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Theorem climcndslem1 15561

Description: Lemma for climcnds 15563: bound the original series by the condensed series. (Contributed by Mario Carneiro, 18-Jul-2014.)

**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
climcndslem1	⊢ ((𝜑 ∧ 𝑁 ∈ ℕ₀) → (seq1( + , 𝐹)‘((2↑(𝑁 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑁))

Distinct variable groups: 𝑘,𝑛,𝐹 𝑘,𝐺,𝑛 𝜑,𝑘,𝑛 Allowed substitution hints: 𝑁(𝑘,𝑛)

Proof of Theorem climcndslem1 Dummy variables 𝑗 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7282	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝑥 + 1) = (0 + 1))
2		0p1e1 12095	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
3	1, 2	eqtrdi 2794	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝑥 + 1) = 1)
4	3	oveq2d 7291	. . . . . . . . 9 ⊢ (𝑥 = 0 → (2↑(𝑥 + 1)) = (2↑1))
5		2cn 12048	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
6		exp1 13788	. . . . . . . . . . 11 ⊢ (2 ∈ ℂ → (2↑1) = 2)
7	5, 6	ax-mp 5	. . . . . . . . . 10 ⊢ (2↑1) = 2
8		df-2 12036	. . . . . . . . . 10 ⊢ 2 = (1 + 1)
9	7, 8	eqtri 2766	. . . . . . . . 9 ⊢ (2↑1) = (1 + 1)
10	4, 9	eqtrdi 2794	. . . . . . . 8 ⊢ (𝑥 = 0 → (2↑(𝑥 + 1)) = (1 + 1))
11	10	oveq1d 7290	. . . . . . 7 ⊢ (𝑥 = 0 → ((2↑(𝑥 + 1)) − 1) = ((1 + 1) − 1))
12		ax-1cn 10929	. . . . . . . 8 ⊢ 1 ∈ ℂ
13	12, 12	pncan3oi 11237	. . . . . . 7 ⊢ ((1 + 1) − 1) = 1
14	11, 13	eqtrdi 2794	. . . . . 6 ⊢ (𝑥 = 0 → ((2↑(𝑥 + 1)) − 1) = 1)
15	14	fveq2d 6778	. . . . 5 ⊢ (𝑥 = 0 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) = (seq1( + , 𝐹)‘1))
16		fveq2 6774	. . . . 5 ⊢ (𝑥 = 0 → (seq0( + , 𝐺)‘𝑥) = (seq0( + , 𝐺)‘0))
17	15, 16	breq12d 5087	. . . 4 ⊢ (𝑥 = 0 → ((seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑥) ↔ (seq1( + , 𝐹)‘1) ≤ (seq0( + , 𝐺)‘0)))
18	17	imbi2d 341	. . 3 ⊢ (𝑥 = 0 → ((𝜑 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑥)) ↔ (𝜑 → (seq1( + , 𝐹)‘1) ≤ (seq0( + , 𝐺)‘0))))
19		oveq1 7282	. . . . . . 7 ⊢ (𝑥 = 𝑗 → (𝑥 + 1) = (𝑗 + 1))
20	19	oveq2d 7291	. . . . . 6 ⊢ (𝑥 = 𝑗 → (2↑(𝑥 + 1)) = (2↑(𝑗 + 1)))
21	20	fvoveq1d 7297	. . . . 5 ⊢ (𝑥 = 𝑗 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) = (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)))
22		fveq2 6774	. . . . 5 ⊢ (𝑥 = 𝑗 → (seq0( + , 𝐺)‘𝑥) = (seq0( + , 𝐺)‘𝑗))
23	21, 22	breq12d 5087	. . . 4 ⊢ (𝑥 = 𝑗 → ((seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑥) ↔ (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑗)))
24	23	imbi2d 341	. . 3 ⊢ (𝑥 = 𝑗 → ((𝜑 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑥)) ↔ (𝜑 → (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑗))))
25		oveq1 7282	. . . . . . 7 ⊢ (𝑥 = (𝑗 + 1) → (𝑥 + 1) = ((𝑗 + 1) + 1))
26	25	oveq2d 7291	. . . . . 6 ⊢ (𝑥 = (𝑗 + 1) → (2↑(𝑥 + 1)) = (2↑((𝑗 + 1) + 1)))
27	26	fvoveq1d 7297	. . . . 5 ⊢ (𝑥 = (𝑗 + 1) → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) = (seq1( + , 𝐹)‘((2↑((𝑗 + 1) + 1)) − 1)))
28		fveq2 6774	. . . . 5 ⊢ (𝑥 = (𝑗 + 1) → (seq0( + , 𝐺)‘𝑥) = (seq0( + , 𝐺)‘(𝑗 + 1)))
29	27, 28	breq12d 5087	. . . 4 ⊢ (𝑥 = (𝑗 + 1) → ((seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑥) ↔ (seq1( + , 𝐹)‘((2↑((𝑗 + 1) + 1)) − 1)) ≤ (seq0( + , 𝐺)‘(𝑗 + 1))))
30	29	imbi2d 341	. . 3 ⊢ (𝑥 = (𝑗 + 1) → ((𝜑 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑥)) ↔ (𝜑 → (seq1( + , 𝐹)‘((2↑((𝑗 + 1) + 1)) − 1)) ≤ (seq0( + , 𝐺)‘(𝑗 + 1)))))
31		oveq1 7282	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑥 + 1) = (𝑁 + 1))
32	31	oveq2d 7291	. . . . . 6 ⊢ (𝑥 = 𝑁 → (2↑(𝑥 + 1)) = (2↑(𝑁 + 1)))
