝑁 → (2↑𝑥) = (2↑𝑁))
2524fveq2d 6910 . . . . . 6 (𝑥 = 𝑁 → (seq1( + , 𝐹)‘(2↑𝑥)) = (seq1( + , 𝐹)‘(2↑𝑁)))
2625oveq2d 7446 . . . . 5 (𝑥 = 𝑁 → (2 · (seq1( + , 𝐹)‘(2↑𝑥))) = (2 · (seq1( + , 𝐹)‘(2↑𝑁))))
2723, 26breq12d 5160 . . . 4 (𝑥 = 𝑁 → ((seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥))) ↔ (seq1( + , 𝐺)‘𝑁) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑁)))))
2827imbi2d 340 . . 3 (𝑥 = 𝑁 → ((𝜑 → (seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥)))) ↔ (𝜑 → (seq1( + , 𝐺)‘𝑁) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑁))))))
29 fveq2 6906 . . . . . . . 8 (𝑘 = 1 → (𝐹𝑘) = (𝐹‘1))
3029breq2d 5159 . . . . . . 7 (𝑘 = 1 → (0 ≤ (𝐹𝑘) ↔ 0 ≤ (𝐹‘1)))
31 climcnds.2 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 0 ≤ (𝐹𝑘))
3231ralrimiva 3143 . . . . . . 7 (𝜑 → ∀𝑘 ∈ ℕ 0 ≤ (𝐹𝑘))
33 1nn 12274 . . . . . . . 8 1 ∈ ℕ
3433a1i 11 . . . . . . 7 (𝜑 → 1 ∈ ℕ)
3530, 32, 34rspcdva 3622 . . . . . 6 (𝜑 → 0 ≤ (𝐹‘1))
36 fveq2 6906 . . . . . . . . 9 (𝑘 = 2 → (𝐹𝑘) = (𝐹‘2))
3736eleq1d 2823 . . . . . . . 8 (𝑘 = 2 → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘2) ∈ ℝ))
38 climcnds.1 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℝ)
3938ralrimiva 3143 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ ℕ (𝐹𝑘) ∈ ℝ)
40 2nn 12336 . . . . . . . . 9 2 ∈ ℕ
4140a1i 11 . . . . . . . 8 (𝜑 → 2 ∈ ℕ)
4237, 39, 41rspcdva 3622 . . . . . . 7 (𝜑 → (𝐹‘2) ∈ ℝ)
4329eleq1d 2823 . . . . . . . 8 (𝑘 = 1 → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘1) ∈ ℝ))
4443, 39, 34rspcdva 3622 . . . . . . 7 (𝜑 → (𝐹‘1) ∈ ℝ)
4542, 44addge02d 11849 . . . . . 6 (𝜑 → (0 ≤ (𝐹‘1) ↔ (𝐹‘2) ≤ ((𝐹‘1) + (𝐹‘2))))
4635, 45mpbid 232 . . . . 5 (𝜑 → (𝐹‘2) ≤ ((𝐹‘1) + (𝐹‘2)))
4744, 42readdcld 11287 . . . . . 6 (𝜑 → ((𝐹‘1) + (𝐹‘2)) ∈ ℝ)
4841nnrpd 13072 . . . . . 6 (𝜑 → 2 ∈ ℝ+)
4942, 47, 48lemul2d 13118 . . . . 5 (𝜑 → ((𝐹‘2) ≤ ((𝐹‘1) + (𝐹‘2)) ↔ (2 · (𝐹‘2)) ≤ (2 · ((𝐹‘1) + (𝐹‘2)))))
5046, 49mpbid 232 . . . 4 (𝜑 → (2 · (𝐹‘2)) ≤ (2 · ((𝐹‘1) + (𝐹‘2))))
51 1z 12644 . . . . 5 1 ∈ ℤ
52 fveq2 6906 . . . . . . 7 (𝑛 = 1 → (𝐺𝑛) = (𝐺‘1))
53 oveq2 7438 . . . . . . . . 9 (𝑛 = 1 → (2↑𝑛) = (2↑1))
5453, 5eqtrdi 2790 . . . . . . . 8 (𝑛 = 1 → (2↑𝑛) = 2)
5554fveq2d 6910 . . . . . . . 8 (𝑛 = 1 → (𝐹‘(2↑𝑛)) = (𝐹‘2))
5654, 55oveq12d 7448 . . . . . . 7 (𝑛 = 1 → ((2↑𝑛) · (𝐹‘(2↑𝑛))) = (2 · (𝐹‘2)))
5752, 56eqeq12d 2750 . . . . . 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 3143 . . . . . 6 (𝜑 → ∀𝑛 ∈ ℕ0 (𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))
60 1nn0 12539 . . . . . . 7 1 ∈ ℕ0
6160a1i 11 . . . . . 6 (𝜑 → 1 ∈ ℕ0)
6257, 59, 61rspcdva 3622 . . . . 5 (𝜑 → (𝐺‘1) = (2 · (𝐹‘2)))
6351, 62seq1i 14052 . . . 4 (𝜑 → (seq1( + , 𝐺)‘1) = (2 · (𝐹‘2)))
64 nnuz 12918 . . . . . 6 ℕ = (ℤ‘1)
65 df-2 12326 . . . . . 6 2 = (1 + 1)
66 eqidd 2735 . . . . . . 7 (𝜑 → (𝐹‘1) = (𝐹‘1))
6751, 66seq1i 14052 . . . . . 6 (𝜑 → (seq1( + , 𝐹)‘1) = (𝐹‘1))
68 eqidd 2735 . . . . . 6 (𝜑 → (𝐹‘2) = (𝐹‘2))
6964, 34, 65, 67, 68seqp1d 14055 . . . . 5 (𝜑 → (seq1( + , 𝐹)‘2) = ((𝐹‘1) + (𝐹‘2)))
7069oveq2d 7446 . . . 4 (𝜑 → (2 · (seq1( + , 𝐹)‘2)) = (2 · ((𝐹‘1) + (𝐹‘2))))
7150, 63, 703brtr4d 5179 . . 3 (𝜑 → (seq1( + , 𝐺)‘1) ≤ (2 · (seq1( + , 𝐹)‘2)))
72 fveq2 6906 . . . . . . . . . . 11 (𝑛 = (𝑗 + 1) → (𝐺𝑛) = (𝐺‘(𝑗 + 1)))
73 oveq2 7438 . . . . . . . . . . . 12 (𝑛 = (𝑗 + 1) → (2↑𝑛) = (2↑(𝑗 + 1)))
7473fveq2d 6910 . . . . . . . . . . . 12 (𝑛 = (𝑗 + 1) → (𝐹‘(2↑𝑛)) = (𝐹‘(2↑(𝑗 + 1))))
