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Theorem climcndslem2 15038
Description: Lemma for climcnds 15039: 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 6538 . . . . 5 (𝑥 = 1 → (seq1( + , 𝐺)‘𝑥) = (seq1( + , 𝐺)‘1))
2 oveq2 7024 . . . . . . . 8 (𝑥 = 1 → (2↑𝑥) = (2↑1))
3 2cn 11560 . . . . . . . . 9 2 ∈ ℂ
4 exp1 13285 . . . . . . . . 9 (2 ∈ ℂ → (2↑1) = 2)
53, 4ax-mp 5 . . . . . . . 8 (2↑1) = 2
62, 5syl6eq 2847 . . . . . . 7 (𝑥 = 1 → (2↑𝑥) = 2)
76fveq2d 6542 . . . . . 6 (𝑥 = 1 → (seq1( + , 𝐹)‘(2↑𝑥)) = (seq1( + , 𝐹)‘2))
87oveq2d 7032 . . . . 5 (𝑥 = 1 → (2 · (seq1( + , 𝐹)‘(2↑𝑥))) = (2 · (seq1( + , 𝐹)‘2)))
91, 8breq12d 4975 . . . 4 (𝑥 = 1 → ((seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥))) ↔ (seq1( + , 𝐺)‘1) ≤ (2 · (seq1( + , 𝐹)‘2))))
109imbi2d 342 . . 3 (𝑥 = 1 → ((𝜑 → (seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥)))) ↔ (𝜑 → (seq1( + , 𝐺)‘1) ≤ (2 · (seq1( + , 𝐹)‘2)))))
11 fveq2 6538 . . . . 5 (𝑥 = 𝑗 → (seq1( + , 𝐺)‘𝑥) = (seq1( + , 𝐺)‘𝑗))
12 oveq2 7024 . . . . . . 7 (𝑥 = 𝑗 → (2↑𝑥) = (2↑𝑗))
1312fveq2d 6542 . . . . . 6 (𝑥 = 𝑗 → (seq1( + , 𝐹)‘(2↑𝑥)) = (seq1( + , 𝐹)‘(2↑𝑗)))
1413oveq2d 7032 . . . . 5 (𝑥 = 𝑗 → (2 · (seq1( + , 𝐹)‘(2↑𝑥))) = (2 · (seq1( + , 𝐹)‘(2↑𝑗))))
1511, 14breq12d 4975 . . . 4 (𝑥 = 𝑗 → ((seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥))) ↔ (seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗)))))
1615imbi2d 342 . . 3 (𝑥 = 𝑗 → ((𝜑 → (seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥)))) ↔ (𝜑 → (seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗))))))
17 fveq2 6538 . . . . 5 (𝑥 = (𝑗 + 1) → (seq1( + , 𝐺)‘𝑥) = (seq1( + , 𝐺)‘(𝑗 + 1)))
18 oveq2 7024 . . . . . . 7 (𝑥 = (𝑗 + 1) → (2↑𝑥) = (2↑(𝑗 + 1)))
1918fveq2d 6542 . . . . . 6 (𝑥 = (𝑗 + 1) → (seq1( + , 𝐹)‘(2↑𝑥)) = (seq1( + , 𝐹)‘(2↑(𝑗 + 1))))
2019oveq2d 7032 . . . . 5 (𝑥 = (𝑗 + 1) → (2 · (seq1( + , 𝐹)‘(2↑𝑥))) = (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1)))))
2117, 20breq12d 4975 . . . 4 (𝑥 = (𝑗 + 1) → ((seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥))) ↔ (seq1( + , 𝐺)‘(𝑗 + 1)) ≤ (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1))))))
2221imbi2d 342 . . 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 6538 . . . . 5 (𝑥 = 𝑁 → (seq1( + , 𝐺)‘𝑥) = (seq1( + , 𝐺)‘𝑁))
24 oveq2 7024 . . . . . . 7 (𝑥 = 𝑁 → (2↑𝑥) = (2↑𝑁))
2524fveq2d 6542 . . . . . 6 (𝑥 = 𝑁 → (seq1( + , 𝐹)‘(2↑𝑥)) = (seq1( + , 𝐹)‘(2↑𝑁)))
2625oveq2d 7032 . . . . 5 (𝑥 = 𝑁 → (2 · (seq1( + , 𝐹)‘(2↑𝑥))) = (2 · (seq1( + , 𝐹)‘(2↑𝑁))))
2723, 26breq12d 4975 . . . 4 (𝑥 = 𝑁 → ((seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥))) ↔ (seq1( + , 𝐺)‘𝑁) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑁)))))
2827imbi2d 342 . . 3 (𝑥 = 𝑁 → ((𝜑 → (seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥)))) ↔ (𝜑 → (seq1( + , 𝐺)‘𝑁) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑁))))))
29 fveq2 6538 . . . . . . . 8 (𝑘 = 1 → (𝐹𝑘) = (𝐹‘1))
3029breq2d 4974 . . . . . . 7 (𝑘 = 1 → (0 ≤ (𝐹𝑘) ↔ 0 ≤ (𝐹‘1)))
31 climcnds.2 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 0 ≤ (𝐹𝑘))
3231ralrimiva 3149 . . . . . . 7 (𝜑 → ∀𝑘 ∈ ℕ 0 ≤ (𝐹𝑘))
33 1nn 11497 . . . . . . . 8 1 ∈ ℕ
3433a1i 11 . . . . . . 7 (𝜑 → 1 ∈ ℕ)
3530, 32, 34rspcdva 3565 . . . . . 6 (𝜑 → 0 ≤ (𝐹‘1))
36 fveq2 6538 . . . . . . . . 9 (𝑘 = 2 → (𝐹𝑘) = (𝐹‘2))
3736eleq1d 2867 . . . . . . . 8 (𝑘 = 2 → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘2) ∈ ℝ))
38 climcnds.1 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℝ)
3938ralrimiva 3149 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ ℕ (𝐹𝑘) ∈ ℝ)
40 2nn 11558 . . . . . . . . 9 2 ∈ ℕ
