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Theorem climcndslem1 15821

Description: Lemma for climcnds 15823: 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 7421	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝑥 + 1) = (0 + 1))
2		0p1e1 12358	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
3	1, 2	eqtrdi 2784	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝑥 + 1) = 1)
4	3	oveq2d 7430	. . . . . . . . 9 ⊢ (𝑥 = 0 → (2↑(𝑥 + 1)) = (2↑1))
5		2cn 12311	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
6		exp1 14058	. . . . . . . . . . 11 ⊢ (2 ∈ ℂ → (2↑1) = 2)
7	5, 6	ax-mp 5	. . . . . . . . . 10 ⊢ (2↑1) = 2
8		df-2 12299	. . . . . . . . . 10 ⊢ 2 = (1 + 1)
9	7, 8	eqtri 2756	. . . . . . . . 9 ⊢ (2↑1) = (1 + 1)
10	4, 9	eqtrdi 2784	. . . . . . . 8 ⊢ (𝑥 = 0 → (2↑(𝑥 + 1)) = (1 + 1))
11	10	oveq1d 7429	. . . . . . 7 ⊢ (𝑥 = 0 → ((2↑(𝑥 + 1)) − 1) = ((1 + 1) − 1))
12		ax-1cn 11190	. . . . . . . 8 ⊢ 1 ∈ ℂ
13	12, 12	pncan3oi 11500	. . . . . . 7 ⊢ ((1 + 1) − 1) = 1
14	11, 13	eqtrdi 2784	. . . . . 6 ⊢ (𝑥 = 0 → ((2↑(𝑥 + 1)) − 1) = 1)
15	14	fveq2d 6895	. . . . 5 ⊢ (𝑥 = 0 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) = (seq1( + , 𝐹)‘1))
16		fveq2 6891	. . . . 5 ⊢ (𝑥 = 0 → (seq0( + , 𝐺)‘𝑥) = (seq0( + , 𝐺)‘0))
17	15, 16	breq12d 5155	. . . 4 ⊢ (𝑥 = 0 → ((seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑥) ↔ (seq1( + , 𝐹)‘1) ≤ (seq0( + , 𝐺)‘0)))
18	17	imbi2d 340	. . 3 ⊢ (𝑥 = 0 → ((𝜑 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑥)) ↔ (𝜑 → (seq1( + , 𝐹)‘1) ≤ (seq0( + , 𝐺)‘0))))
19		oveq1 7421	. . . . . . 7 ⊢ (𝑥 = 𝑗 → (𝑥 + 1) = (𝑗 + 1))
20	19	oveq2d 7430	. . . . . 6 ⊢ (𝑥 = 𝑗 → (2↑(𝑥 + 1)) = (2↑(𝑗 + 1)))
21	20	fvoveq1d 7436	. . . . 5 ⊢ (𝑥 = 𝑗 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) = (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)))
22		fveq2 6891	. . . . 5 ⊢ (𝑥 = 𝑗 → (seq0( + , 𝐺)‘𝑥) = (seq0( + , 𝐺)‘𝑗))
23	21, 22	breq12d 5155	. . . 4 ⊢ (𝑥 = 𝑗 → ((seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑥) ↔ (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑗)))
24	23	imbi2d 340	. . 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 7421	. . . . . . 7 ⊢ (𝑥 = (𝑗 + 1) → (𝑥 + 1) = ((𝑗 + 1) + 1))
26	25	oveq2d 7430	. . . . . 6 ⊢ (𝑥 = (𝑗 + 1) → (2↑(𝑥 + 1)) = (2↑((𝑗 + 1) + 1)))
27	26	fvoveq1d 7436	. . . . 5 ⊢ (𝑥 = (𝑗 + 1) → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) = (seq1( + , 𝐹)‘((2↑((𝑗 + 1) + 1)) − 1)))
28		fveq2 6891	. . . . 5 ⊢ (𝑥 = (𝑗 + 1) → (seq0( + , 𝐺)‘𝑥) = (seq0( + , 𝐺)‘(𝑗 + 1)))
29	27, 28	breq12d 5155	. . . 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 340	. . 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 7421	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑥 + 1) = (𝑁 + 1))
32	31	oveq2d 7430	. . . . . 6 ⊢ (𝑥 = 𝑁 → (2↑(𝑥 + 1)) = (2↑(𝑁 + 1)))
33	32	fvoveq1d 7436	. . . . 5 ⊢ (𝑥 = 𝑁 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) = (seq1( + , 𝐹)‘((2↑(𝑁 + 1)) − 1)))
34		fveq2 6891	. . . . 5 ⊢ (𝑥 = 𝑁 → (seq0( + , 𝐺)‘𝑥) = (seq0( + , 𝐺)‘𝑁))
35	33, 34	breq12d 5155	. . . 4 ⊢ (𝑥 = 𝑁 → ((seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑥) ↔ (seq1( + , 𝐹)‘((2↑(𝑁 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑁)))
36	35	imbi2d 340	. . 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 6891	. . . . . . . 8 ⊢ (𝑘 = 1 → (𝐹‘𝑘) = (𝐹‘1))
38	37	eleq1d 2814	. . . . . . 7 ⊢ (𝑘 = 1 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘1) ∈ ℝ))
39		climcnds.1	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ)
40	39	ralrimiva 3142	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℝ)