7573, 74oveq12d 7448 . . . . . . . . . . 11 (𝑛 = (𝑗 + 1) → ((2↑𝑛) · (𝐹‘(2↑𝑛))) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))))
7672, 75eqeq12d 2750 . . . . . . . . . 10 (𝑛 = (𝑗 + 1) → ((𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))) ↔ (𝐺‘(𝑗 + 1)) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1))))))
7759adantr 480 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ∀𝑛 ∈ ℕ0 (𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))
78 peano2nn 12275 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
7978adantl 481 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ)
8079nnnn0d 12584 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ0)
8176, 77, 80rspcdva 3622 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐺‘(𝑗 + 1)) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))))
82 nnnn0 12530 . . . . . . . . . . . . 13 (𝑗 ∈ ℕ → 𝑗 ∈ ℕ0)
8382adantl 481 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ0)
84 expp1 14105 . . . . . . . . . . . 12 ((2 ∈ ℂ ∧ 𝑗 ∈ ℕ0) → (2↑(𝑗 + 1)) = ((2↑𝑗) · 2))
853, 83, 84sylancr 587 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) = ((2↑𝑗) · 2))
86 nnexpcl 14111 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑗 ∈ ℕ0) → (2↑𝑗) ∈ ℕ)
8740, 82, 86sylancr 587 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ → (2↑𝑗) ∈ ℕ)
8887adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℕ)
8988nncnd 12279 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℂ)
90 mulcom 11238 . . . . . . . . . . . 12 (((2↑𝑗) ∈ ℂ ∧ 2 ∈ ℂ) → ((2↑𝑗) · 2) = (2 · (2↑𝑗)))
9189, 3, 90sylancl 586 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · 2) = (2 · (2↑𝑗)))
9285, 91eqtrd 2774 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) = (2 · (2↑𝑗)))
9392oveq1d 7445 . . . . . . . . 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 6906 . . . . . . . . . . . . 13 (𝑘 = (2↑(𝑗 + 1)) → (𝐹𝑘) = (𝐹‘(2↑(𝑗 + 1))))
9695eleq1d 2823 . . . . . . . . . . . 12 (𝑘 = (2↑(𝑗 + 1)) → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ))
9739adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ∀𝑘 ∈ ℕ (𝐹𝑘) ∈ ℝ)
98 nnexpcl 14111 . . . . . . . . . . . . 13 ((2 ∈ ℕ ∧ (𝑗 + 1) ∈ ℕ0) → (2↑(𝑗 + 1)) ∈ ℕ)
9940, 80, 98sylancr 587 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ ℕ)
10096, 97, 99rspcdva 3622 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ)
101100recnd 11286 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℂ)
10294, 89, 101mulassd 11281 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ((2 · (2↑𝑗)) · (𝐹‘(2↑(𝑗 + 1)))) = (2 · ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1))))))
10381, 93, 1023eqtrd 2778 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐺‘(𝑗 + 1)) = (2 · ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1))))))
10488nnnn0d 12584 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℕ0)
105 hashfz1 14381 . . . . . . . . . . . . . . 15 ((2↑𝑗) ∈ ℕ0 → (♯‘(1...(2↑𝑗))) = (2↑𝑗))
106104, 105syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (♯‘(1...(2↑𝑗))) = (2↑𝑗))
107106, 89eqeltrd 2838 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (♯‘(1...(2↑𝑗))) ∈ ℂ)
108 fzfid 14010 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (((2↑𝑗) + 1)...(2↑(𝑗 + 1))) ∈ Fin)
109 hashcl 14391 . . . . . . . . . . . . . . 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 12586 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∈ ℂ)