4140a1i 11 . . . . . . . 8 (𝜑 → 2 ∈ ℕ)
4237, 39, 41rspcdva 3565 . . . . . . 7 (𝜑 → (𝐹‘2) ∈ ℝ)
4329eleq1d 2867 . . . . . . . 8 (𝑘 = 1 → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘1) ∈ ℝ))
4443, 39, 34rspcdva 3565 . . . . . . 7 (𝜑 → (𝐹‘1) ∈ ℝ)
4542, 44addge02d 11077 . . . . . 6 (𝜑 → (0 ≤ (𝐹‘1) ↔ (𝐹‘2) ≤ ((𝐹‘1) + (𝐹‘2))))
4635, 45mpbid 233 . . . . 5 (𝜑 → (𝐹‘2) ≤ ((𝐹‘1) + (𝐹‘2)))
4744, 42readdcld 10516 . . . . . 6 (𝜑 → ((𝐹‘1) + (𝐹‘2)) ∈ ℝ)
4841nnrpd 12279 . . . . . 6 (𝜑 → 2 ∈ ℝ+)
4942, 47, 48lemul2d 12325 . . . . 5 (𝜑 → ((𝐹‘2) ≤ ((𝐹‘1) + (𝐹‘2)) ↔ (2 · (𝐹‘2)) ≤ (2 · ((𝐹‘1) + (𝐹‘2)))))
5046, 49mpbid 233 . . . 4 (𝜑 → (2 · (𝐹‘2)) ≤ (2 · ((𝐹‘1) + (𝐹‘2))))
51 1z 11861 . . . . 5 1 ∈ ℤ
52 fveq2 6538 . . . . . . 7 (𝑛 = 1 → (𝐺𝑛) = (𝐺‘1))
53 oveq2 7024 . . . . . . . . 9 (𝑛 = 1 → (2↑𝑛) = (2↑1))
5453, 5syl6eq 2847 . . . . . . . 8 (𝑛 = 1 → (2↑𝑛) = 2)
5554fveq2d 6542 . . . . . . . 8 (𝑛 = 1 → (𝐹‘(2↑𝑛)) = (𝐹‘2))
5654, 55oveq12d 7034 . . . . . . 7 (𝑛 = 1 → ((2↑𝑛) · (𝐹‘(2↑𝑛))) = (2 · (𝐹‘2)))
5752, 56eqeq12d 2810 . . . . . 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 3149 . . . . . 6 (𝜑 → ∀𝑛 ∈ ℕ0 (𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))
60 1nn0 11761 . . . . . . 7 1 ∈ ℕ0
6160a1i 11 . . . . . 6 (𝜑 → 1 ∈ ℕ0)
6257, 59, 61rspcdva 3565 . . . . 5 (𝜑 → (𝐺‘1) = (2 · (𝐹‘2)))
6351, 62seq1i 13233 . . . 4 (𝜑 → (seq1( + , 𝐺)‘1) = (2 · (𝐹‘2)))
64 nnuz 12130 . . . . . 6 ℕ = (ℤ‘1)
65 df-2 11548 . . . . . 6 2 = (1 + 1)
66 eqidd 2796 . . . . . . 7 (𝜑 → (𝐹‘1) = (𝐹‘1))
6751, 66seq1i 13233 . . . . . 6 (𝜑 → (seq1( + , 𝐹)‘1) = (𝐹‘1))
68 eqidd 2796 . . . . . 6 (𝜑 → (𝐹‘2) = (𝐹‘2))
6964, 33, 65, 67, 68seqp1i 13236 . . . . 5 (𝜑 → (seq1( + , 𝐹)‘2) = ((𝐹‘1) + (𝐹‘2)))
7069oveq2d 7032 . . . 4 (𝜑 → (2 · (seq1( + , 𝐹)‘2)) = (2 · ((𝐹‘1) + (𝐹‘2))))
7150, 63, 703brtr4d 4994 . . 3 (𝜑 → (seq1( + , 𝐺)‘1) ≤ (2 · (seq1( + , 𝐹)‘2)))
72 fveq2 6538 . . . . . . . . . . 11 (𝑛 = (𝑗 + 1) → (𝐺𝑛) = (𝐺‘(𝑗 + 1)))
73 oveq2 7024 . . . . . . . . . . . 12 (𝑛 = (𝑗 + 1) → (2↑𝑛) = (2↑(𝑗 + 1)))
7473fveq2d 6542 . . . . . . . . . . . 12 (𝑛 = (𝑗 + 1) → (𝐹‘(2↑𝑛)) = (𝐹‘(2↑(𝑗 + 1))))
7573, 74oveq12d 7034 . . . . . . . . . . 11 (𝑛 = (𝑗 + 1) → ((2↑𝑛) · (𝐹‘(2↑𝑛))) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))))
7672, 75eqeq12d 2810 . . . . . . . . . 10 (𝑛 = (𝑗 + 1) → ((𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))) ↔ (𝐺‘(𝑗 + 1)) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1))))))
7759adantr 481 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ∀𝑛 ∈ ℕ0 (𝐺𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))
78 peano2nn 11498 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
7978adantl 482 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ)
8079nnnn0d 11803 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ0)
8176, 77, 80rspcdva 3565 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐺‘(𝑗 + 1)) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))))
82 nnnn0 11752 . . . . . . . . . . . . 13 (𝑗 ∈ ℕ → 𝑗 ∈ ℕ0)
8382adantl 482 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ0)
84 expp1 13286 . . . . . . . . . . . 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 13292 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑗 ∈ ℕ0) → (2↑𝑗) ∈ ℕ)
8740, 82, 86sylancr 587 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ → (2↑𝑗) ∈ ℕ)
8887adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℕ)
8988nncnd 11502 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℂ)
90 mulcom 10469 . . . . . . . . . . . 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 2831 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) = (2 · (2↑𝑗)))