41		1nn 12247	. . . . . . . 8 ⊢ 1 ∈ ℕ
42	41	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℕ)
43	38, 40, 42	rspcdva 3609	. . . . . 6 ⊢ (𝜑 → (𝐹‘1) ∈ ℝ)
44	43	leidd 11804	. . . . 5 ⊢ (𝜑 → (𝐹‘1) ≤ (𝐹‘1))
45	43	recnd 11266	. . . . . 6 ⊢ (𝜑 → (𝐹‘1) ∈ ℂ)
46	45	mullidd 11256	. . . . 5 ⊢ (𝜑 → (1 · (𝐹‘1)) = (𝐹‘1))
47	44, 46	breqtrrd 5170	. . . 4 ⊢ (𝜑 → (𝐹‘1) ≤ (1 · (𝐹‘1)))
48		1z 12616	. . . . 5 ⊢ 1 ∈ ℤ
49		eqidd 2729	. . . . 5 ⊢ (𝜑 → (𝐹‘1) = (𝐹‘1))
50	48, 49	seq1i 14006	. . . 4 ⊢ (𝜑 → (seq1( + , 𝐹)‘1) = (𝐹‘1))
51		0z 12593	. . . . 5 ⊢ 0 ∈ ℤ
52		fveq2 6891	. . . . . . 7 ⊢ (𝑛 = 0 → (𝐺‘𝑛) = (𝐺‘0))
53		oveq2 7422	. . . . . . . . 9 ⊢ (𝑛 = 0 → (2↑𝑛) = (2↑0))
54		exp0 14056	. . . . . . . . . 10 ⊢ (2 ∈ ℂ → (2↑0) = 1)
55	5, 54	ax-mp 5	. . . . . . . . 9 ⊢ (2↑0) = 1
56	53, 55	eqtrdi 2784	. . . . . . . 8 ⊢ (𝑛 = 0 → (2↑𝑛) = 1)
57	56	fveq2d 6895	. . . . . . . 8 ⊢ (𝑛 = 0 → (𝐹‘(2↑𝑛)) = (𝐹‘1))
58	56, 57	oveq12d 7432	. . . . . . 7 ⊢ (𝑛 = 0 → ((2↑𝑛) · (𝐹‘(2↑𝑛))) = (1 · (𝐹‘1)))
59	52, 58	eqeq12d 2744	. . . . . 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 3142	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ₀ (𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))
62		0nn0 12511	. . . . . . 7 ⊢ 0 ∈ ℕ₀
63	62	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℕ₀)
64	59, 61, 63	rspcdva 3609	. . . . 5 ⊢ (𝜑 → (𝐺‘0) = (1 · (𝐹‘1)))
65	51, 64	seq1i 14006	. . . 4 ⊢ (𝜑 → (seq0( + , 𝐺)‘0) = (1 · (𝐹‘1)))
66	47, 50, 65	3brtr4d 5174	. . 3 ⊢ (𝜑 → (seq1( + , 𝐹)‘1) ≤ (seq0( + , 𝐺)‘0))
67		fzfid 13964	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1)) ∈ Fin)
68		simpl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 𝜑)
69		2nn 12309	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ
70		peano2nn0 12536	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ₀ → (𝑗 + 1) ∈ ℕ₀)
71	70	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝑗 + 1) ∈ ℕ₀)
72		nnexpcl 14065	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℕ ∧ (𝑗 + 1) ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℕ)
73	69, 71, 72	sylancr 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℕ)
74		elfzuz 13523	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1)) → 𝑘 ∈ (ℤ_≥‘(2↑(𝑗 + 1))))
75		eluznn 12926	. . . . . . . . . . 11 ⊢ (((2↑(𝑗 + 1)) ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) → 𝑘 ∈ ℕ)
76	73, 74, 75	syl2an 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) → 𝑘 ∈ ℕ)
77	68, 76, 39	syl2an2r 684	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) → (𝐹‘𝑘) ∈ ℝ)
78		fveq2 6891	. . . . . . . . . . . 12 ⊢ (𝑘 = (2↑(𝑗 + 1)) → (𝐹‘𝑘) = (𝐹‘(2↑(𝑗 + 1))))
79	78	eleq1d 2814	. . . . . . . . . . 11 ⊢ (𝑘 = (2↑(𝑗 + 1)) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ))
80	40	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℝ)
81	79, 80, 73	rspcdva 3609	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ)
82	81	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ)
83		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) → 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1))))
84		simplll 774	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...𝑛)) → 𝜑)
85	73	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) → (2↑(𝑗 + 1)) ∈ ℕ)
86		elfzuz 13523	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ((2↑(𝑗 + 1))...𝑛) → 𝑘 ∈ (ℤ_≥‘(2↑(𝑗 + 1))))
87	85, 86, 75	syl2an 595	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...𝑛)) → 𝑘 ∈ ℕ)
88	84, 87, 39	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...𝑛)) → (𝐹‘𝑘) ∈ ℝ)
89		simplll 774	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...(𝑛 − 1))) → 𝜑)