33	32	fvoveq1d 7297	. . . . 5 ⊢ (𝑥 = 𝑁 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) = (seq1( + , 𝐹)‘((2↑(𝑁 + 1)) − 1)))
34		fveq2 6774	. . . . 5 ⊢ (𝑥 = 𝑁 → (seq0( + , 𝐺)‘𝑥) = (seq0( + , 𝐺)‘𝑁))
35	33, 34	breq12d 5087	. . . 4 ⊢ (𝑥 = 𝑁 → ((seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑥) ↔ (seq1( + , 𝐹)‘((2↑(𝑁 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑁)))
36	35	imbi2d 341	. . 3 ⊢ (𝑥 = 𝑁 → ((𝜑 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑥)) ↔ (𝜑 → (seq1( + , 𝐹)‘((2↑(𝑁 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑁))))
37		fveq2 6774	. . . . . . . 8 ⊢ (𝑘 = 1 → (𝐹‘𝑘) = (𝐹‘1))
38	37	eleq1d 2823	. . . . . . 7 ⊢ (𝑘 = 1 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘1) ∈ ℝ))
39		climcnds.1	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ)
40	39	ralrimiva 3103	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℝ)
41		1nn 11984	. . . . . . . 8 ⊢ 1 ∈ ℕ
42	41	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℕ)
43	38, 40, 42	rspcdva 3562	. . . . . 6 ⊢ (𝜑 → (𝐹‘1) ∈ ℝ)
44	43	leidd 11541	. . . . 5 ⊢ (𝜑 → (𝐹‘1) ≤ (𝐹‘1))
45	43	recnd 11003	. . . . . 6 ⊢ (𝜑 → (𝐹‘1) ∈ ℂ)
46	45	mulid2d 10993	. . . . 5 ⊢ (𝜑 → (1 · (𝐹‘1)) = (𝐹‘1))
47	44, 46	breqtrrd 5102	. . . 4 ⊢ (𝜑 → (𝐹‘1) ≤ (1 · (𝐹‘1)))
48		1z 12350	. . . . 5 ⊢ 1 ∈ ℤ
49		eqidd 2739	. . . . 5 ⊢ (𝜑 → (𝐹‘1) = (𝐹‘1))
50	48, 49	seq1i 13735	. . . 4 ⊢ (𝜑 → (seq1( + , 𝐹)‘1) = (𝐹‘1))
51		0z 12330	. . . . 5 ⊢ 0 ∈ ℤ
52		fveq2 6774	. . . . . . 7 ⊢ (𝑛 = 0 → (𝐺‘𝑛) = (𝐺‘0))
53		oveq2 7283	. . . . . . . . 9 ⊢ (𝑛 = 0 → (2↑𝑛) = (2↑0))
54		exp0 13786	. . . . . . . . . 10 ⊢ (2 ∈ ℂ → (2↑0) = 1)
55	5, 54	ax-mp 5	. . . . . . . . 9 ⊢ (2↑0) = 1
56	53, 55	eqtrdi 2794	. . . . . . . 8 ⊢ (𝑛 = 0 → (2↑𝑛) = 1)
57	56	fveq2d 6778	. . . . . . . 8 ⊢ (𝑛 = 0 → (𝐹‘(2↑𝑛)) = (𝐹‘1))
58	56, 57	oveq12d 7293	. . . . . . 7 ⊢ (𝑛 = 0 → ((2↑𝑛) · (𝐹‘(2↑𝑛))) = (1 · (𝐹‘1)))
59	52, 58	eqeq12d 2754	. . . . . 6 ⊢ (𝑛 = 0 → ((𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))) ↔ (𝐺‘0) = (1 · (𝐹‘1))))
60		climcnds.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))
61	60	ralrimiva 3103	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ₀ (𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))
62		0nn0 12248	. . . . . . 7 ⊢ 0 ∈ ℕ₀
63	62	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℕ₀)
64	59, 61, 63	rspcdva 3562	. . . . 5 ⊢ (𝜑 → (𝐺‘0) = (1 · (𝐹‘1)))
65	51, 64	seq1i 13735	. . . 4 ⊢ (𝜑 → (seq0( + , 𝐺)‘0) = (1 · (𝐹‘1)))
66	47, 50, 65	3brtr4d 5106	. . 3 ⊢ (𝜑 → (seq1( + , 𝐹)‘1) ≤ (seq0( + , 𝐺)‘0))
67		fzfid 13693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1)) ∈ Fin)
68		simpl 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 𝜑)
69		2nn 12046	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ
70		peano2nn0 12273	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ₀ → (𝑗 + 1) ∈ ℕ₀)
71	70	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝑗 + 1) ∈ ℕ₀)
72		nnexpcl 13795	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℕ ∧ (𝑗 + 1) ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℕ)
73	69, 71, 72	sylancr 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℕ)
74		elfzuz 13252	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1)) → 𝑘 ∈ (ℤ_≥‘(2↑(𝑗 + 1))))
75		eluznn 12658	. . . . . . . . . . 11 ⊢ (((2↑(𝑗 + 1)) ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) → 𝑘 ∈ ℕ)
76	73, 74, 75	syl2an 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) → 𝑘 ∈ ℕ)
77	68, 76, 39	syl2an2r 682	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) → (𝐹‘𝑘) ∈ ℝ)
78		fveq2 6774	. . . . . . . . . . . 12 ⊢ (𝑘 = (2↑(𝑗 + 1)) → (𝐹‘𝑘) = (𝐹‘(2↑(𝑗 + 1))))