112 simpr 484 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
113112nnzd 12637 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℤ)
114 uzid 12890 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ𝑗))
115 peano2uz 12940 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (ℤ𝑗) → (𝑗 + 1) ∈ (ℤ𝑗))
116 2re 12337 . . . . . . . . . . . . . . . . . . 19 2 ∈ ℝ
117 1le2 12472 . . . . . . . . . . . . . . . . . . 19 1 ≤ 2
118 leexp2a 14208 . . . . . . . . . . . . . . . . . . 19 ((2 ∈ ℝ ∧ 1 ≤ 2 ∧ (𝑗 + 1) ∈ (ℤ𝑗)) → (2↑𝑗) ≤ (2↑(𝑗 + 1)))
119116, 117, 118mp3an12 1450 . . . . . . . . . . . . . . . . . 18 ((𝑗 + 1) ∈ (ℤ𝑗) → (2↑𝑗) ≤ (2↑(𝑗 + 1)))
120113, 114, 115, 1194syl 19 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ≤ (2↑(𝑗 + 1)))
12188, 64eleqtrdi 2848 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ (ℤ‘1))
12299nnzd 12637 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ ℤ)
123 elfz5 13552 . . . . . . . . . . . . . . . . . 18 (((2↑𝑗) ∈ (ℤ‘1) ∧ (2↑(𝑗 + 1)) ∈ ℤ) → ((2↑𝑗) ∈ (1...(2↑(𝑗 + 1))) ↔ (2↑𝑗) ≤ (2↑(𝑗 + 1))))
124121, 122, 123syl2anc 584 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) ∈ (1...(2↑(𝑗 + 1))) ↔ (2↑𝑗) ≤ (2↑(𝑗 + 1))))
125120, 124mpbird 257 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ (1...(2↑(𝑗 + 1))))
126 fzsplit 13586 . . . . . . . . . . . . . . . 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 6910 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (♯‘(1...(2↑(𝑗 + 1)))) = (♯‘((1...(2↑𝑗)) ∪ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))))))
12989times2d 12507 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · 2) = ((2↑𝑗) + (2↑𝑗)))
13085, 129eqtrd 2774 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) = ((2↑𝑗) + (2↑𝑗)))
13199nnnn0d 12584 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ ℕ0)
132 hashfz1 14381 . . . . . . . . . . . . . . . 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 7445 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → ((♯‘(1...(2↑𝑗))) + (2↑𝑗)) = ((2↑𝑗) + (2↑𝑗)))
135130, 133, 1343eqtr4d 2784 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (♯‘(1...(2↑(𝑗 + 1)))) = ((♯‘(1...(2↑𝑗))) + (2↑𝑗)))
136 fzfid 14010 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (1...(2↑𝑗)) ∈ Fin)
13788nnred 12278 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℝ)
138137ltp1d 12195 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) < ((2↑𝑗) + 1))
139 fzdisj 13587 . . . . . . . . . . . . . . . 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 14417 . . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (♯‘((1...(2↑𝑗)) ∪ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))))) = ((♯‘(1...(2↑𝑗))) + (♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1))))))
143128, 135, 1423eqtr3d 2782 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ((♯‘(1...(2↑𝑗))) + (2↑𝑗)) = ((♯‘(1...(2↑𝑗))) + (♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1))))))
144107, 89, 111, 143addcanad 11463 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) = (♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))))
145144oveq1d 7445 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) = ((♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) · (𝐹‘(2↑(𝑗 + 1)))))
