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Theorem climcndslem2 15800

Description: Lemma for climcnds 15801: 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	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ)
climcnds.2	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝐹‘𝑘))
climcnds.3	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
climcnds.4	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))

**Assertion**
Ref	Expression
climcndslem2	⊢ ((𝜑 ∧ 𝑁 ∈ 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.

Step	Hyp	Ref	Expression
1		fveq2 6890	. . . . 5 ⊢ (𝑥 = 1 → (seq1( + , 𝐺)‘𝑥) = (seq1( + , 𝐺)‘1))
2		oveq2 7419	. . . . . . . 8 ⊢ (𝑥 = 1 → (2↑𝑥) = (2↑1))
3		2cn 12291	. . . . . . . . 9 ⊢ 2 ∈ ℂ
4		exp1 14037	. . . . . . . . 9 ⊢ (2 ∈ ℂ → (2↑1) = 2)
5	3, 4	ax-mp 5	. . . . . . . 8 ⊢ (2↑1) = 2
6	2, 5	eqtrdi 2786	. . . . . . 7 ⊢ (𝑥 = 1 → (2↑𝑥) = 2)
7	6	fveq2d 6894	. . . . . 6 ⊢ (𝑥 = 1 → (seq1( + , 𝐹)‘(2↑𝑥)) = (seq1( + , 𝐹)‘2))
8	7	oveq2d 7427	. . . . 5 ⊢ (𝑥 = 1 → (2 · (seq1( + , 𝐹)‘(2↑𝑥))) = (2 · (seq1( + , 𝐹)‘2)))
9	1, 8	breq12d 5160	. . . 4 ⊢ (𝑥 = 1 → ((seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥))) ↔ (seq1( + , 𝐺)‘1) ≤ (2 · (seq1( + , 𝐹)‘2))))
10	9	imbi2d 339	. . 3 ⊢ (𝑥 = 1 → ((𝜑 → (seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥)))) ↔ (𝜑 → (seq1( + , 𝐺)‘1) ≤ (2 · (seq1( + , 𝐹)‘2)))))
11		fveq2 6890	. . . . 5 ⊢ (𝑥 = 𝑗 → (seq1( + , 𝐺)‘𝑥) = (seq1( + , 𝐺)‘𝑗))
12		oveq2 7419	. . . . . . 7 ⊢ (𝑥 = 𝑗 → (2↑𝑥) = (2↑𝑗))
13	12	fveq2d 6894	. . . . . 6 ⊢ (𝑥 = 𝑗 → (seq1( + , 𝐹)‘(2↑𝑥)) = (seq1( + , 𝐹)‘(2↑𝑗)))
14	13	oveq2d 7427	. . . . 5 ⊢ (𝑥 = 𝑗 → (2 · (seq1( + , 𝐹)‘(2↑𝑥))) = (2 · (seq1( + , 𝐹)‘(2↑𝑗))))
15	11, 14	breq12d 5160	. . . 4 ⊢ (𝑥 = 𝑗 → ((seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥))) ↔ (seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗)))))
16	15	imbi2d 339	. . 3 ⊢ (𝑥 = 𝑗 → ((𝜑 → (seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥)))) ↔ (𝜑 → (seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗))))))
17		fveq2 6890	. . . . 5 ⊢ (𝑥 = (𝑗 + 1) → (seq1( + , 𝐺)‘𝑥) = (seq1( + , 𝐺)‘(𝑗 + 1)))
18		oveq2 7419	. . . . . . 7 ⊢ (𝑥 = (𝑗 + 1) → (2↑𝑥) = (2↑(𝑗 + 1)))
19	18	fveq2d 6894	. . . . . 6 ⊢ (𝑥 = (𝑗 + 1) → (seq1( + , 𝐹)‘(2↑𝑥)) = (seq1( + , 𝐹)‘(2↑(𝑗 + 1))))
20	19	oveq2d 7427	. . . . 5 ⊢ (𝑥 = (𝑗 + 1) → (2 · (seq1( + , 𝐹)‘(2↑𝑥))) = (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1)))))
21	17, 20	breq12d 5160	. . . 4 ⊢ (𝑥 = (𝑗 + 1) → ((seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥))) ↔ (seq1( + , 𝐺)‘(𝑗 + 1)) ≤ (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1))))))
22	21	imbi2d 339	. . 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 6890	. . . . 5 ⊢ (𝑥 = 𝑁 → (seq1( + , 𝐺)‘𝑥) = (seq1( + , 𝐺)‘𝑁))
24		oveq2 7419	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (2↑𝑥) = (2↑𝑁))
25	24	fveq2d 6894	. . . . . 6 ⊢ (𝑥 = 𝑁 → (seq1( + , 𝐹)‘(2↑𝑥)) = (seq1( + , 𝐹)‘(2↑𝑁)))
26	25	oveq2d 7427	. . . . 5 ⊢ (𝑥 = 𝑁 → (2 · (seq1( + , 𝐹)‘(2↑𝑥))) = (2 · (seq1( + , 𝐹)‘(2↑𝑁))))
27	23, 26	breq12d 5160	. . . 4 ⊢ (𝑥 = 𝑁 → ((seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥))) ↔ (seq1( + , 𝐺)‘𝑁) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑁)))))
28	27	imbi2d 339	. . 3 ⊢ (𝑥 = 𝑁 → ((𝜑 → (seq1( + , 𝐺)‘𝑥) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑥)))) ↔ (𝜑 → (seq1( + , 𝐺)‘𝑁) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑁))))))