79	78	eleq1d 2823	. . . . . . . . . . 11 ⊢ (𝑘 = (2↑(𝑗 + 1)) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ))
80	40	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℝ)
81	79, 80, 73	rspcdva 3562	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ)
82	81	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ)
83		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) → 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1))))
84		simplll 772	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...𝑛)) → 𝜑)
85	73	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) → (2↑(𝑗 + 1)) ∈ ℕ)
86		elfzuz 13252	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ((2↑(𝑗 + 1))...𝑛) → 𝑘 ∈ (ℤ_≥‘(2↑(𝑗 + 1))))
87	85, 86, 75	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...𝑛)) → 𝑘 ∈ ℕ)
88	84, 87, 39	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...𝑛)) → (𝐹‘𝑘) ∈ ℝ)
89		simplll 772	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...(𝑛 − 1))) → 𝜑)
90		elfzuz 13252	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ((2↑(𝑗 + 1))...(𝑛 − 1)) → 𝑘 ∈ (ℤ_≥‘(2↑(𝑗 + 1))))
91	85, 90, 75	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...(𝑛 − 1))) → 𝑘 ∈ ℕ)
92		climcnds.3	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
93	89, 91, 92	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...(𝑛 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
94	83, 88, 93	monoord2 13754	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) → (𝐹‘𝑛) ≤ (𝐹‘(2↑(𝑗 + 1))))
95	94	ralrimiva 3103	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ∀𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))(𝐹‘𝑛) ≤ (𝐹‘(2↑(𝑗 + 1))))
96		fveq2 6774	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
97	96	breq1d 5084	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → ((𝐹‘𝑛) ≤ (𝐹‘(2↑(𝑗 + 1))) ↔ (𝐹‘𝑘) ≤ (𝐹‘(2↑(𝑗 + 1)))))
98	97	rspccva 3560	. . . . . . . . . 10 ⊢ ((∀𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))(𝐹‘𝑛) ≤ (𝐹‘(2↑(𝑗 + 1))) ∧ 𝑘 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) → (𝐹‘𝑘) ≤ (𝐹‘(2↑(𝑗 + 1))))
99	95, 74, 98	syl2an 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) → (𝐹‘𝑘) ≤ (𝐹‘(2↑(𝑗 + 1))))
100	67, 77, 82, 99	fsumle 15511	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘) ≤ Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘(2↑(𝑗 + 1))))
101		fzfid 13693	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (1...((2↑(𝑗 + 1)) − 1)) ∈ Fin)
102		hashcl 14071	. . . . . . . . . . . . 13 ⊢ ((1...((2↑(𝑗 + 1)) − 1)) ∈ Fin → (♯‘(1...((2↑(𝑗 + 1)) − 1))) ∈ ℕ₀)
103	101, 102	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (♯‘(1...((2↑(𝑗 + 1)) − 1))) ∈ ℕ₀)
104	103	nn0cnd 12295	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (♯‘(1...((2↑(𝑗 + 1)) − 1))) ∈ ℂ)
105	73	nnred 11988	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℝ)
106	105	recnd 11003	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℂ)
107		hashcl 14071	. . . . . . . . . . . . 13 ⊢ (((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1)) ∈ Fin → (♯‘((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) ∈ ℕ₀)
108	67, 107	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (♯‘((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) ∈ ℕ₀)
109	108	nn0cnd 12295	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (♯‘((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) ∈ ℂ)
110		2z 12352	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℤ
111		zexpcl 13797	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ ℤ ∧ (𝑗 + 1) ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℤ)
