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Theorem climcndslem1 15885

Description: Lemma for climcnds 15887: 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 7438	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝑥 + 1) = (0 + 1))
2		0p1e1 12388	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
3	1, 2	eqtrdi 2793	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝑥 + 1) = 1)
4	3	oveq2d 7447	. . . . . . . . 9 ⊢ (𝑥 = 0 → (2↑(𝑥 + 1)) = (2↑1))
5		2cn 12341	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
6		exp1 14108	. . . . . . . . . . 11 ⊢ (2 ∈ ℂ → (2↑1) = 2)
7	5, 6	ax-mp 5	. . . . . . . . . 10 ⊢ (2↑1) = 2
8		df-2 12329	. . . . . . . . . 10 ⊢ 2 = (1 + 1)
9	7, 8	eqtri 2765	. . . . . . . . 9 ⊢ (2↑1) = (1 + 1)
10	4, 9	eqtrdi 2793	. . . . . . . 8 ⊢ (𝑥 = 0 → (2↑(𝑥 + 1)) = (1 + 1))
11	10	oveq1d 7446	. . . . . . 7 ⊢ (𝑥 = 0 → ((2↑(𝑥 + 1)) − 1) = ((1 + 1) − 1))
12		ax-1cn 11213	. . . . . . . 8 ⊢ 1 ∈ ℂ
13	12, 12	pncan3oi 11524	. . . . . . 7 ⊢ ((1 + 1) − 1) = 1
14	11, 13	eqtrdi 2793	. . . . . 6 ⊢ (𝑥 = 0 → ((2↑(𝑥 + 1)) − 1) = 1)
15	14	fveq2d 6910	. . . . 5 ⊢ (𝑥 = 0 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) = (seq1( + , 𝐹)‘1))
16		fveq2 6906	. . . . 5 ⊢ (𝑥 = 0 → (seq0( + , 𝐺)‘𝑥) = (seq0( + , 𝐺)‘0))
17	15, 16	breq12d 5156	. . . 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 7438	. . . . . . 7 ⊢ (𝑥 = 𝑗 → (𝑥 + 1) = (𝑗 + 1))
20	19	oveq2d 7447	. . . . . 6 ⊢ (𝑥 = 𝑗 → (2↑(𝑥 + 1)) = (2↑(𝑗 + 1)))
21	20	fvoveq1d 7453	. . . . 5 ⊢ (𝑥 = 𝑗 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) = (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)))
22		fveq2 6906	. . . . 5 ⊢ (𝑥 = 𝑗 → (seq0( + , 𝐺)‘𝑥) = (seq0( + , 𝐺)‘𝑗))
23	21, 22	breq12d 5156	. . . 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 7438	. . . . . . 7 ⊢ (𝑥 = (𝑗 + 1) → (𝑥 + 1) = ((𝑗 + 1) + 1))
26	25	oveq2d 7447	. . . . . 6 ⊢ (𝑥 = (𝑗 + 1) → (2↑(𝑥 + 1)) = (2↑((𝑗 + 1) + 1)))
27	26	fvoveq1d 7453	. . . . 5 ⊢ (𝑥 = (𝑗 + 1) → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) = (seq1( + , 𝐹)‘((2↑((𝑗 + 1) + 1)) − 1)))
28		fveq2 6906	. . . . 5 ⊢ (𝑥 = (𝑗 + 1) → (seq0( + , 𝐺)‘𝑥) = (seq0( + , 𝐺)‘(𝑗 + 1)))
29	27, 28	breq12d 5156	. . . 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 7438	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑥 + 1) = (𝑁 + 1))
32	31	oveq2d 7447	. . . . . 6 ⊢ (𝑥 = 𝑁 → (2↑(𝑥 + 1)) = (2↑(𝑁 + 1)))
33	32	fvoveq1d 7453	. . . . 5 ⊢ (𝑥 = 𝑁 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) = (seq1( + , 𝐹)‘((2↑(𝑁 + 1)) − 1)))
34		fveq2 6906	. . . . 5 ⊢ (𝑥 = 𝑁 → (seq0( + , 𝐺)‘𝑥) = (seq0( + , 𝐺)‘𝑁))
35	33, 34	breq12d 5156	. . . 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 6906	. . . . . . . 8 ⊢ (𝑘 = 1 → (𝐹‘𝑘) = (𝐹‘1))
38	37	eleq1d 2826	. . . . . . 7 ⊢ (𝑘 = 1 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘1) ∈ ℝ))
39		climcnds.1	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ)
40	39	ralrimiva 3146	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℝ)
41		1nn 12277	. . . . . . . 8 ⊢ 1 ∈ ℕ
42	41	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℕ)
43	38, 40, 42	rspcdva 3623	. . . . . 6 ⊢ (𝜑 → (𝐹‘1) ∈ ℝ)
44	43	leidd 11829	. . . . 5 ⊢ (𝜑 → (𝐹‘1) ≤ (𝐹‘1))
45	43	recnd 11289	. . . . . 6 ⊢ (𝜑 → (𝐹‘1) ∈ ℂ)
46	45	mullidd 11279	. . . . 5 ⊢ (𝜑 → (1 · (𝐹‘1)) = (𝐹‘1))
47	44, 46	breqtrrd 5171	. . . 4 ⊢ (𝜑 → (𝐹‘1) ≤ (1 · (𝐹‘1)))
48		1z 12647	. . . . 5 ⊢ 1 ∈ ℤ
49		eqidd 2738	. . . . 5 ⊢ (𝜑 → (𝐹‘1) = (𝐹‘1))
50	48, 49	seq1i 14056	. . . 4 ⊢ (𝜑 → (seq1( + , 𝐹)‘1) = (𝐹‘1))
51		0z 12624	. . . . 5 ⊢ 0 ∈ ℤ
52		fveq2 6906	. . . . . . 7 ⊢ (𝑛 = 0 → (𝐺‘𝑛) = (𝐺‘0))