146 fsumconst 15822 . . . . . . . . . . . 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 584 . . . . . . . . . . 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 2777 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) = Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘(2↑(𝑗 + 1))))
149100adantr 480 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ)
150 simpl 482 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → 𝜑)
151 peano2nn 12275 . . . . . . . . . . . . . 14 ((2↑𝑗) ∈ ℕ → ((2↑𝑗) + 1) ∈ ℕ)
15288, 151syl 17 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) + 1) ∈ ℕ)
153 elfzuz 13556 . . . . . . . . . . . . 13 (𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))) → 𝑘 ∈ (ℤ‘((2↑𝑗) + 1)))
154 eluznn 12957 . . . . . . . . . . . . 13 ((((2↑𝑗) + 1) ∈ ℕ ∧ 𝑘 ∈ (ℤ‘((2↑𝑗) + 1))) → 𝑘 ∈ ℕ)
155152, 153, 154syl2an 596 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → 𝑘 ∈ ℕ)
156150, 155, 38syl2an2r 685 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (𝐹𝑘) ∈ ℝ)
157 elfzuz3 13557 . . . . . . . . . . . . . . 15 (𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))) → (2↑(𝑗 + 1)) ∈ (ℤ𝑛))
158157adantl 481 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (2↑(𝑗 + 1)) ∈ (ℤ𝑛))
159 simplll 775 . . . . . . . . . . . . . . 15 ((((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...(2↑(𝑗 + 1)))) → 𝜑)
160 elfzuz 13556 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))) → 𝑛 ∈ (ℤ‘((2↑𝑗) + 1)))
161 eluznn 12957 . . . . . . . . . . . . . . . . 17 ((((2↑𝑗) + 1) ∈ ℕ ∧ 𝑛 ∈ (ℤ‘((2↑𝑗) + 1))) → 𝑛 ∈ ℕ)
162152, 160, 161syl2an 596 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → 𝑛 ∈ ℕ)
163 elfzuz 13556 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (𝑛...(2↑(𝑗 + 1))) → 𝑘 ∈ (ℤ𝑛))
164 eluznn 12957 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑘 ∈ ℕ)
165162, 163, 164syl2an 596 . . . . . . . . . . . . . . 15 ((((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...(2↑(𝑗 + 1)))) → 𝑘 ∈ ℕ)
166159, 165, 38syl2anc 584 . . . . . . . . . . . . . 14 ((((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...(2↑(𝑗 + 1)))) → (𝐹𝑘) ∈ ℝ)
167 simplll 775 . . . . . . . . . . . . . . 15 ((((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...((2↑(𝑗 + 1)) − 1))) → 𝜑)
168 elfzuz 13556 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (𝑛...((2↑(𝑗 + 1)) − 1)) → 𝑘 ∈ (ℤ𝑛))
169162, 168, 164syl2an 596 . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . 14 ((((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...((2↑(𝑗 + 1)) − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
172158, 166, 171monoord2 14070 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑛))
173172ralrimiva 3143 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ∀𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑛))
174 fveq2 6906 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
175174breq2d 5159 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → ((𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑛) ↔ (𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑘)))
176175rspccva 3620 . . . . . . . . . . . 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 580 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑘))
178108, 149, 156, 177fsumle 15831 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘(2↑(𝑗 + 1))) ≤ Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘))