112	110, 71, 111	sylancr 587	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℤ)
113		2re 12047	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℝ
114		1le2 12182	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ≤ 2
115		nn0p1nn 12272	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ ℕ₀ → (𝑗 + 1) ∈ ℕ)
116	115	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝑗 + 1) ∈ ℕ)
117		nnuz 12621	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℕ = (ℤ_≥‘1)
118	116, 117	eleqtrdi 2849	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝑗 + 1) ∈ (ℤ_≥‘1))
119		leexp2a 13890	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2 ∈ ℝ ∧ 1 ≤ 2 ∧ (𝑗 + 1) ∈ (ℤ_≥‘1)) → (2↑1) ≤ (2↑(𝑗 + 1)))
120	113, 114, 118, 119	mp3an12i 1464	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑1) ≤ (2↑(𝑗 + 1)))
121	7, 120	eqbrtrrid 5110	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 2 ≤ (2↑(𝑗 + 1)))
122	110	eluz1i 12590	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2↑(𝑗 + 1)) ∈ (ℤ_≥‘2) ↔ ((2↑(𝑗 + 1)) ∈ ℤ ∧ 2 ≤ (2↑(𝑗 + 1))))
123	112, 121, 122	sylanbrc 583	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ (ℤ_≥‘2))
124		uz2m1nn 12663	. . . . . . . . . . . . . . . . . 18 ⊢ ((2↑(𝑗 + 1)) ∈ (ℤ_≥‘2) → ((2↑(𝑗 + 1)) − 1) ∈ ℕ)
125	123, 124	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) − 1) ∈ ℕ)
126	125, 117	eleqtrdi 2849	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) − 1) ∈ (ℤ_≥‘1))
127		peano2zm 12363	. . . . . . . . . . . . . . . . . 18 ⊢ ((2↑(𝑗 + 1)) ∈ ℤ → ((2↑(𝑗 + 1)) − 1) ∈ ℤ)
128	112, 127	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) − 1) ∈ ℤ)
129		peano2nn0 12273	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 + 1) ∈ ℕ₀ → ((𝑗 + 1) + 1) ∈ ℕ₀)
130	71, 129	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((𝑗 + 1) + 1) ∈ ℕ₀)
131		zexpcl 13797	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2 ∈ ℤ ∧ ((𝑗 + 1) + 1) ∈ ℕ₀) → (2↑((𝑗 + 1) + 1)) ∈ ℤ)
132	110, 130, 131	sylancr 587	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑((𝑗 + 1) + 1)) ∈ ℤ)
133		peano2zm 12363	. . . . . . . . . . . . . . . . . 18 ⊢ ((2↑((𝑗 + 1) + 1)) ∈ ℤ → ((2↑((𝑗 + 1) + 1)) − 1) ∈ ℤ)
134	132, 133	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑((𝑗 + 1) + 1)) − 1) ∈ ℤ)
135	112	zred 12426	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℝ)
136	132	zred 12426	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑((𝑗 + 1) + 1)) ∈ ℝ)
137		1red 10976	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 1 ∈ ℝ)
138	71	nn0zd 12424	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝑗 + 1) ∈ ℤ)
139		uzid 12597	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 + 1) ∈ ℤ → (𝑗 + 1) ∈ (ℤ_≥‘(𝑗 + 1)))
140		peano2uz 12641	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 + 1) ∈ (ℤ_≥‘(𝑗 + 1)) → ((𝑗 + 1) + 1) ∈ (ℤ_≥‘(𝑗 + 1)))
141		leexp2a 13890	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ ℝ ∧ 1 ≤ 2 ∧ ((𝑗 + 1) + 1) ∈ (ℤ_≥‘(𝑗 + 1))) → (2↑(𝑗 + 1)) ≤ (2↑((𝑗 + 1) + 1)))
142	113, 114, 141	mp3an12 1450	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑗 + 1) + 1) ∈ (ℤ_≥‘(𝑗 + 1)) → (2↑(𝑗 + 1)) ≤ (2↑((𝑗 + 1) + 1)))
143	138, 139, 140, 142	4syl 19	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) ≤ (2↑((𝑗 + 1) + 1)))
144	135, 136, 137, 143	lesub1dd 11591	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) − 1) ≤ ((2↑((𝑗 + 1) + 1)) − 1))
145		eluz2 12588	. . . . . . . . . . . . . . . . 17 ⊢ (((2↑((𝑗 + 1) + 1)) − 1) ∈ (ℤ_≥‘((2↑(𝑗 + 1)) − 1)) ↔ (((2↑(𝑗 + 1)) − 1) ∈ ℤ ∧ ((2↑((𝑗 + 1) + 1)) − 1) ∈ ℤ ∧ ((2↑(𝑗 + 1)) − 1) ≤ ((2↑((𝑗 + 1) + 1)) − 1)))