9392oveq1d 7031 . . . . . . . . 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 6538 . . . . . . . . . . . . 13 (𝑘 = (2↑(𝑗 + 1)) → (𝐹𝑘) = (𝐹‘(2↑(𝑗 + 1))))
9695eleq1d 2867 . . . . . . . . . . . 12 (𝑘 = (2↑(𝑗 + 1)) → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ))
9739adantr 481 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ∀𝑘 ∈ ℕ (𝐹𝑘) ∈ ℝ)
98 nnexpcl 13292 . . . . . . . . . . . . 13 ((2 ∈ ℕ ∧ (𝑗 + 1) ∈ ℕ0) → (2↑(𝑗 + 1)) ∈ ℕ)
9940, 80, 98sylancr 587 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ ℕ)
10096, 97, 99rspcdva 3565 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ)
101100recnd 10515 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℂ)
10294, 89, 101mulassd 10510 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ((2 · (2↑𝑗)) · (𝐹‘(2↑(𝑗 + 1)))) = (2 · ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1))))))
10381, 93, 1023eqtrd 2835 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐺‘(𝑗 + 1)) = (2 · ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1))))))
10488nnnn0d 11803 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℕ0)
105 hashfz1 13556 . . . . . . . . . . . . . . 15 ((2↑𝑗) ∈ ℕ0 → (♯‘(1...(2↑𝑗))) = (2↑𝑗))
106104, 105syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (♯‘(1...(2↑𝑗))) = (2↑𝑗))
107106, 89eqeltrd 2883 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (♯‘(1...(2↑𝑗))) ∈ ℂ)
108 fzfid 13191 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (((2↑𝑗) + 1)...(2↑(𝑗 + 1))) ∈ Fin)
109 hashcl 13567 . . . . . . . . . . . . . . 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 11805 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∈ ℂ)
112 simpr 485 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
113112nnzd 11935 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℤ)
114 uzid 12108 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ𝑗))
115 peano2uz 12150 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (ℤ𝑗) → (𝑗 + 1) ∈ (ℤ𝑗))
116 2re 11559 . . . . . . . . . . . . . . . . . . 19 2 ∈ ℝ
117 1le2 11694 . . . . . . . . . . . . . . . . . . 19 1 ≤ 2
118 leexp2a 13386 . . . . . . . . . . . . . . . . . . 19 ((2 ∈ ℝ ∧ 1 ≤ 2 ∧ (𝑗 + 1) ∈ (ℤ𝑗)) → (2↑𝑗) ≤ (2↑(𝑗 + 1)))
119116, 117, 118mp3an12 1443 . . . . . . . . . . . . . . . . . 18 ((𝑗 + 1) ∈ (ℤ𝑗) → (2↑𝑗) ≤ (2↑(𝑗 + 1)))
120113, 114, 115, 1194syl 19 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ≤ (2↑(𝑗 + 1)))
12188, 64syl6eleq 2893 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ (ℤ‘1))
12299nnzd 11935 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ ℤ)
123 elfz5 12750 . . . . . . . . . . . . . . . . . 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 258 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ (1...(2↑(𝑗 + 1))))
126 fzsplit 12783 . . . . . . . . . . . . . . . 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 6542 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (♯‘(1...(2↑(𝑗 + 1)))) = (♯‘((1...(2↑𝑗)) ∪ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))))))
12989times2d 11729 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · 2) = ((2↑𝑗) + (2↑𝑗)))
13085, 129eqtrd 2831 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) = ((2↑𝑗) + (2↑𝑗)))
13199nnnn0d 11803 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ ℕ0)
132 hashfz1 13556 . . . . . . . . . . . . . . . 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 7031 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → ((♯‘(1...(2↑𝑗))) + (2↑𝑗)) = ((2↑𝑗) + (2↑𝑗)))
135130, 133, 1343eqtr4d 2841 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (♯‘(1...(2↑(𝑗 + 1)))) = ((♯‘(1...(2↑𝑗))) + (2↑𝑗)))