179148, 178eqbrtrd 5169 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) ≤ Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘))
180137, 100remulcld 11288 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) ∈ ℝ)
181108, 156fsumrecl 15766 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘) ∈ ℝ)
182 2rp 13036 . . . . . . . . . . 11 2 ∈ ℝ+
183182a1i 11 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → 2 ∈ ℝ+)
184180, 181, 183lemul2d 13118 . . . . . . . . 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 232 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (2 · ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1))))) ≤ (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)))
186103, 185eqbrtrd 5169 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (𝐺‘(𝑗 + 1)) ≤ (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)))
187 1zzd 12645 . . . . . . . . . 10 (𝜑 → 1 ∈ ℤ)
188 nnnn0 12530 . . . . . . . . . . 11 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
189 simpr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
190 nnexpcl 14111 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
19140, 189, 190sylancr 587 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
192191nnred 12278 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℝ)
193 fveq2 6906 . . . . . . . . . . . . . . 15 (𝑘 = (2↑𝑛) → (𝐹𝑘) = (𝐹‘(2↑𝑛)))
194193eleq1d 2823 . . . . . . . . . . . . . 14 (𝑘 = (2↑𝑛) → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘(2↑𝑛)) ∈ ℝ))
19539adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ0) → ∀𝑘 ∈ ℕ (𝐹𝑘) ∈ ℝ)
196194, 195, 191rspcdva 3622 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ0) → (𝐹‘(2↑𝑛)) ∈ ℝ)
197192, 196remulcld 11288 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ0) → ((2↑𝑛) · (𝐹‘(2↑𝑛))) ∈ ℝ)
19858, 197eqeltrd 2838 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → (𝐺𝑛) ∈ ℝ)
199188, 198sylan2 593 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ ℝ)
20064, 187, 199serfre 14068 . . . . . . . . 9 (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
201200ffvelcdmda 7103 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (seq1( + , 𝐺)‘𝑗) ∈ ℝ)
20272eleq1d 2823 . . . . . . . . 9 (𝑛 = (𝑗 + 1) → ((𝐺𝑛) ∈ ℝ ↔ (𝐺‘(𝑗 + 1)) ∈ ℝ))
203199ralrimiva 3143 . . . . . . . . . 10 (𝜑 → ∀𝑛 ∈ ℕ (𝐺𝑛) ∈ ℝ)
204203adantr 480 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ∀𝑛 ∈ ℕ (𝐺𝑛) ∈ ℝ)
205202, 204, 79rspcdva 3622 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐺‘(𝑗 + 1)) ∈ ℝ)
20664, 187, 38serfre 14068 . . . . . . . . . 10 (𝜑 → seq1( + , 𝐹):ℕ⟶ℝ)
207 ffvelcdm 7100 . . . . . . . . . 10 ((seq1( + , 𝐹):ℕ⟶ℝ ∧ (2↑𝑗) ∈ ℕ) → (seq1( + , 𝐹)‘(2↑𝑗)) ∈ ℝ)
208206, 87, 207syl2an 596 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘(2↑𝑗)) ∈ ℝ)
209 remulcl 11237 . . . . . . . . 9 ((2 ∈ ℝ ∧ (seq1( + , 𝐹)‘(2↑𝑗)) ∈ ℝ) → (2 · (seq1( + , 𝐹)‘(2↑𝑗))) ∈ ℝ)
210116, 208, 209sylancr 587 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (2 · (seq1( + , 𝐹)‘(2↑𝑗))) ∈ ℝ)
211 remulcl 11237 . . . . . . . . 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 587 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)) ∈ ℝ)
213 le2add 11742 . . . . . . . 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 839 . . . . . . 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 694 . . . . . 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, 64eleqtrdi 2848 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ‘1))