29		fveq2 6890	. . . . . . . 8 ⊢ (𝑘 = 1 → (𝐹‘𝑘) = (𝐹‘1))
30	29	breq2d 5159	. . . . . . 7 ⊢ (𝑘 = 1 → (0 ≤ (𝐹‘𝑘) ↔ 0 ≤ (𝐹‘1)))
31		climcnds.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝐹‘𝑘))
32	31	ralrimiva 3144	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ ℕ 0 ≤ (𝐹‘𝑘))
33		1nn 12227	. . . . . . . 8 ⊢ 1 ∈ ℕ
34	33	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℕ)
35	30, 32, 34	rspcdva 3612	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝐹‘1))
36		fveq2 6890	. . . . . . . . 9 ⊢ (𝑘 = 2 → (𝐹‘𝑘) = (𝐹‘2))
37	36	eleq1d 2816	. . . . . . . 8 ⊢ (𝑘 = 2 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘2) ∈ ℝ))
38		climcnds.1	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ)
39	38	ralrimiva 3144	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℝ)
40		2nn 12289	. . . . . . . . 9 ⊢ 2 ∈ ℕ
41	40	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℕ)
42	37, 39, 41	rspcdva 3612	. . . . . . 7 ⊢ (𝜑 → (𝐹‘2) ∈ ℝ)
43	29	eleq1d 2816	. . . . . . . 8 ⊢ (𝑘 = 1 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘1) ∈ ℝ))
44	43, 39, 34	rspcdva 3612	. . . . . . 7 ⊢ (𝜑 → (𝐹‘1) ∈ ℝ)
45	42, 44	addge02d 11807	. . . . . 6 ⊢ (𝜑 → (0 ≤ (𝐹‘1) ↔ (𝐹‘2) ≤ ((𝐹‘1) + (𝐹‘2))))
46	35, 45	mpbid 231	. . . . 5 ⊢ (𝜑 → (𝐹‘2) ≤ ((𝐹‘1) + (𝐹‘2)))
47	44, 42	readdcld 11247	. . . . . 6 ⊢ (𝜑 → ((𝐹‘1) + (𝐹‘2)) ∈ ℝ)
48	41	nnrpd 13018	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ⁺)
49	42, 47, 48	lemul2d 13064	. . . . 5 ⊢ (𝜑 → ((𝐹‘2) ≤ ((𝐹‘1) + (𝐹‘2)) ↔ (2 · (𝐹‘2)) ≤ (2 · ((𝐹‘1) + (𝐹‘2)))))
50	46, 49	mpbid 231	. . . 4 ⊢ (𝜑 → (2 · (𝐹‘2)) ≤ (2 · ((𝐹‘1) + (𝐹‘2))))
51		1z 12596	. . . . 5 ⊢ 1 ∈ ℤ
52		fveq2 6890	. . . . . . 7 ⊢ (𝑛 = 1 → (𝐺‘𝑛) = (𝐺‘1))
53		oveq2 7419	. . . . . . . . 9 ⊢ (𝑛 = 1 → (2↑𝑛) = (2↑1))
54	53, 5	eqtrdi 2786	. . . . . . . 8 ⊢ (𝑛 = 1 → (2↑𝑛) = 2)
55	54	fveq2d 6894	. . . . . . . 8 ⊢ (𝑛 = 1 → (𝐹‘(2↑𝑛)) = (𝐹‘2))
56	54, 55	oveq12d 7429	. . . . . . 7 ⊢ (𝑛 = 1 → ((2↑𝑛) · (𝐹‘(2↑𝑛))) = (2 · (𝐹‘2)))
57	52, 56	eqeq12d 2746	. . . . . 6 ⊢ (𝑛 = 1 → ((𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))) ↔ (𝐺‘1) = (2 · (𝐹‘2))))
58		climcnds.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))
59	58	ralrimiva 3144	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ₀ (𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))
60		1nn0 12492	. . . . . . 7 ⊢ 1 ∈ ℕ₀
61	60	a1i 11	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℕ₀)
62	57, 59, 61	rspcdva 3612	. . . . 5 ⊢ (𝜑 → (𝐺‘1) = (2 · (𝐹‘2)))
63	51, 62	seq1i 13984	. . . 4 ⊢ (𝜑 → (seq1( + , 𝐺)‘1) = (2 · (𝐹‘2)))
64		nnuz 12869	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
65		df-2 12279	. . . . . 6 ⊢ 2 = (1 + 1)
66		eqidd 2731	. . . . . . 7 ⊢ (𝜑 → (𝐹‘1) = (𝐹‘1))
67	51, 66	seq1i 13984	. . . . . 6 ⊢ (𝜑 → (seq1( + , 𝐹)‘1) = (𝐹‘1))
68		eqidd 2731	. . . . . 6 ⊢ (𝜑 → (𝐹‘2) = (𝐹‘2))
69	64, 34, 65, 67, 68	seqp1d 13987	. . . . 5 ⊢ (𝜑 → (seq1( + , 𝐹)‘2) = ((𝐹‘1) + (𝐹‘2)))
70	69	oveq2d 7427	. . . 4 ⊢ (𝜑 → (2 · (seq1( + , 𝐹)‘2)) = (2 · ((𝐹‘1) + (𝐹‘2))))
71	50, 63, 70	3brtr4d 5179	. . 3 ⊢ (𝜑 → (seq1( + , 𝐺)‘1) ≤ (2 · (seq1( + , 𝐹)‘2)))
72		fveq2 6890	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑗 + 1) → (𝐺‘𝑛) = (𝐺‘(𝑗 + 1)))
73		oveq2 7419	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑗 + 1) → (2↑𝑛) = (2↑(𝑗 + 1)))
74	73	fveq2d 6894	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑗 + 1) → (𝐹‘(2↑𝑛)) = (𝐹‘(2↑(𝑗 + 1))))