136 fzfid 13191 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (1...(2↑𝑗)) ∈ Fin)
13788nnred 11501 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℝ)
138137ltp1d 11418 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) < ((2↑𝑗) + 1))
139 fzdisj 12784 . . . . . . . . . . . . . . . 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 13591 . . . . . . . . . . . . . . 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 1364 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → (♯‘((1...(2↑𝑗)) ∪ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))))) = ((♯‘(1...(2↑𝑗))) + (♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1))))))
143128, 135, 1423eqtr3d 2839 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ((♯‘(1...(2↑𝑗))) + (2↑𝑗)) = ((♯‘(1...(2↑𝑗))) + (♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1))))))
144107, 89, 111, 143addcanad 10692 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (2↑𝑗) = (♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))))
145144oveq1d 7031 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) = ((♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) · (𝐹‘(2↑(𝑗 + 1)))))
146 fsumconst 14978 . . . . . . . . . . . 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 2834 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) = Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘(2↑(𝑗 + 1))))
149100adantr 481 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ)
150 simpl 483 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → 𝜑)
151 peano2nn 11498 . . . . . . . . . . . . . 14 ((2↑𝑗) ∈ ℕ → ((2↑𝑗) + 1) ∈ ℕ)
15288, 151syl 17 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) + 1) ∈ ℕ)
153 elfzuz 12754 . . . . . . . . . . . . 13 (𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))) → 𝑘 ∈ (ℤ‘((2↑𝑗) + 1)))
154 eluznn 12167 . . . . . . . . . . . . 13 ((((2↑𝑗) + 1) ∈ ℕ ∧ 𝑘 ∈ (ℤ‘((2↑𝑗) + 1))) → 𝑘 ∈ ℕ)
155152, 153, 154syl2an 595 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → 𝑘 ∈ ℕ)
156150, 155, 38syl2an2r 681 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (𝐹𝑘) ∈ ℝ)
157 elfzuz3 12755 . . . . . . . . . . . . . . 15 (𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))) → (2↑(𝑗 + 1)) ∈ (ℤ𝑛))
158157adantl 482 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (2↑(𝑗 + 1)) ∈ (ℤ𝑛))
159 simplll 771 . . . . . . . . . . . . . . 15 ((((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...(2↑(𝑗 + 1)))) → 𝜑)
160 elfzuz 12754 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))) → 𝑛 ∈ (ℤ‘((2↑𝑗) + 1)))
161 eluznn 12167 . . . . . . . . . . . . . . . . 17 ((((2↑𝑗) + 1) ∈ ℕ ∧ 𝑛 ∈ (ℤ‘((2↑𝑗) + 1))) → 𝑛 ∈ ℕ)
162152, 160, 161syl2an 595 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → 𝑛 ∈ ℕ)
163 elfzuz 12754 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (𝑛...(2↑(𝑗 + 1))) → 𝑘 ∈ (ℤ𝑛))
164 eluznn 12167 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑘 ∈ ℕ)
165162, 163, 164syl2an 595 . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . 15 ((((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...((2↑(𝑗 + 1)) − 1))) → 𝜑)
168 elfzuz 12754 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (𝑛...((2↑(𝑗 + 1)) − 1)) → 𝑘 ∈ (ℤ𝑛))
169162, 168, 164syl2an 595 . . . . . . . . . . . . . . 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 13251 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑛))
173172ralrimiva 3149 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ∀𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑛))
174 fveq2 6538 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