146	128, 134, 144, 145	syl3anbrc 1342	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑((𝑗 + 1) + 1)) − 1) ∈ (ℤ_≥‘((2↑(𝑗 + 1)) − 1)))
147		elfzuzb 13250	. . . . . . . . . . . . . . . 16 ⊢ (((2↑(𝑗 + 1)) − 1) ∈ (1...((2↑((𝑗 + 1) + 1)) − 1)) ↔ (((2↑(𝑗 + 1)) − 1) ∈ (ℤ_≥‘1) ∧ ((2↑((𝑗 + 1) + 1)) − 1) ∈ (ℤ_≥‘((2↑(𝑗 + 1)) − 1))))
148	126, 146, 147	sylanbrc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) − 1) ∈ (1...((2↑((𝑗 + 1) + 1)) − 1)))
149		fzsplit 13282	. . . . . . . . . . . . . . 15 ⊢ (((2↑(𝑗 + 1)) − 1) ∈ (1...((2↑((𝑗 + 1) + 1)) − 1)) → (1...((2↑((𝑗 + 1) + 1)) − 1)) = ((1...((2↑(𝑗 + 1)) − 1)) ∪ ((((2↑(𝑗 + 1)) − 1) + 1)...((2↑((𝑗 + 1) + 1)) − 1))))
150	148, 149	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (1...((2↑((𝑗 + 1) + 1)) − 1)) = ((1...((2↑(𝑗 + 1)) − 1)) ∪ ((((2↑(𝑗 + 1)) − 1) + 1)...((2↑((𝑗 + 1) + 1)) − 1))))
151		npcan 11230	. . . . . . . . . . . . . . . . 17 ⊢ (((2↑(𝑗 + 1)) ∈ ℂ ∧ 1 ∈ ℂ) → (((2↑(𝑗 + 1)) − 1) + 1) = (2↑(𝑗 + 1)))
152	106, 12, 151	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (((2↑(𝑗 + 1)) − 1) + 1) = (2↑(𝑗 + 1)))
153	152	oveq1d 7290	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((((2↑(𝑗 + 1)) − 1) + 1)...((2↑((𝑗 + 1) + 1)) − 1)) = ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1)))
154	153	uneq2d 4097	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((1...((2↑(𝑗 + 1)) − 1)) ∪ ((((2↑(𝑗 + 1)) − 1) + 1)...((2↑((𝑗 + 1) + 1)) − 1))) = ((1...((2↑(𝑗 + 1)) − 1)) ∪ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))))
155	150, 154	eqtrd 2778	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (1...((2↑((𝑗 + 1) + 1)) − 1)) = ((1...((2↑(𝑗 + 1)) − 1)) ∪ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))))
156	155	fveq2d 6778	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (♯‘(1...((2↑((𝑗 + 1) + 1)) − 1))) = (♯‘((1...((2↑(𝑗 + 1)) − 1)) ∪ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1)))))
157		expp1 13789	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℂ ∧ (𝑗 + 1) ∈ ℕ₀) → (2↑((𝑗 + 1) + 1)) = ((2↑(𝑗 + 1)) · 2))
158	5, 71, 157	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑((𝑗 + 1) + 1)) = ((2↑(𝑗 + 1)) · 2))
159	106	times2d 12217	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) · 2) = ((2↑(𝑗 + 1)) + (2↑(𝑗 + 1))))
160	158, 159	eqtrd 2778	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑((𝑗 + 1) + 1)) = ((2↑(𝑗 + 1)) + (2↑(𝑗 + 1))))
161	160	oveq1d 7290	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑((𝑗 + 1) + 1)) − 1) = (((2↑(𝑗 + 1)) + (2↑(𝑗 + 1))) − 1))
162		1cnd 10970	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 1 ∈ ℂ)
163	106, 106, 162	addsubd 11353	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (((2↑(𝑗 + 1)) + (2↑(𝑗 + 1))) − 1) = (((2↑(𝑗 + 1)) − 1) + (2↑(𝑗 + 1))))
164	161, 163	eqtrd 2778	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑((𝑗 + 1) + 1)) − 1) = (((2↑(𝑗 + 1)) − 1) + (2↑(𝑗 + 1))))
165		uztrn 12600	. . . . . . . . . . . . . . . . 17 ⊢ ((((2↑((𝑗 + 1) + 1)) − 1) ∈ (ℤ_≥‘((2↑(𝑗 + 1)) − 1)) ∧ ((2↑(𝑗 + 1)) − 1) ∈ (ℤ_≥‘1)) → ((2↑((𝑗 + 1) + 1)) − 1) ∈ (ℤ_≥‘1))
166	146, 126, 165	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑((𝑗 + 1) + 1)) − 1) ∈ (ℤ_≥‘1))
167	166, 117	eleqtrrdi 2850	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑((𝑗 + 1) + 1)) − 1) ∈ ℕ)
168	167	nnnn0d 12293	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑((𝑗 + 1) + 1)) − 1) ∈ ℕ₀)
169		hashfz1 14060	. . . . . . . . . . . . . 14 ⊢ (((2↑((𝑗 + 1) + 1)) − 1) ∈ ℕ₀ → (♯‘(1...((2↑((𝑗 + 1) + 1)) − 1))) = ((2↑((𝑗 + 1) + 1)) − 1))
