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Theorem climcndslem2 15886
Description: Lemma for climcnds 15887: 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 7439 . . . . . . . 8 (𝑥 = 1 → (2↑𝑥) = (2↑1))
3 2cn 12341 . . . . . . . . 9 2 ∈ ℂ
4 exp1 14108 . . . . . . . . 9 (2 ∈ ℂ → (2↑1) = 2)
53, 4ax-mp 5 . . . . . . . 8 (2↑1) = 2
62, 5eqtrdi 2793 . . . . . . 7 (𝑥 = 1 → (2↑𝑥) = 2)
76fveq2d 6910 . . . . . 6 (𝑥 = 1 → (seq1( + , 𝐹)‘(2↑𝑥)) = (seq1( + , 𝐹)‘2))
87oveq2d 7447 . . . . 5 (𝑥 = 1 → (2 · (seq1( + , 𝐹)‘(2↑𝑥))) = (2 · (seq1( + , 𝐹)‘2)))
91, 8breq12d 5156 . . . 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 7439 . . . . . . 7 (𝑥 = 𝑗 → (2↑𝑥) = (2↑𝑗))
1312fveq2d 6910 . . . . . 6 (𝑥 = 𝑗 → (seq1( + , 𝐹)‘(2↑𝑥)) = (seq1( + , 𝐹)‘(2↑𝑗)))
1413oveq2d 7447 . . . . 5 (𝑥 = 𝑗 → (2 · (seq1( + , 𝐹)‘(2↑𝑥))) = (2 · (seq1( + , 𝐹)‘(2↑𝑗))))
1511, 14breq12d 5156 . . . 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 7439 . . . . . . 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 7447 . . . . 5 (𝑥 = (𝑗 + 1) → (2 · (seq1( + , 𝐹)‘(2↑𝑥))) = (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1)))))
2117, 20breq12d 5156 . . . 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 7439 . . . . . . 7 (𝑥 = 𝑁 → (2↑𝑥) = (2↑𝑁))
2524fveq2d 6910 . . . . . 6 (𝑥 = 𝑁 → (seq1( + , 𝐹)‘(2↑𝑥)) = (seq1( + , 𝐹)‘(2↑𝑁)))
2625oveq2d 7447 . . . . 5 (𝑥 = 𝑁 → (2 · (seq1( + , 𝐹)‘(2↑𝑥))) = (2 · (seq1( + , 𝐹)‘(2↑𝑁))))
2723, 26breq12d 5156 . . . 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 5155 . . . . . . 7 (𝑘 = 1 → (0 ≤ (𝐹𝑘) ↔ 0 ≤ (𝐹‘1)))
31 climcnds.2 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 0 ≤ (𝐹𝑘))
3231ralrimiva 3146 . . . . . . 7 (𝜑 → ∀𝑘 ∈ ℕ 0 ≤ (𝐹𝑘))
33 1nn 12277 . . . . . . . 8 1 ∈ ℕ
3433a1i 11 . . . . . . 7 (𝜑 → 1 ∈ ℕ)
3530, 32, 34rspcdva 3623 . . . . . 6 (𝜑 → 0 ≤ (𝐹‘1))
36 fveq2 6906 . . . . . . . . 9 (𝑘 = 2 → (𝐹𝑘) = (𝐹‘2))
3736eleq1d 2826 . . . . . . . 8 (𝑘 = 2 → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘2) ∈ ℝ))
38 climcnds.1 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℝ)
3938ralrimiva 3146 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ ℕ (𝐹𝑘) ∈ ℝ)
40 2nn 12339 . . . . . . . . 9 2 ∈ ℕ
4140a1i 11 . . . . . . . 8 (𝜑 → 2 ∈ ℕ)
4237, 39, 41rspcdva 3623 . . . . . . 7 (𝜑 → (𝐹‘2) ∈ ℝ)
4329eleq1d 2826 . . . . . . . 8 (𝑘 = 1 → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘1) ∈ ℝ))
4443, 39, 34rspcdva 3623 . . . . . . 7 (𝜑 → (𝐹‘1) ∈ ℝ)
4542, 44addge02d 11852 . . . . . 6 (𝜑 → (0 ≤ (𝐹‘1) ↔ (𝐹‘2) ≤ ((𝐹‘1) + (𝐹‘2))))
4635, 45mpbid 232 . . . . 5 (𝜑 → (𝐹‘2) ≤ ((𝐹‘1) + (𝐹‘2)))
4744, 42readdcld 11290 . . . . . 6 (𝜑 → ((𝐹‘1) + (𝐹‘2)) ∈ ℝ)
4841nnrpd 13075 . . . . . 6 (𝜑 → 2 ∈ ℝ+)
4942, 47, 48lemul2d 13121 . . . . 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 12647 . . . . 5 1 ∈ ℤ
52 fveq2 6906 . . . . . . 7 (𝑛 = 1 → (𝐺𝑛) = (𝐺‘1))
53 oveq2 7439 . . . . . . . . 9 (𝑛 = 1 → (2↑𝑛) = (2↑1))
5453, 5eqtrdi 2793 . . . . . . . 8 (𝑛 = 1 → (2↑𝑛) = 2)
5554fveq2d 6910 . . . . . . . 8 (𝑛 = 1 → (𝐹‘(2↑𝑛)) = (𝐹‘2))
5654, 55oveq12d 7449 . . . . . . 7 (𝑛 = 1 → ((2↑𝑛) · (𝐹‘(2↑𝑛))) = (2 · (𝐹‘2)))
5752, 56eqeq12d 2753 . . . . . 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 3146 . . . . . 6 (𝜑 → ∀𝑛 ∈ ℕ0 (𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))
60 1nn0 12542 . . . . . . 7 1 ∈ ℕ0
6160a1i 11 . . . . . 6 (𝜑 → 1 ∈ ℕ0)
6257, 59, 61rspcdva 3623 . . . . 5 (𝜑 → (𝐺‘1) = (2 · (𝐹‘2)))