90		elfzuz 13523	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ((2↑(𝑗 + 1))...(𝑛 − 1)) → 𝑘 ∈ (ℤ_≥‘(2↑(𝑗 + 1))))
91	85, 90, 75	syl2an 595	. . . . . . . . . . . . 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 583	. . . . . . . . . . . 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 14024	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) → (𝐹‘𝑛) ≤ (𝐹‘(2↑(𝑗 + 1))))
95	94	ralrimiva 3142	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ∀𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))(𝐹‘𝑛) ≤ (𝐹‘(2↑(𝑗 + 1))))
96		fveq2 6891	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
97	96	breq1d 5152	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → ((𝐹‘𝑛) ≤ (𝐹‘(2↑(𝑗 + 1))) ↔ (𝐹‘𝑘) ≤ (𝐹‘(2↑(𝑗 + 1)))))
98	97	rspccva 3607	. . . . . . . . . 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 595	. . . . . . . . 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 15771	. . . . . . . 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 13964	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (1...((2↑(𝑗 + 1)) − 1)) ∈ Fin)
102		hashcl 14341	. . . . . . . . . . . . 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 12558	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (♯‘(1...((2↑(𝑗 + 1)) − 1))) ∈ ℂ)
105	73	nnred 12251	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℝ)
106	105	recnd 11266	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℂ)
107		hashcl 14341	. . . . . . . . . . . . 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 12558	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (♯‘((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) ∈ ℂ)
110		2z 12618	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℤ
111		zexpcl 14067	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ ℤ ∧ (𝑗 + 1) ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℤ)
112	110, 71, 111	sylancr 586	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℤ)
113		2re 12310	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℝ
114		1le2 12445	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ≤ 2
115		nn0p1nn 12535	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ ℕ₀ → (𝑗 + 1) ∈ ℕ)
116	115	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝑗 + 1) ∈ ℕ)
117		nnuz 12889	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℕ = (ℤ_≥‘1)
118	116, 117	eleqtrdi 2839	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝑗 + 1) ∈ (ℤ_≥‘1))
119		leexp2a 14162	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2 ∈ ℝ ∧ 1 ≤ 2 ∧ (𝑗 + 1) ∈ (ℤ_≥‘1)) → (2↑1) ≤ (2↑(𝑗 + 1)))
120	113, 114, 118, 119	mp3an12i 1462	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑1) ≤ (2↑(𝑗 + 1)))
121	7, 120	eqbrtrrid 5178	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 2 ≤ (2↑(𝑗 + 1)))
122	110	eluz1i 12854	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2↑(𝑗 + 1)) ∈ (ℤ_≥‘2) ↔ ((2↑(𝑗 + 1)) ∈ ℤ ∧ 2 ≤ (2↑(𝑗 + 1))))
123	112, 121, 122	sylanbrc 582	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ (ℤ_≥‘2))
124		uz2m1nn 12931	. . . . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) − 1) ∈ (ℤ_≥‘1))
127		peano2zm 12629	. . . . . . . . . . . . . . . . . 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 12536	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 + 1) ∈ ℕ₀ → ((𝑗 + 1) + 1) ∈ ℕ₀)
130	71, 129	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((𝑗 + 1) + 1) ∈ ℕ₀)
131		zexpcl 14067	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2 ∈ ℤ ∧ ((𝑗 + 1) + 1) ∈ ℕ₀) → (2↑((𝑗 + 1) + 1)) ∈ ℤ)
132	110, 130, 131	sylancr 586	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑((𝑗 + 1) + 1)) ∈ ℤ)