75	73, 74	oveq12d 7429	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑗 + 1) → ((2↑𝑛) · (𝐹‘(2↑𝑛))) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))))
76	72, 75	eqeq12d 2746	. . . . . . . . . 10 ⊢ (𝑛 = (𝑗 + 1) → ((𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))) ↔ (𝐺‘(𝑗 + 1)) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1))))))
77	59	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∀𝑛 ∈ ℕ₀ (𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))
78		peano2nn 12228	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
79	78	adantl 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ)
80	79	nnnn0d 12536	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ₀)
81	76, 77, 80	rspcdva 3612	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐺‘(𝑗 + 1)) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))))
82		nnnn0 12483	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℕ₀)
83	82	adantl 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ₀)
84		expp1 14038	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℂ ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) = ((2↑𝑗) · 2))
85	3, 83, 84	sylancr 585	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) = ((2↑𝑗) · 2))
86		nnexpcl 14044	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) → (2↑𝑗) ∈ ℕ)
87	40, 82, 86	sylancr 585	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → (2↑𝑗) ∈ ℕ)
88	87	adantl 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℕ)
89	88	nncnd 12232	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℂ)
90		mulcom 11198	. . . . . . . . . . . 12 ⊢ (((2↑𝑗) ∈ ℂ ∧ 2 ∈ ℂ) → ((2↑𝑗) · 2) = (2 · (2↑𝑗)))
91	89, 3, 90	sylancl 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((2↑𝑗) · 2) = (2 · (2↑𝑗)))
92	85, 91	eqtrd 2770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) = (2 · (2↑𝑗)))
93	92	oveq1d 7426	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))) = ((2 · (2↑𝑗)) · (𝐹‘(2↑(𝑗 + 1)))))
94	3	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 2 ∈ ℂ)
95		fveq2 6890	. . . . . . . . . . . . 13 ⊢ (𝑘 = (2↑(𝑗 + 1)) → (𝐹‘𝑘) = (𝐹‘(2↑(𝑗 + 1))))
96	95	eleq1d 2816	. . . . . . . . . . . 12 ⊢ (𝑘 = (2↑(𝑗 + 1)) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ))
97	39	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℝ)
98		nnexpcl 14044	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℕ ∧ (𝑗 + 1) ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℕ)
99	40, 80, 98	sylancr 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ ℕ)
100	96, 97, 99	rspcdva 3612	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ)
101	100	recnd 11246	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℂ)
102	94, 89, 101	mulassd 11241	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((2 · (2↑𝑗)) · (𝐹‘(2↑(𝑗 + 1)))) = (2 · ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1))))))
103	81, 93, 102	3eqtrd 2774	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐺‘(𝑗 + 1)) = (2 · ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1))))))
104	88	nnnn0d 12536	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℕ₀)
105		hashfz1 14310	. . . . . . . . . . . . . . 15 ⊢ ((2↑𝑗) ∈ ℕ₀ → (♯‘(1...(2↑𝑗))) = (2↑𝑗))
106	104, 105	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (♯‘(1...(2↑𝑗))) = (2↑𝑗))
107	106, 89	eqeltrd 2831	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (♯‘(1...(2↑𝑗))) ∈ ℂ)
108		fzfid 13942	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((2↑𝑗) + 1)...(2↑(𝑗 + 1))) ∈ Fin)