6351, 62seq1i 14056 . . . 4 (𝜑 → (seq1( + , 𝐺)‘1) = (2 · (𝐹‘2)))
64 nnuz 12921 . . . . . 6 ℕ = (ℤ‘1)
65 df-2 12329 . . . . . 6 2 = (1 + 1)
66 eqidd 2738 . . . . . . 7 (𝜑 → (𝐹‘1) = (𝐹‘1))
6751, 66seq1i 14056 . . . . . 6 (𝜑 → (seq1( + , 𝐹)‘1) = (𝐹‘1))
68 eqidd 2738 . . . . . 6 (𝜑 → (𝐹‘2) = (𝐹‘2))
6964, 34, 65, 67, 68seqp1d 14059 . . . . 5 (𝜑 → (seq1( + , 𝐹)‘2) = ((𝐹‘1) + (𝐹‘2)))
7069oveq2d 7447 . . . 4 (𝜑 → (2 · (seq1( + , 𝐹)‘2)) = (2 · ((𝐹‘1) + (𝐹‘2))))
7150, 63, 703brtr4d 5175 . . 3 (𝜑 → (seq1( + , 𝐺)‘1) ≤ (2 · (seq1( + , 𝐹)‘2)))
72 fveq2 6906 . . . . . . . . . . 11 (𝑛 = (𝑗 + 1) → (𝐺𝑛) = (𝐺‘(𝑗 + 1)))
73 oveq2 7439 . . . . . . . . . . . 12 (𝑛 = (𝑗 + 1) → (2↑𝑛) = (2↑(𝑗 + 1)))
7473fveq2d 6910 . . . . . . . . . . . 12 (𝑛 = (𝑗 + 1) → (𝐹‘(2↑𝑛)) = (𝐹‘(2↑(𝑗 + 1))))
7573, 74oveq12d 7449 . . . . . . . . . . 11 (𝑛 = (𝑗 + 1) → ((2↑𝑛) · (𝐹‘(2↑𝑛))) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))))
7672, 75eqeq12d 2753 . . . . . . . . . 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 12278 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
7978adantl 481 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ)
8079nnnn0d 12587 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ0)
8176, 77, 80rspcdva 3623 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐺‘(𝑗 + 1)) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))))
82 nnnn0 12533 . . . . . . . . . . . . 13 (𝑗 ∈ ℕ → 𝑗 ∈ ℕ0)
8382adantl 481 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ0)
84 expp1 14109 . . . . . . . . . . . 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 14115 . . . . . . . . . . . . . . 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 12282 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℂ)
90 mulcom 11241 . . . . . . . . . . . 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 2777 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) = (2 · (2↑𝑗)))
9392oveq1d 7446 . . . . . . . . 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 2826 . . . . . . . . . . . 12 (𝑘 = (2↑(𝑗 + 1)) → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ))
9739adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ∀𝑘 ∈ ℕ (𝐹𝑘) ∈ ℝ)
98 nnexpcl 14115 . . . . . . . . . . . . 13 ((2 ∈ ℕ ∧ (𝑗 + 1) ∈ ℕ0) → (2↑(𝑗 + 1)) ∈ ℕ)
9940, 80, 98sylancr 587 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ ℕ)
10096, 97, 99rspcdva 3623 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ)
101100recnd 11289 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℂ)
10294, 89, 101mulassd 11284 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ((2 · (2↑𝑗)) · (𝐹‘(2↑(𝑗 + 1)))) = (2 · ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1))))))
10381, 93, 1023eqtrd 2781 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐺‘(𝑗 + 1)) = (2 · ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1))))))
10488nnnn0d 12587 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℕ0)
105 hashfz1 14385 . . . . . . . . . . . . . . 15 ((2↑𝑗) ∈ ℕ0 → (♯‘(1...(2↑𝑗))) = (2↑𝑗))
106104, 105syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (♯‘(1...(2↑𝑗))) = (2↑𝑗))
107106, 89eqeltrd 2841 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (♯‘(1...(2↑𝑗))) ∈ ℂ)
108 fzfid 14014 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (((2↑𝑗) + 1)...(2↑(𝑗 + 1))) ∈ Fin)
109 hashcl 14395 . . . . . . . . . . . . . . 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 12589 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∈ ℂ)