133		peano2zm 12629	. . . . . . . . . . . . . . . . . 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 12690	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℝ)
136	132	zred 12690	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑((𝑗 + 1) + 1)) ∈ ℝ)
137		1red 11239	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 1 ∈ ℝ)
138	71	nn0zd 12608	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝑗 + 1) ∈ ℤ)
139		uzid 12861	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 + 1) ∈ ℤ → (𝑗 + 1) ∈ (ℤ_≥‘(𝑗 + 1)))
140		peano2uz 12909	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 + 1) ∈ (ℤ_≥‘(𝑗 + 1)) → ((𝑗 + 1) + 1) ∈ (ℤ_≥‘(𝑗 + 1)))
141		leexp2a 14162	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ ℝ ∧ 1 ≤ 2 ∧ ((𝑗 + 1) + 1) ∈ (ℤ_≥‘(𝑗 + 1))) → (2↑(𝑗 + 1)) ≤ (2↑((𝑗 + 1) + 1)))
142	113, 114, 141	mp3an12 1448	. . . . . . . . . . . . . . . . . . 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 11854	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) − 1) ≤ ((2↑((𝑗 + 1) + 1)) − 1))
145		eluz2 12852	. . . . . . . . . . . . . . . . 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 1341	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑((𝑗 + 1) + 1)) − 1) ∈ (ℤ_≥‘((2↑(𝑗 + 1)) − 1)))
147		elfzuzb 13521	. . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) − 1) ∈ (1...((2↑((𝑗 + 1) + 1)) − 1)))
149		fzsplit 13553	. . . . . . . . . . . . . . 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 11493	. . . . . . . . . . . . . . . . 17 ⊢ (((2↑(𝑗 + 1)) ∈ ℂ ∧ 1 ∈ ℂ) → (((2↑(𝑗 + 1)) − 1) + 1) = (2↑(𝑗 + 1)))
152	106, 12, 151	sylancl 585	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (((2↑(𝑗 + 1)) − 1) + 1) = (2↑(𝑗 + 1)))
153	152	oveq1d 7429	. . . . . . . . . . . . . . 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 4159	. . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . 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 6895	. . . . . . . . . . . 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 14059	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℂ ∧ (𝑗 + 1) ∈ ℕ₀) → (2↑((𝑗 + 1) + 1)) = ((2↑(𝑗 + 1)) · 2))
158	5, 71, 157	sylancr 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑((𝑗 + 1) + 1)) = ((2↑(𝑗 + 1)) · 2))
159	106	times2d 12480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) · 2) = ((2↑(𝑗 + 1)) + (2↑(𝑗 + 1))))
160	158, 159	eqtrd 2768	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑((𝑗 + 1) + 1)) = ((2↑(𝑗 + 1)) + (2↑(𝑗 + 1))))
161	160	oveq1d 7429	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑((𝑗 + 1) + 1)) − 1) = (((2↑(𝑗 + 1)) + (2↑(𝑗 + 1))) − 1))
162		1cnd 11233	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 1 ∈ ℂ)
163	106, 106, 162	addsubd 11616	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (((2↑(𝑗 + 1)) + (2↑(𝑗 + 1))) − 1) = (((2↑(𝑗 + 1)) − 1) + (2↑(𝑗 + 1))))
164	161, 163	eqtrd 2768	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑((𝑗 + 1) + 1)) − 1) = (((2↑(𝑗 + 1)) − 1) + (2↑(𝑗 + 1))))
165		uztrn 12864	. . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑((𝑗 + 1) + 1)) − 1) ∈ (ℤ_≥‘1))
167	166, 117	eleqtrrdi 2840	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑((𝑗 + 1) + 1)) − 1) ∈ ℕ)
168	167	nnnn0d 12556	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑((𝑗 + 1) + 1)) − 1) ∈ ℕ₀)
169		hashfz1 14331	. . . . . . . . . . . . . 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 12556	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) − 1) ∈ ℕ₀)
172		hashfz1 14331	. . . . . . . . . . . . . . 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 7429	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((♯‘(1...((2↑(𝑗 + 1)) − 1))) + (2↑(𝑗 + 1))) = (((2↑(𝑗 + 1)) − 1) + (2↑(𝑗 + 1))))