109		hashcl 14320	. . . . . . . . . . . . . . 15 ⊢ ((((2↑𝑗) + 1)...(2↑(𝑗 + 1))) ∈ Fin → (♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∈ ℕ₀)
110	108, 109	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∈ ℕ₀)
111	110	nn0cnd 12538	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∈ ℂ)
112		simpr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
113	112	nnzd 12589	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℤ)
114		uzid 12841	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ_≥‘𝑗))
115		peano2uz 12889	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (ℤ_≥‘𝑗) → (𝑗 + 1) ∈ (ℤ_≥‘𝑗))
116		2re 12290	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℝ
117		1le2 12425	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ≤ 2
118		leexp2a 14141	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2 ∈ ℝ ∧ 1 ≤ 2 ∧ (𝑗 + 1) ∈ (ℤ_≥‘𝑗)) → (2↑𝑗) ≤ (2↑(𝑗 + 1)))
119	116, 117, 118	mp3an12 1449	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 + 1) ∈ (ℤ_≥‘𝑗) → (2↑𝑗) ≤ (2↑(𝑗 + 1)))
120	113, 114, 115, 119	4syl 19	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2↑𝑗) ≤ (2↑(𝑗 + 1)))
121	88, 64	eleqtrdi 2841	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2↑𝑗) ∈ (ℤ_≥‘1))
122	99	nnzd 12589	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ ℤ)
123		elfz5 13497	. . . . . . . . . . . . . . . . . 18 ⊢ (((2↑𝑗) ∈ (ℤ_≥‘1) ∧ (2↑(𝑗 + 1)) ∈ ℤ) → ((2↑𝑗) ∈ (1...(2↑(𝑗 + 1))) ↔ (2↑𝑗) ≤ (2↑(𝑗 + 1))))
124	121, 122, 123	syl2anc 582	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((2↑𝑗) ∈ (1...(2↑(𝑗 + 1))) ↔ (2↑𝑗) ≤ (2↑(𝑗 + 1))))
125	120, 124	mpbird 256	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2↑𝑗) ∈ (1...(2↑(𝑗 + 1))))
126		fzsplit 13531	. . . . . . . . . . . . . . . 16 ⊢ ((2↑𝑗) ∈ (1...(2↑(𝑗 + 1))) → (1...(2↑(𝑗 + 1))) = ((1...(2↑𝑗)) ∪ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))))
127	125, 126	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (1...(2↑(𝑗 + 1))) = ((1...(2↑𝑗)) ∪ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))))
128	127	fveq2d 6894	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (♯‘(1...(2↑(𝑗 + 1)))) = (♯‘((1...(2↑𝑗)) ∪ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))))))
129	89	times2d 12460	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((2↑𝑗) · 2) = ((2↑𝑗) + (2↑𝑗)))
130	85, 129	eqtrd 2770	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) = ((2↑𝑗) + (2↑𝑗)))
131	99	nnnn0d 12536	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ ℕ₀)
132		hashfz1 14310	. . . . . . . . . . . . . . . 16 ⊢ ((2↑(𝑗 + 1)) ∈ ℕ₀ → (♯‘(1...(2↑(𝑗 + 1)))) = (2↑(𝑗 + 1)))
133	131, 132	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (♯‘(1...(2↑(𝑗 + 1)))) = (2↑(𝑗 + 1)))
134	106	oveq1d 7426	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((♯‘(1...(2↑𝑗))) + (2↑𝑗)) = ((2↑𝑗) + (2↑𝑗)))
135	130, 133, 134	3eqtr4d 2780	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (♯‘(1...(2↑(𝑗 + 1)))) = ((♯‘(1...(2↑𝑗))) + (2↑𝑗)))
136		fzfid 13942	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (1...(2↑𝑗)) ∈ Fin)
137	88	nnred 12231	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2↑𝑗) ∈ ℝ)
138	137	ltp1d 12148	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2↑𝑗) < ((2↑𝑗) + 1))
139		fzdisj 13532	. . . . . . . . . . . . . . . 16 ⊢ ((2↑𝑗) < ((2↑𝑗) + 1) → ((1...(2↑𝑗)) ∩ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) = ∅)