112 simpr 484 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
113112nnzd 12640 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℤ)
114 uzid 12893 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ𝑗))
115 peano2uz 12943 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (ℤ𝑗) → (𝑗 + 1) ∈ (ℤ𝑗))
116 2re 12340 . . . . . . . . . . . . . . . . . . 19 2 ∈ ℝ
117 1le2 12475 . . . . . . . . . . . . . . . . . . 19 1 ≤ 2
118 leexp2a 14212 . . . . . . . . . . . . . . . . . . 19 ((2 ∈ ℝ ∧ 1 ≤ 2 ∧ (𝑗 + 1) ∈ (ℤ𝑗)) → (2↑𝑗) ≤ (2↑(𝑗 + 1)))
119116, 117, 118mp3an12 1453 . . . . . . . . . . . . . . . . . 18 ((𝑗 + 1) ∈ (ℤ𝑗) → (2↑𝑗) ≤ (2↑(𝑗 + 1)))
120113, 114, 115, 1194syl 19 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ≤ (2↑(𝑗 + 1)))
12188, 64eleqtrdi 2851 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ (ℤ‘1))
12299nnzd 12640 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ ℤ)
123 elfz5 13556 . . . . . . . . . . . . . . . . . 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 13590 . . . . . . . . . . . . . . . 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 12510 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · 2) = ((2↑𝑗) + (2↑𝑗)))
13085, 129eqtrd 2777 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) = ((2↑𝑗) + (2↑𝑗)))
13199nnnn0d 12587 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ ℕ0)
132 hashfz1 14385 . . . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → ((♯‘(1...(2↑𝑗))) + (2↑𝑗)) = ((2↑𝑗) + (2↑𝑗)))
135130, 133, 1343eqtr4d 2787 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (♯‘(1...(2↑(𝑗 + 1)))) = ((♯‘(1...(2↑𝑗))) + (2↑𝑗)))
136 fzfid 14014 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (1...(2↑𝑗)) ∈ Fin)
13788nnred 12281 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℝ)
138137ltp1d 12198 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) < ((2↑𝑗) + 1))
139 fzdisj 13591 . . . . . . . . . . . . . . . 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 14421 . . . . . . . . . . . . . . 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 1373 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (♯‘((1...(2↑𝑗)) ∪ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))))) = ((♯‘(1...(2↑𝑗))) + (♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1))))))
143128, 135, 1423eqtr3d 2785 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ((♯‘(1...(2↑𝑗))) + (2↑𝑗)) = ((♯‘(1...(2↑𝑗))) + (♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1))))))
144107, 89, 111, 143addcanad 11466 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) = (♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))))
145144oveq1d 7446 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) = ((♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) · (𝐹‘(2↑(𝑗 + 1)))))
146 fsumconst 15826 . . . . . . . . . . . 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 2780 . . . . . . . . . 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 12278 . . . . . . . . . . . . . 14 ((2↑𝑗) ∈ ℕ → ((2↑𝑗) + 1) ∈ ℕ)
15288, 151syl 17 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) + 1) ∈ ℕ)
153 elfzuz 13560 . . . . . . . . . . . . 13 (𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))) → 𝑘 ∈ (ℤ‘((2↑𝑗) + 1)))