170	168, 169	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (♯‘(1...((2↑((𝑗 + 1) + 1)) − 1))) = ((2↑((𝑗 + 1) + 1)) − 1))
171	125	nnnn0d 12293	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) − 1) ∈ ℕ₀)
172		hashfz1 14060	. . . . . . . . . . . . . . 15 ⊢ (((2↑(𝑗 + 1)) − 1) ∈ ℕ₀ → (♯‘(1...((2↑(𝑗 + 1)) − 1))) = ((2↑(𝑗 + 1)) − 1))
173	171, 172	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (♯‘(1...((2↑(𝑗 + 1)) − 1))) = ((2↑(𝑗 + 1)) − 1))
174	173	oveq1d 7290	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((♯‘(1...((2↑(𝑗 + 1)) − 1))) + (2↑(𝑗 + 1))) = (((2↑(𝑗 + 1)) − 1) + (2↑(𝑗 + 1))))
175	164, 170, 174	3eqtr4d 2788	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (♯‘(1...((2↑((𝑗 + 1) + 1)) − 1))) = ((♯‘(1...((2↑(𝑗 + 1)) − 1))) + (2↑(𝑗 + 1))))
176	105	ltm1d 11907	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) − 1) < (2↑(𝑗 + 1)))
177		fzdisj 13283	. . . . . . . . . . . . . 14 ⊢ (((2↑(𝑗 + 1)) − 1) < (2↑(𝑗 + 1)) → ((1...((2↑(𝑗 + 1)) − 1)) ∩ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) = ∅)
178	176, 177	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((1...((2↑(𝑗 + 1)) − 1)) ∩ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) = ∅)
179		hashun 14097	. . . . . . . . . . . . 13 ⊢ (((1...((2↑(𝑗 + 1)) − 1)) ∈ Fin ∧ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1)) ∈ Fin ∧ ((1...((2↑(𝑗 + 1)) − 1)) ∩ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) = ∅) → (♯‘((1...((2↑(𝑗 + 1)) − 1)) ∪ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1)))) = ((♯‘(1...((2↑(𝑗 + 1)) − 1))) + (♯‘((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1)))))
180	101, 67, 178, 179	syl3anc 1370	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (♯‘((1...((2↑(𝑗 + 1)) − 1)) ∪ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1)))) = ((♯‘(1...((2↑(𝑗 + 1)) − 1))) + (♯‘((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1)))))
181	156, 175, 180	3eqtr3d 2786	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((♯‘(1...((2↑(𝑗 + 1)) − 1))) + (2↑(𝑗 + 1))) = ((♯‘(1...((2↑(𝑗 + 1)) − 1))) + (♯‘((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1)))))
182	104, 106, 109, 181	addcanad 11180	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) = (♯‘((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))))
183	182	oveq1d 7290	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))) = ((♯‘((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) · (𝐹‘(2↑(𝑗 + 1)))))
184		fveq2 6774	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑗 + 1) → (𝐺‘𝑛) = (𝐺‘(𝑗 + 1)))
185		oveq2 7283	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑗 + 1) → (2↑𝑛) = (2↑(𝑗 + 1)))
186	185	fveq2d 6778	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑗 + 1) → (𝐹‘(2↑𝑛)) = (𝐹‘(2↑(𝑗 + 1))))
187	185, 186	oveq12d 7293	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑗 + 1) → ((2↑𝑛) · (𝐹‘(2↑𝑛))) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))))
188	184, 187	eqeq12d 2754	. . . . . . . . . 10 ⊢ (𝑛 = (𝑗 + 1) → ((𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))) ↔ (𝐺‘(𝑗 + 1)) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1))))))
189	61	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ∀𝑛 ∈ ℕ₀ (𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))
190	188, 189, 71	rspcdva 3562	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐺‘(𝑗 + 1)) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))))
191	81	recnd 11003	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℂ)