140	138, 139	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((1...(2↑𝑗)) ∩ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) = ∅)
141		hashun 14346	. . . . . . . . . . . . . . 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))))))
142	136, 108, 140, 141	syl3anc 1369	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (♯‘((1...(2↑𝑗)) ∪ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))))) = ((♯‘(1...(2↑𝑗))) + (♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1))))))
143	128, 135, 142	3eqtr3d 2778	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((♯‘(1...(2↑𝑗))) + (2↑𝑗)) = ((♯‘(1...(2↑𝑗))) + (♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1))))))
144	107, 89, 111, 143	addcanad 11423	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2↑𝑗) = (♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))))
145	144	oveq1d 7426	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) = ((♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) · (𝐹‘(2↑(𝑗 + 1)))))
146		fsumconst 15740	. . . . . . . . . . . 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)))))
147	108, 101, 146	syl2anc 582	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘(2↑(𝑗 + 1))) = ((♯‘(((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) · (𝐹‘(2↑(𝑗 + 1)))))
148	145, 147	eqtr4d 2773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) = Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘(2↑(𝑗 + 1))))
149	100	adantr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ)
150		simpl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝜑)
151		peano2nn 12228	. . . . . . . . . . . . . 14 ⊢ ((2↑𝑗) ∈ ℕ → ((2↑𝑗) + 1) ∈ ℕ)
152	88, 151	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((2↑𝑗) + 1) ∈ ℕ)
153		elfzuz 13501	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))) → 𝑘 ∈ (ℤ_≥‘((2↑𝑗) + 1)))
154		eluznn 12906	. . . . . . . . . . . . 13 ⊢ ((((2↑𝑗) + 1) ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘((2↑𝑗) + 1))) → 𝑘 ∈ ℕ)
155	152, 153, 154	syl2an 594	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → 𝑘 ∈ ℕ)
156	150, 155, 38	syl2an2r 681	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (𝐹‘𝑘) ∈ ℝ)
157		elfzuz3 13502	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))) → (2↑(𝑗 + 1)) ∈ (ℤ_≥‘𝑛))
158	157	adantl 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (2↑(𝑗 + 1)) ∈ (ℤ_≥‘𝑛))
159		simplll 771	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...(2↑(𝑗 + 1)))) → 𝜑)
160		elfzuz 13501	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1))) → 𝑛 ∈ (ℤ_≥‘((2↑𝑗) + 1)))
161		eluznn 12906	. . . . . . . . . . . . . . . . 17 ⊢ ((((2↑𝑗) + 1) ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘((2↑𝑗) + 1))) → 𝑛 ∈ ℕ)
162	152, 160, 161	syl2an 594	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → 𝑛 ∈ ℕ)
163		elfzuz 13501	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (𝑛...(2↑(𝑗 + 1))) → 𝑘 ∈ (ℤ_≥‘𝑛))
164		eluznn 12906	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑘 ∈ ℕ)
165	162, 163, 164	syl2an 594	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...(2↑(𝑗 + 1)))) → 𝑘 ∈ ℕ)
166	159, 165, 38	syl2anc 582	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...(2↑(𝑗 + 1)))) → (𝐹‘𝑘) ∈ ℝ)
167		simplll 771	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...((2↑(𝑗 + 1)) − 1))) → 𝜑)
168		elfzuz 13501	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (𝑛...((2↑(𝑗 + 1)) − 1)) → 𝑘 ∈ (ℤ_≥‘𝑛))