154 eluznn 12960 . . . . . . . . . . . . 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 13561 . . . . . . . . . . . . . . 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 13560 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))) → 𝑛 ∈ (ℤ‘((2↑𝑗) + 1)))
161 eluznn 12960 . . . . . . . . . . . . . . . . 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 13560 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (𝑛...(2↑(𝑗 + 1))) → 𝑘 ∈ (ℤ𝑛))
164 eluznn 12960 . . . . . . . . . . . . . . . 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 13560 . . . . . . . . . . . . . . . 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 14074 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑛))
173172ralrimiva 3146 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ∀𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑛))
174 fveq2 6906 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
175174breq2d 5155 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → ((𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑛) ↔ (𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑘)))
176175rspccva 3621 . . . . . . . . . . . 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 15835 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘(2↑(𝑗 + 1))) ≤ Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘))
179148, 178eqbrtrd 5165 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) ≤ Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘))
180137, 100remulcld 11291 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) ∈ ℝ)
181108, 156fsumrecl 15770 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘) ∈ ℝ)
182 2rp 13039 . . . . . . . . . . 11 2 ∈ ℝ+
183182a1i 11 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → 2 ∈ ℝ+)
184180, 181, 183lemul2d 13121 . . . . . . . . 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 5165 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (𝐺‘(𝑗 + 1)) ≤ (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)))
187 1zzd 12648 . . . . . . . . . 10 (𝜑 → 1 ∈ ℤ)
188 nnnn0 12533 . . . . . . . . . . 11 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
189 simpr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
190 nnexpcl 14115 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
19140, 189, 190sylancr 587 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
192191nnred 12281 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℝ)
193 fveq2 6906 . . . . . . . . . . . . . . 15 (𝑘 = (2↑𝑛) → (𝐹𝑘) = (𝐹‘(2↑𝑛)))
194193eleq1d 2826 . . . . . . . . . . . . . 14 (𝑘 = (2↑𝑛) → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘(2↑𝑛)) ∈ ℝ))
19539adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ0) → ∀𝑘 ∈ ℕ (𝐹𝑘) ∈ ℝ)
196194, 195, 191rspcdva 3623 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ0) → (𝐹‘(2↑𝑛)) ∈ ℝ)
197192, 196remulcld 11291 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ0) → ((2↑𝑛) · (𝐹‘(2↑𝑛))) ∈ ℝ)
19858, 197eqeltrd 2841 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → (𝐺𝑛) ∈ ℝ)
199188, 198sylan2 593 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ ℝ)
20064, 187, 199serfre 14072 . . . . . . . . 9 (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
201200ffvelcdmda 7104 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (seq1( + , 𝐺)‘𝑗) ∈ ℝ)
20272eleq1d 2826 . . . . . . . . 9 (𝑛 = (𝑗 + 1) → ((𝐺𝑛) ∈ ℝ ↔ (𝐺‘(𝑗 + 1)) ∈ ℝ))
203199ralrimiva 3146 . . . . . . . . . 10 (𝜑 → ∀𝑛 ∈ ℕ (𝐺𝑛) ∈ ℝ)