175	164, 170, 174	3eqtr4d 2778	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (♯‘(1...((2↑((𝑗 + 1) + 1)) − 1))) = ((♯‘(1...((2↑(𝑗 + 1)) − 1))) + (2↑(𝑗 + 1))))
176	105	ltm1d 12170	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) − 1) < (2↑(𝑗 + 1)))
177		fzdisj 13554	. . . . . . . . . . . . . 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 14367	. . . . . . . . . . . . 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 1369	. . . . . . . . . . . 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 2776	. . . . . . . . . . 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 11443	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) = (♯‘((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))))
183	182	oveq1d 7429	. . . . . . . . 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 6891	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑗 + 1) → (𝐺‘𝑛) = (𝐺‘(𝑗 + 1)))
185		oveq2 7422	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑗 + 1) → (2↑𝑛) = (2↑(𝑗 + 1)))
186	185	fveq2d 6895	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑗 + 1) → (𝐹‘(2↑𝑛)) = (𝐹‘(2↑(𝑗 + 1))))
187	185, 186	oveq12d 7432	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑗 + 1) → ((2↑𝑛) · (𝐹‘(2↑𝑛))) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))))
188	184, 187	eqeq12d 2744	. . . . . . . . . 10 ⊢ (𝑛 = (𝑗 + 1) → ((𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))) ↔ (𝐺‘(𝑗 + 1)) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1))))))
189	61	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ∀𝑛 ∈ ℕ₀ (𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))
190	188, 189, 71	rspcdva 3609	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐺‘(𝑗 + 1)) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))))
191	81	recnd 11266	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℂ)
192		fsumconst 15762	. . . . . . . . . 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 583	. . . . . . . . 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 2778	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐺‘(𝑗 + 1)) = Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘(2↑(𝑗 + 1))))
195	100, 194	breqtrrd 5170	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘) ≤ (𝐺‘(𝑗 + 1)))
196		elfznn 13556	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1)) → 𝑘 ∈ ℕ)
197	68, 196, 39	syl2an 595	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))) → (𝐹‘𝑘) ∈ ℝ)
198	101, 197	fsumrecl 15706	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) ∈ ℝ)
199	67, 77	fsumrecl 15706	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘) ∈ ℝ)
200		nn0uz 12888	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
201		0zd 12594	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℤ)
202		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ ℕ₀)
203		nnexpcl 14065	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
204	69, 202, 203	sylancr 586	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
205	204	nnred 12251	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℝ)
206		fveq2 6891	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (2↑𝑛) → (𝐹‘𝑘) = (𝐹‘(2↑𝑛)))
207	206	eleq1d 2814	. . . . . . . . . . . . 13 ⊢ (𝑘 = (2↑𝑛) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(2↑𝑛)) ∈ ℝ))
208	40	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℝ)
209	207, 208, 204	rspcdva 3609	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐹‘(2↑𝑛)) ∈ ℝ)