169	162, 168, 164	syl2an 594	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...((2↑(𝑗 + 1)) − 1))) → 𝑘 ∈ ℕ)
170		climcnds.3	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
171	167, 169, 170	syl2anc 582	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ (𝑛...((2↑(𝑗 + 1)) − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
172	158, 166, 171	monoord2 14003	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹‘𝑛))
173	172	ralrimiva 3144	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∀𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹‘𝑛))
174		fveq2 6890	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
175	174	breq2d 5159	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → ((𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹‘𝑛) ↔ (𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹‘𝑘)))
176	175	rspccva 3610	. . . . . . . . . . . 12 ⊢ ((∀𝑛 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹‘𝑛) ∧ 𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹‘𝑘))
177	173, 176	sylan 578	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))) → (𝐹‘(2↑(𝑗 + 1))) ≤ (𝐹‘𝑘))
178	108, 149, 156, 177	fsumle 15749	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘(2↑(𝑗 + 1))) ≤ Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘𝑘))
179	148, 178	eqbrtrd 5169	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) ≤ Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘𝑘))
180	137, 100	remulcld 11248	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) ∈ ℝ)
181	108, 156	fsumrecl 15684	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘𝑘) ∈ ℝ)
182		2rp 12983	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ⁺
183	182	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 2 ∈ ℝ⁺)
184	180, 181, 183	lemul2d 13064	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1)))) ≤ Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘𝑘) ↔ (2 · ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1))))) ≤ (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘𝑘))))
185	179, 184	mpbid 231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2 · ((2↑𝑗) · (𝐹‘(2↑(𝑗 + 1))))) ≤ (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘𝑘)))
186	103, 185	eqbrtrd 5169	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐺‘(𝑗 + 1)) ≤ (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘𝑘)))
187		1zzd 12597	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℤ)
188		nnnn0 12483	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
189		simpr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ ℕ₀)
190		nnexpcl 14044	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
191	40, 189, 190	sylancr 585	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
192	191	nnred 12231	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℝ)
193		fveq2 6890	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (2↑𝑛) → (𝐹‘𝑘) = (𝐹‘(2↑𝑛)))
194	193	eleq1d 2816	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (2↑𝑛) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(2↑𝑛)) ∈ ℝ))
195	39	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℝ)
196	194, 195, 191	rspcdva 3612	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐹‘(2↑𝑛)) ∈ ℝ)
197	192, 196	remulcld 11248	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((2↑𝑛) · (𝐹‘(2↑𝑛))) ∈ ℝ)
198	58, 197	eqeltrd 2831	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) ∈ ℝ)
199	188, 198	sylan2 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ℝ)