204203adantr 480 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ∀𝑛 ∈ ℕ (𝐺𝑛) ∈ ℝ)
205202, 204, 79rspcdva 3623 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐺‘(𝑗 + 1)) ∈ ℝ)
20664, 187, 38serfre 14072 . . . . . . . . . 10 (𝜑 → seq1( + , 𝐹):ℕ⟶ℝ)
207 ffvelcdm 7101 . . . . . . . . . 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 11240 . . . . . . . . 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 11240 . . . . . . . . 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 11745 . . . . . . . 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 2851 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ‘1))
217 seqp1 14057 . . . . . . . 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 14014 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (1...(2↑(𝑗 + 1))) ∈ Fin)
220 elfznn 13593 . . . . . . . . . . . 12 (𝑘 ∈ (1...(2↑(𝑗 + 1))) → 𝑘 ∈ ℕ)
22138recnd 11289 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℂ)
222150, 220, 221syl2an 596 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑(𝑗 + 1)))) → (𝐹𝑘) ∈ ℂ)
223140, 127, 219, 222fsumsplit 15777 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...(2↑(𝑗 + 1)))(𝐹𝑘) = (Σ𝑘 ∈ (1...(2↑𝑗))(𝐹𝑘) + Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)))
224 eqidd 2738 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑(𝑗 + 1)))) → (𝐹𝑘) = (𝐹𝑘))
22599, 64eleqtrdi 2851 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ (ℤ‘1))
226224, 225, 222fsumser 15766 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...(2↑(𝑗 + 1)))(𝐹𝑘) = (seq1( + , 𝐹)‘(2↑(𝑗 + 1))))
227 eqidd 2738 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑𝑗))) → (𝐹𝑘) = (𝐹𝑘))
228 elfznn 13593 . . . . . . . . . . . . 13 (𝑘 ∈ (1...(2↑𝑗)) → 𝑘 ∈ ℕ)
229150, 228, 221syl2an 596 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑𝑗))) → (𝐹𝑘) ∈ ℂ)
230227, 121, 229fsumser 15766 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...(2↑𝑗))(𝐹𝑘) = (seq1( + , 𝐹)‘(2↑𝑗)))
231230oveq1d 7446 . . . . . . . . . 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 2785 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘(2↑(𝑗 + 1))) = ((seq1( + , 𝐹)‘(2↑𝑗)) + Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)))
233232oveq2d 7447 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1)))) = (2 · ((seq1( + , 𝐹)‘(2↑𝑗)) + Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘))))
234208recnd 11289 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘(2↑𝑗)) ∈ ℂ)
235181recnd 11289 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘) ∈ ℂ)
23694, 234, 235adddid 11285 . . . . . . . 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 2777 . . . . . . 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 5156 . . . . . 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 12284 . 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 1540  wcel 2108  wral 3061  cun 3949  cin 3950  c0 4333   class class class wbr 5143  wf 6557  cfv 6561  (class class class)co 7431  Fincfn 8985  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160   < clt 11295  cle 11296  cmin 11492  cn 12266  2c2 12321  0cn0 12526  cz 12613  cuz 12878  +crp 13034  ...cfz 13547  seqcseq 14042  cexp 14102  chash 14369  Σcsu 15722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-ico 13393  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723
This theorem is referenced by:  climcnds  15887
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