175174breq2d 4974 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → ((𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑛) ↔ (𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹𝑘)))
176175rspccva 3558 . . . . . . . . . . . 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 14987 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘(2↑(𝑗 + 1))) ≤ Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘))
179148, 178eqbrtrd 4984 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) ≤ Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘))
180137, 100remulcld 10517 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) ∈ ℝ)
181108, 156fsumrecl 14924 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘) ∈ ℝ)
182 2rp 12244 . . . . . . . . . . 11 2 ∈ ℝ+
183182a1i 11 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → 2 ∈ ℝ+)
184180, 181, 183lemul2d 12325 . . . . . . . . 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 233 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (2 · ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1))))) ≤ (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)))
186103, 185eqbrtrd 4984 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (𝐺‘(𝑗 + 1)) ≤ (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)))
187 1zzd 11862 . . . . . . . . . 10 (𝜑 → 1 ∈ ℤ)
188 nnnn0 11752 . . . . . . . . . . 11 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
189 simpr 485 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
190 nnexpcl 13292 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
19140, 189, 190sylancr 587 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
192191nnred 11501 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℝ)
193 fveq2 6538 . . . . . . . . . . . . . . 15 (𝑘 = (2↑𝑛) → (𝐹𝑘) = (𝐹‘(2↑𝑛)))
194193eleq1d 2867 . . . . . . . . . . . . . 14 (𝑘 = (2↑𝑛) → ((𝐹𝑘) ∈ ℝ ↔ (𝐹‘(2↑𝑛)) ∈ ℝ))
19539adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ0) → ∀𝑘 ∈ ℕ (𝐹𝑘) ∈ ℝ)
196194, 195, 191rspcdva 3565 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ0) → (𝐹‘(2↑𝑛)) ∈ ℝ)
197192, 196remulcld 10517 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ0) → ((2↑𝑛) · (𝐹‘(2↑𝑛))) ∈ ℝ)
19858, 197eqeltrd 2883 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → (𝐺𝑛) ∈ ℝ)
199188, 198sylan2 592 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ ℝ)
20064, 187, 199serfre 13249 . . . . . . . . 9 (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
201200ffvelrnda 6716 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (seq1( + , 𝐺)‘𝑗) ∈ ℝ)
20272eleq1d 2867 . . . . . . . . 9 (𝑛 = (𝑗 + 1) → ((𝐺𝑛) ∈ ℝ ↔ (𝐺‘(𝑗 + 1)) ∈ ℝ))
203199ralrimiva 3149 . . . . . . . . . 10 (𝜑 → ∀𝑛 ∈ ℕ (𝐺𝑛) ∈ ℝ)
204203adantr 481 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ∀𝑛 ∈ ℕ (𝐺𝑛) ∈ ℝ)
205202, 204, 79rspcdva 3565 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐺‘(𝑗 + 1)) ∈ ℝ)
20664, 187, 38serfre 13249 . . . . . . . . . 10 (𝜑 → seq1( + , 𝐹):ℕ⟶ℝ)
207 ffvelrn 6714 . . . . . . . . . 10 ((seq1( + , 𝐹):ℕ⟶ℝ ∧ (2↑𝑗) ∈ ℕ) → (seq1( + , 𝐹)‘(2↑𝑗)) ∈ ℝ)
208206, 87, 207syl2an 595 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘(2↑𝑗)) ∈ ℝ)
209 remulcl 10468 . . . . . . . . 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 10468 . . . . . . . . 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 10970 . . . . . . . 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 835 . . . . . . 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 690 . . . . . 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 2893 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ‘1))