200	64, 187, 199	serfre 14001	. . . . . . . . 9 ⊢ (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
201	200	ffvelcdmda 7085	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (seq1( + , 𝐺)‘𝑗) ∈ ℝ)
202	72	eleq1d 2816	. . . . . . . . 9 ⊢ (𝑛 = (𝑗 + 1) → ((𝐺‘𝑛) ∈ ℝ ↔ (𝐺‘(𝑗 + 1)) ∈ ℝ))
203	199	ralrimiva 3144	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐺‘𝑛) ∈ ℝ)
204	203	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∀𝑛 ∈ ℕ (𝐺‘𝑛) ∈ ℝ)
205	202, 204, 79	rspcdva 3612	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐺‘(𝑗 + 1)) ∈ ℝ)
206	64, 187, 38	serfre 14001	. . . . . . . . . 10 ⊢ (𝜑 → seq1( + , 𝐹):ℕ⟶ℝ)
207		ffvelcdm 7082	. . . . . . . . . 10 ⊢ ((seq1( + , 𝐹):ℕ⟶ℝ ∧ (2↑𝑗) ∈ ℕ) → (seq1( + , 𝐹)‘(2↑𝑗)) ∈ ℝ)
208	206, 87, 207	syl2an 594	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘(2↑𝑗)) ∈ ℝ)
209		remulcl 11197	. . . . . . . . 9 ⊢ ((2 ∈ ℝ ∧ (seq1( + , 𝐹)‘(2↑𝑗)) ∈ ℝ) → (2 · (seq1( + , 𝐹)‘(2↑𝑗))) ∈ ℝ)
210	116, 208, 209	sylancr 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2 · (seq1( + , 𝐹)‘(2↑𝑗))) ∈ ℝ)
211		remulcl 11197	. . . . . . . . 9 ⊢ ((2 ∈ ℝ ∧ Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘𝑘) ∈ ℝ) → (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘𝑘)) ∈ ℝ)
212	116, 181, 211	sylancr 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘𝑘)) ∈ ℝ)
213		le2add 11700	. . . . . . . 8 ⊢ ((((seq1( + , 𝐺)‘𝑗) ∈ ℝ ∧ (𝐺‘(𝑗 + 1)) ∈ ℝ) ∧ ((2 · (seq1( + , 𝐹)‘(2↑𝑗))) ∈ ℝ ∧ (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘𝑘)) ∈ ℝ)) → (((seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗))) ∧ (𝐺‘(𝑗 + 1)) ≤ (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘𝑘))) → ((seq1( + , 𝐺)‘𝑗) + (𝐺‘(𝑗 + 1))) ≤ ((2 · (seq1( + , 𝐹)‘(2↑𝑗))) + (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘𝑘)))))
214	201, 205, 210, 212, 213	syl22anc 835	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗))) ∧ (𝐺‘(𝑗 + 1)) ≤ (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘𝑘))) → ((seq1( + , 𝐺)‘𝑗) + (𝐺‘(𝑗 + 1))) ≤ ((2 · (seq1( + , 𝐹)‘(2↑𝑗))) + (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘𝑘)))))
215	186, 214	mpan2d 690	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 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)))(𝐹‘𝑘)))))
216	112, 64	eleqtrdi 2841	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ_≥‘1))
217		seqp1 13985	. . . . . . . 8 ⊢ (𝑗 ∈ (ℤ_≥‘1) → (seq1( + , 𝐺)‘(𝑗 + 1)) = ((seq1( + , 𝐺)‘𝑗) + (𝐺‘(𝑗 + 1))))
218	216, 217	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (seq1( + , 𝐺)‘(𝑗 + 1)) = ((seq1( + , 𝐺)‘𝑗) + (𝐺‘(𝑗 + 1))))
219		fzfid 13942	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (1...(2↑(𝑗 + 1))) ∈ Fin)
220		elfznn 13534	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...(2↑(𝑗 + 1))) → 𝑘 ∈ ℕ)
221	38	recnd 11246	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
222	150, 220, 221	syl2an 594	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑(𝑗 + 1)))) → (𝐹‘𝑘) ∈ ℂ)
223	140, 127, 219, 222	fsumsplit 15691	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...(2↑(𝑗 + 1)))(𝐹‘𝑘) = (Σ𝑘 ∈ (1...(2↑𝑗))(𝐹‘𝑘) + Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘𝑘)))
224		eqidd 2731	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑(𝑗 + 1)))) → (𝐹‘𝑘) = (𝐹‘𝑘))
225	99, 64	eleqtrdi 2841	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2↑(𝑗 + 1)) ∈ (ℤ_≥‘1))
226	224, 225, 222	fsumser 15680	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...(2↑(𝑗 + 1)))(𝐹‘𝑘) = (seq1( + , 𝐹)‘(2↑(𝑗 + 1))))