192		fsumconst 15502	. . . . . . . . . 10 ⊢ ((((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1)) ∈ Fin ∧ (𝐹‘(2↑(𝑗 + 1))) ∈ ℂ) → Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘(2↑(𝑗 + 1))) = ((♯‘((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) · (𝐹‘(2↑(𝑗 + 1)))))
193	67, 191, 192	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘(2↑(𝑗 + 1))) = ((♯‘((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) · (𝐹‘(2↑(𝑗 + 1)))))
194	183, 190, 193	3eqtr4d 2788	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐺‘(𝑗 + 1)) = Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘(2↑(𝑗 + 1))))
195	100, 194	breqtrrd 5102	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘) ≤ (𝐺‘(𝑗 + 1)))
196		elfznn 13285	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1)) → 𝑘 ∈ ℕ)
197	68, 196, 39	syl2an 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))) → (𝐹‘𝑘) ∈ ℝ)
198	101, 197	fsumrecl 15446	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) ∈ ℝ)
199	67, 77	fsumrecl 15446	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘) ∈ ℝ)
200		nn0uz 12620	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
201		0zd 12331	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℤ)
202		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ ℕ₀)
203		nnexpcl 13795	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
204	69, 202, 203	sylancr 587	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
205	204	nnred 11988	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℝ)
206		fveq2 6774	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (2↑𝑛) → (𝐹‘𝑘) = (𝐹‘(2↑𝑛)))
207	206	eleq1d 2823	. . . . . . . . . . . . 13 ⊢ (𝑘 = (2↑𝑛) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(2↑𝑛)) ∈ ℝ))
208	40	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℝ)
209	207, 208, 204	rspcdva 3562	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐹‘(2↑𝑛)) ∈ ℝ)
210	205, 209	remulcld 11005	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((2↑𝑛) · (𝐹‘(2↑𝑛))) ∈ ℝ)
211	60, 210	eqeltrd 2839	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) ∈ ℝ)
212	200, 201, 211	serfre 13752	. . . . . . . . 9 ⊢ (𝜑 → seq0( + , 𝐺):ℕ₀⟶ℝ)
213	212	ffvelrnda 6961	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (seq0( + , 𝐺)‘𝑗) ∈ ℝ)
214	135, 81	remulcld 11005	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))) ∈ ℝ)
215	190, 214	eqeltrd 2839	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐺‘(𝑗 + 1)) ∈ ℝ)
216		le2add 11457	. . . . . . . 8 ⊢ (((Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) ∈ ℝ ∧ Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘) ∈ ℝ) ∧ ((seq0( + , 𝐺)‘𝑗) ∈ ℝ ∧ (𝐺‘(𝑗 + 1)) ∈ ℝ)) → ((Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) ≤ (seq0( + , 𝐺)‘𝑗) ∧ Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘) ≤ (𝐺‘(𝑗 + 1))) → (Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘)) ≤ ((seq0( + , 𝐺)‘𝑗) + (𝐺‘(𝑗 + 1)))))
217	198, 199, 213, 215, 216	syl22anc 836	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) ≤ (seq0( + , 𝐺)‘𝑗) ∧ Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘) ≤ (𝐺‘(𝑗 + 1))) → (Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘)) ≤ ((seq0( + , 𝐺)‘𝑗) + (𝐺‘(𝑗 + 1)))))
218	195, 217	mpan2d 691	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) ≤ (seq0( + , 𝐺)‘𝑗) → (Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘)) ≤ ((seq0( + , 𝐺)‘𝑗) + (𝐺‘(𝑗 + 1)))))
219		eqidd 2739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))) → (𝐹‘𝑘) = (𝐹‘𝑘))
220	39	recnd 11003	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
221	68, 196, 220	syl2an 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))) → (𝐹‘𝑘) ∈ ℂ)