217 seqp1 14053 . . . . . . . 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 14010 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (1...(2↑(𝑗 + 1))) ∈ Fin)
220 elfznn 13589 . . . . . . . . . . . 12 (𝑘 ∈ (1...(2↑(𝑗 + 1))) → 𝑘 ∈ ℕ)
22138recnd 11286 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℂ)
222150, 220, 221syl2an 596 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑(𝑗 + 1)))) → (𝐹𝑘) ∈ ℂ)
223140, 127, 219, 222fsumsplit 15773 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...(2↑(𝑗 + 1)))(𝐹𝑘) = (Σ𝑘 ∈ (1...(2↑𝑗))(𝐹𝑘) + Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)))
224 eqidd 2735 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑(𝑗 + 1)))) → (𝐹𝑘) = (𝐹𝑘))
22599, 64eleqtrdi 2848 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ (ℤ‘1))
226224, 225, 222fsumser 15762 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...(2↑(𝑗 + 1)))(𝐹𝑘) = (seq1( + , 𝐹)‘(2↑(𝑗 + 1))))
227 eqidd 2735 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑𝑗))) → (𝐹𝑘) = (𝐹𝑘))
228 elfznn 13589 . . . . . . . . . . . . 13 (𝑘 ∈ (1...(2↑𝑗)) → 𝑘 ∈ ℕ)
229150, 228, 221syl2an 596 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑𝑗))) → (𝐹𝑘) ∈ ℂ)
230227, 121, 229fsumser 15762 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...(2↑𝑗))(𝐹𝑘) = (seq1( + , 𝐹)‘(2↑𝑗)))
231230oveq1d 7445 . . . . . . . . . 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 2782 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘(2↑(𝑗 + 1))) = ((seq1( + , 𝐹)‘(2↑𝑗)) + Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)))
233232oveq2d 7446 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1)))) = (2 · ((seq1( + , 𝐹)‘(2↑𝑗)) + Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘))))
234208recnd 11286 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘(2↑𝑗)) ∈ ℂ)
235181recnd 11286 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘) ∈ ℂ)
23694, 234, 235adddid 11282 . . . . . . . 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 2774 . . . . . . 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 5160 . . . . . 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 259 . . . . 5 ((𝜑𝑗 ∈ ℕ) → ((seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗))) → (seq1( + , 𝐺)‘(𝑗 + 1)) ≤ (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1))))))
240239expcom 413 . . . 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 12281 . 2 (𝑁 ∈ ℕ → (𝜑 → (seq1( + , 𝐺)‘𝑁) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑁)))))
243242impcom 407 1 ((𝜑𝑁 ∈ ℕ) → (seq1( + , 𝐺)‘𝑁) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑁))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058  cun 3960  cin 3961  c0 4338   class class class wbr 5147  wf 6558  cfv 6562  (class class class)co 7430  Fincfn 8983  cc 11150  cr 11151  0cc0 11152  1c1 11153   + caddc 11155   · cmul 11157   < clt 11292  cle 11293  cmin 11489  cn 12263  2c2 12318  0cn0 12523  cz 12610  cuz 12875  +crp 13031  ...cfz 13543  seqcseq 14038  cexp 14098  chash 14365  Σcsu 15718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-oi 9547  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-rp 13032  df-ico 13389  df-fz 13544  df-fzo 13691  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-sum 15719
This theorem is referenced by:  climcnds  15883
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