227		eqidd 2731	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑𝑗))) → (𝐹‘𝑘) = (𝐹‘𝑘))
228		elfznn 13534	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...(2↑𝑗)) → 𝑘 ∈ ℕ)
229	150, 228, 221	syl2an 594	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...(2↑𝑗))) → (𝐹‘𝑘) ∈ ℂ)
230	227, 121, 229	fsumser 15680	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...(2↑𝑗))(𝐹‘𝑘) = (seq1( + , 𝐹)‘(2↑𝑗)))
231	230	oveq1d 7426	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (Σ𝑘 ∈ (1...(2↑𝑗))(𝐹‘𝑘) + Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘𝑘)) = ((seq1( + , 𝐹)‘(2↑𝑗)) + Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘𝑘)))
232	223, 226, 231	3eqtr3d 2778	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘(2↑(𝑗 + 1))) = ((seq1( + , 𝐹)‘(2↑𝑗)) + Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘𝑘)))
233	232	oveq2d 7427	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1)))) = (2 · ((seq1( + , 𝐹)‘(2↑𝑗)) + Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘𝑘))))
234	208	recnd 11246	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (seq1( + , 𝐹)‘(2↑𝑗)) ∈ ℂ)
235	181	recnd 11246	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘𝑘) ∈ ℂ)
236	94, 234, 235	adddid 11242	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2 · ((seq1( + , 𝐹)‘(2↑𝑗)) + Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘𝑘))) = ((2 · (seq1( + , 𝐹)‘(2↑𝑗))) + (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘𝑘))))
237	233, 236	eqtrd 2770	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1)))) = ((2 · (seq1( + , 𝐹)‘(2↑𝑗))) + (2 · Σ𝑘 ∈ (((2↑𝑗) + 1)...(2↑(𝑗 + 1)))(𝐹‘𝑘))))
238	218, 237	breq12d 5160	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 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)))(𝐹‘𝑘)))))
239	215, 238	sylibrd 258	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗))) → (seq1( + , 𝐺)‘(𝑗 + 1)) ≤ (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1))))))
240	239	expcom 412	. . . 4 ⊢ (𝑗 ∈ ℕ → (𝜑 → ((seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗))) → (seq1( + , 𝐺)‘(𝑗 + 1)) ≤ (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1)))))))
241	240	a2d 29	. . 3 ⊢ (𝑗 ∈ ℕ → ((𝜑 → (seq1( + , 𝐺)‘𝑗) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑗)))) → (𝜑 → (seq1( + , 𝐺)‘(𝑗 + 1)) ≤ (2 · (seq1( + , 𝐹)‘(2↑(𝑗 + 1)))))))
242	10, 16, 22, 28, 71, 241	nnind 12234	. 2 ⊢ (𝑁 ∈ ℕ → (𝜑 → (seq1( + , 𝐺)‘𝑁) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑁)))))
243	242	impcom 406	1 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (seq1( + , 𝐺)‘𝑁) ≤ (2 · (seq1( + , 𝐹)‘(2↑𝑁))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ∀wral 3059 ∪ cun 3945 ∩ cin 3946 ∅c0 4321 class class class wbr 5147 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 Fincfn 8941 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 < clt 11252 ≤ cle 11253 − cmin 11448 ℕcn 12216 2c2 12271 ℕ₀cn0 12476 ℤcz 12562 ℤ_≥cuz 12826 ℝ⁺crp 12978 ...cfz 13488 seqcseq 13970 ↑cexp 14031 ♯chash 14294 Σcsu 15636

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-oadd 8472 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-oi 9507 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-ico 13334 df-fz 13489 df-fzo 13632 df-seq 13971 df-exp 14032 df-hash 14295 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-sum 15637

This theorem is referenced by: climcnds 15801

W3C validator