217 seqp1 13234 . . . . . . . 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 13191 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (1...(2↑(𝑗 + 1))) ∈ Fin)
220 elfznn 12786 . . . . . . . . . . . 12 (𝑘 ∈ (1...(2↑(𝑗 + 1))) → 𝑘 ∈ ℕ)
22138recnd 10515 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℂ)
222150, 220, 221syl2an 595 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑(𝑗 + 1)))) → (𝐹𝑘) ∈ ℂ)
223140, 127, 219, 222fsumsplit 14930 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...(2↑(𝑗 + 1)))(𝐹𝑘) = (Σ𝑘 ∈ (1...(2↑𝑗))(𝐹𝑘) + Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)))
224 eqidd 2796 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑(𝑗 + 1)))) → (𝐹𝑘) = (𝐹𝑘))
22599, 64syl6eleq 2893 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ (ℤ‘1))
226224, 225, 222fsumser 14920 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...(2↑(𝑗 + 1)))(𝐹𝑘) = (seq1( + , 𝐹)‘(2↑(𝑗 + 1))))
227 eqidd 2796 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑𝑗))) → (𝐹𝑘) = (𝐹𝑘))
228 elfznn 12786 . . . . . . . . . . . . 13 (𝑘 ∈ (1...(2↑𝑗)) → 𝑘 ∈ ℕ)
229150, 228, 221syl2an 595 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑𝑗))) → (𝐹𝑘) ∈ ℂ)
230227, 121, 229fsumser 14920 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...(2↑𝑗))(𝐹𝑘) = (seq1( + , 𝐹)‘(2↑𝑗)))
231230oveq1d 7031 . . . . . . . . . 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 2839 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘(2↑(𝑗 + 1))) = ((seq1( + , 𝐹)‘(2↑𝑗)) + Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘)))
233232oveq2d 7032 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1)))) = (2 · ((seq1( + , 𝐹)‘(2↑𝑗)) + Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘))))
234208recnd 10515 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘(2↑𝑗)) ∈ ℂ)
235181recnd 10515 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹𝑘) ∈ ℂ)
23694, 234, 235adddid 10511 . . . . . . . 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 2831 . . . . . . 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 4975 . . . . . 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 260 . . . . 5 ((𝜑𝑗 ∈ ℕ) → ((seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗))) → (seq1( + , 𝐺)‘(𝑗 + 1)) ≤ (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1))))))
240239expcom 414 . . . 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 11504 . 2 (𝑁 ∈ ℕ → (𝜑 → (seq1( + , 𝐺)‘𝑁) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑁)))))
243242impcom 408 1 ((𝜑𝑁 ∈ ℕ) → (seq1( + , 𝐺)‘𝑁) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑁))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wcel 2081  wral 3105  cun 3857  cin 3858  c0 4211   class class class wbr 4962  wf 6221  cfv 6225  (class class class)co 7016  Fincfn 8357  cc 10381  cr 10382  0cc0 10383  1c1 10384   + caddc 10386   · cmul 10388   < clt 10521  cle 10522  cmin 10717  cn 11486  2c2 11540  0cn0 11745  cz 11829  cuz 12093  +crp 12239  ...cfz 12742  seqcseq 13219  cexp 13279  chash 13540  Σcsu 14876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460  ax-pre-sup 10461
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-sup 8752  df-oi 8820  df-dju 9176  df-card 9214  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-div 11146  df-nn 11487  df-2 11548  df-3 11549  df-n0 11746  df-z 11830  df-uz 12094  df-rp 12240  df-ico 12594  df-fz 12743  df-fzo 12884  df-seq 13220  df-exp 13280  df-hash 13541  df-cj 14292  df-re 14293  df-im 14294  df-sqrt 14428  df-abs 14429  df-clim 14679  df-sum 14877
This theorem is referenced by:  climcnds  15039
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