210	205, 209	remulcld 11268	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((2↑𝑛) · (𝐹‘(2↑𝑛))) ∈ ℝ)
211	60, 210	eqeltrd 2829	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) ∈ ℝ)
212	200, 201, 211	serfre 14022	. . . . . . . . 9 ⊢ (𝜑 → seq0( + , 𝐺):ℕ₀⟶ℝ)
213	212	ffvelcdmda 7088	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (seq0( + , 𝐺)‘𝑗) ∈ ℝ)
214	135, 81	remulcld 11268	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))) ∈ ℝ)
215	190, 214	eqeltrd 2829	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐺‘(𝑗 + 1)) ∈ ℝ)
216		le2add 11720	. . . . . . . 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 838	. . . . . . 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 693	. . . . . 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 2729	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))) → (𝐹‘𝑘) = (𝐹‘𝑘))
220	39	recnd 11266	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
221	68, 196, 220	syl2an 595	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))) → (𝐹‘𝑘) ∈ ℂ)
222	219, 126, 221	fsumser 15702	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) = (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)))
223	222	eqcomd 2734	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)) = Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘))
224	223	breq1d 5152	. . . . . 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 2729	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ (1...((2↑((𝑗 + 1) + 1)) − 1))) → (𝐹‘𝑘) = (𝐹‘𝑘))
226		elfznn 13556	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...((2↑((𝑗 + 1) + 1)) − 1)) → 𝑘 ∈ ℕ)
227	68, 226, 220	syl2an 595	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ (1...((2↑((𝑗 + 1) + 1)) − 1))) → (𝐹‘𝑘) ∈ ℂ)
228	225, 166, 227	fsumser 15702	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → Σ𝑘 ∈ (1...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘) = (seq1( + , 𝐹)‘((2↑((𝑗 + 1) + 1)) − 1)))
229		fzfid 13964	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (1...((2↑((𝑗 + 1) + 1)) − 1)) ∈ Fin)
230	178, 155, 229, 227	fsumsplit 15713	. . . . . . . 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 2770	. . . . . . 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 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 𝑗 ∈ ℕ₀)
233	232, 200	eleqtrdi 2839	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 𝑗 ∈ (ℤ_≥‘0))
234		seqp1 14007	. . . . . . . 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 5155	. . . . . 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 413	. . . 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 12681	. 2 ⊢ (𝑁 ∈ ℕ₀ → (𝜑 → (seq1( + , 𝐹)‘((2↑(𝑁 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑁)))
241	240	impcom 407	1 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ₀) → (seq1( + , 𝐹)‘((2↑(𝑁 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3057 ∪ cun 3943 ∩ cin 3944 ∅c0 4318 class class class wbr 5142 ‘cfv 6542 (class class class)co 7414 Fincfn 8957 ℂcc 11130 ℝcr 11131 0cc0 11132 1c1 11133 + caddc 11135 · cmul 11137 < clt 11272 ≤ cle 11273 − cmin 11468 ℕcn 12236 2c2 12291 ℕ₀cn0 12496 ℤcz 12582 ℤ_≥cuz 12846 ...cfz 13510 seqcseq 13992 ↑cexp 14052 ♯chash 14315 Σcsu 15658

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9459 df-oi 9527 df-dju 9918 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-n0 12497 df-z 12583 df-uz 12847 df-rp 13001 df-ico 13356 df-fz 13511 df-fzo 13654 df-seq 13993 df-exp 14053 df-hash 14316 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15458 df-sum 15659

This theorem is referenced by: climcnds 15823

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