222	219, 126, 221	fsumser 15442	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) = (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)))
223	222	eqcomd 2744	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)) = Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘))
224	223	breq1d 5084	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑗) ↔ Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) ≤ (seq0( + , 𝐺)‘𝑗)))
225		eqidd 2739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ (1...((2↑((𝑗 + 1) + 1)) − 1))) → (𝐹‘𝑘) = (𝐹‘𝑘))
226		elfznn 13285	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...((2↑((𝑗 + 1) + 1)) − 1)) → 𝑘 ∈ ℕ)
227	68, 226, 220	syl2an 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ (1...((2↑((𝑗 + 1) + 1)) − 1))) → (𝐹‘𝑘) ∈ ℂ)
228	225, 166, 227	fsumser 15442	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → Σ𝑘 ∈ (1...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘) = (seq1( + , 𝐹)‘((2↑((𝑗 + 1) + 1)) − 1)))
229		fzfid 13693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (1...((2↑((𝑗 + 1) + 1)) − 1)) ∈ Fin)
230	178, 155, 229, 227	fsumsplit 15453	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → Σ𝑘 ∈ (1...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘) = (Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘)))
231	228, 230	eqtr3d 2780	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (seq1( + , 𝐹)‘((2↑((𝑗 + 1) + 1)) − 1)) = (Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘)))
232		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 𝑗 ∈ ℕ₀)
233	232, 200	eleqtrdi 2849	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 𝑗 ∈ (ℤ_≥‘0))
234		seqp1 13736	. . . . . . . 8 ⊢ (𝑗 ∈ (ℤ_≥‘0) → (seq0( + , 𝐺)‘(𝑗 + 1)) = ((seq0( + , 𝐺)‘𝑗) + (𝐺‘(𝑗 + 1))))
235	233, 234	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (seq0( + , 𝐺)‘(𝑗 + 1)) = ((seq0( + , 𝐺)‘𝑗) + (𝐺‘(𝑗 + 1))))
236	231, 235	breq12d 5087	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((seq1( + , 𝐹)‘((2↑((𝑗 + 1) + 1)) − 1)) ≤ (seq0( + , 𝐺)‘(𝑗 + 1)) ↔ (Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘)) ≤ ((seq0( + , 𝐺)‘𝑗) + (𝐺‘(𝑗 + 1)))))
237	218, 224, 236	3imtr4d 294	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑗) → (seq1( + , 𝐹)‘((2↑((𝑗 + 1) + 1)) − 1)) ≤ (seq0( + , 𝐺)‘(𝑗 + 1))))
238	237	expcom 414	. . . 4 ⊢ (𝑗 ∈ ℕ₀ → (𝜑 → ((seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑗) → (seq1( + , 𝐹)‘((2↑((𝑗 + 1) + 1)) − 1)) ≤ (seq0( + , 𝐺)‘(𝑗 + 1)))))
239	238	a2d 29	. . 3 ⊢ (𝑗 ∈ ℕ₀ → ((𝜑 → (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑗)) → (𝜑 → (seq1( + , 𝐹)‘((2↑((𝑗 + 1) + 1)) − 1)) ≤ (seq0( + , 𝐺)‘(𝑗 + 1)))))
240	18, 24, 30, 36, 66, 239	nn0ind 12415	. 2 ⊢ (𝑁 ∈ ℕ₀ → (𝜑 → (seq1( + , 𝐹)‘((2↑(𝑁 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑁)))
241	240	impcom 408	1 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ₀) → (seq1( + , 𝐹)‘((2↑(𝑁 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∪ cun 3885 ∩ cin 3886 ∅c0 4256 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 Fincfn 8733 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 ℕcn 11973 2c2 12028 ℕ₀cn0 12233 ℤcz 12319 ℤ_≥cuz 12582 ...cfz 13239 seqcseq 13721 ↑cexp 13782 ♯chash 14044 Σcsu 15397

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-oadd 8301 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-oi 9269 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-ico 13085 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-sum 15398

This theorem is referenced by: climcnds 15563

W3C validator