53		oveq2 7439	. . . . . . . . 9 ⊢ (𝑛 = 0 → (2↑𝑛) = (2↑0))
54		exp0 14106	. . . . . . . . . 10 ⊢ (2 ∈ ℂ → (2↑0) = 1)
55	5, 54	ax-mp 5	. . . . . . . . 9 ⊢ (2↑0) = 1
56	53, 55	eqtrdi 2793	. . . . . . . 8 ⊢ (𝑛 = 0 → (2↑𝑛) = 1)
57	56	fveq2d 6910	. . . . . . . 8 ⊢ (𝑛 = 0 → (𝐹‘(2↑𝑛)) = (𝐹‘1))
58	56, 57	oveq12d 7449	. . . . . . 7 ⊢ (𝑛 = 0 → ((2↑𝑛) · (𝐹‘(2↑𝑛))) = (1 · (𝐹‘1)))
59	52, 58	eqeq12d 2753	. . . . . 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 3146	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ₀ (𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))
62		0nn0 12541	. . . . . . 7 ⊢ 0 ∈ ℕ₀
63	62	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℕ₀)
64	59, 61, 63	rspcdva 3623	. . . . 5 ⊢ (𝜑 → (𝐺‘0) = (1 · (𝐹‘1)))
65	51, 64	seq1i 14056	. . . 4 ⊢ (𝜑 → (seq0( + , 𝐺)‘0) = (1 · (𝐹‘1)))
66	47, 50, 65	3brtr4d 5175	. . 3 ⊢ (𝜑 → (seq1( + , 𝐹)‘1) ≤ (seq0( + , 𝐺)‘0))
67		fzfid 14014	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1)) ∈ Fin)
68		simpl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 𝜑)
69		2nn 12339	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ
70		peano2nn0 12566	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ₀ → (𝑗 + 1) ∈ ℕ₀)
71	70	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝑗 + 1) ∈ ℕ₀)
72		nnexpcl 14115	. . . . . . . . . . . 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 13560	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1)) → 𝑘 ∈ (ℤ_≥‘(2↑(𝑗 + 1))))
75		eluznn 12960	. . . . . . . . . . 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 685	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) → (𝐹‘𝑘) ∈ ℝ)
78		fveq2 6906	. . . . . . . . . . . 12 ⊢ (𝑘 = (2↑(𝑗 + 1)) → (𝐹‘𝑘) = (𝐹‘(2↑(𝑗 + 1))))
79	78	eleq1d 2826	. . . . . . . . . . 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 3623	. . . . . . . . . 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 775	. . . . . . . . . . . . 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 13560	. . . . . . . . . . . . . 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 775	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...(𝑛 − 1))) → 𝜑)
90		elfzuz 13560	. . . . . . . . . . . . . 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 14074	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) → (𝐹‘𝑛) ≤ (𝐹‘(2↑(𝑗 + 1))))
95	94	ralrimiva 3146	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ∀𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))(𝐹‘𝑛) ≤ (𝐹‘(2↑(𝑗 + 1))))
96		fveq2 6906	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
97	96	breq1d 5153	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → ((𝐹‘𝑛) ≤ (𝐹‘(2↑(𝑗 + 1))) ↔ (𝐹‘𝑘) ≤ (𝐹‘(2↑(𝑗 + 1)))))
98	97	rspccva 3621	. . . . . . . . . 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 15835	. . . . . . . 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 14014	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (1...((2↑(𝑗 + 1)) − 1)) ∈ Fin)
102		hashcl 14395	. . . . . . . . . . . . 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 12589	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (♯‘(1...((2↑(𝑗 + 1)) − 1))) ∈ ℂ)
105	73	nnred 12281	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℝ)
106	105	recnd 11289	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℂ)
107		hashcl 14395	. . . . . . . . . . . . 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 12589	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (♯‘((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) ∈ ℂ)
110		2z 12649	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℤ
111		zexpcl 14117	. . . . . . . . . . . . . . . . . . . 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 12340	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℝ
114		1le2 12475	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ≤ 2
115		nn0p1nn 12565	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ ℕ₀ → (𝑗 + 1) ∈ ℕ)
116	115	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝑗 + 1) ∈ ℕ)
117		nnuz 12921	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℕ = (ℤ_≥‘1)
118	116, 117	eleqtrdi 2851	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝑗 + 1) ∈ (ℤ_≥‘1))
119		leexp2a 14212	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2 ∈ ℝ ∧ 1 ≤ 2 ∧ (𝑗 + 1) ∈ (ℤ_≥‘1)) → (2↑1) ≤ (2↑(𝑗 + 1)))
120	113, 114, 118, 119	mp3an12i 1467	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑1) ≤ (2↑(𝑗 + 1)))
121	7, 120	eqbrtrrid 5179	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 2 ≤ (2↑(𝑗 + 1)))
122	110	eluz1i 12886	. . . . . . . . . . . . . . . . . . 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 12965	. . . . . . . . . . . . . . . . . 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 2851	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) − 1) ∈ (ℤ_≥‘1))
127		peano2zm 12660	. . . . . . . . . . . . . . . . . 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 12566	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 + 1) ∈ ℕ₀ → ((𝑗 + 1) + 1) ∈ ℕ₀)
130	71, 129	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((𝑗 + 1) + 1) ∈ ℕ₀)
131		zexpcl 14117	. . . . . . . . . . . . . . . . . . 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 12660	. . . . . . . . . . . . . . . . . 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 12722	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℝ)
136	132	zred 12722	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑((𝑗 + 1) + 1)) ∈ ℝ)
137		1red 11262	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 1 ∈ ℝ)
138	71	nn0zd 12639	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝑗 + 1) ∈ ℤ)
139		uzid 12893	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 + 1) ∈ ℤ → (𝑗 + 1) ∈ (ℤ_≥‘(𝑗 + 1)))
140		peano2uz 12943	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 + 1) ∈ (ℤ_≥‘(𝑗 + 1)) → ((𝑗 + 1) + 1) ∈ (ℤ_≥‘(𝑗 + 1)))
141		leexp2a 14212	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ ℝ ∧ 1 ≤ 2 ∧ ((𝑗 + 1) + 1) ∈ (ℤ_≥‘(𝑗 + 1))) → (2↑(𝑗 + 1)) ≤ (2↑((𝑗 + 1) + 1)))
142	113, 114, 141	mp3an12 1453	. . . . . . . . . . . . . . . . . . 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 11879	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) − 1) ≤ ((2↑((𝑗 + 1) + 1)) − 1))
145		eluz2 12884	. . . . . . . . . . . . . . . . 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 1344	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑((𝑗 + 1) + 1)) − 1) ∈ (ℤ_≥‘((2↑(𝑗 + 1)) − 1)))
147		elfzuzb 13558	. . . . . . . . . . . . . . . 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 13590	. . . . . . . . . . . . . . 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 11517	. . . . . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . . . 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 4168	. . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . 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 6910	. . . . . . . . . . . 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 14109	. . . . . . . . . . . . . . . . 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 12510	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) · 2) = ((2↑(𝑗 + 1)) + (2↑(𝑗 + 1))))
160	158, 159	eqtrd 2777	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑((𝑗 + 1) + 1)) = ((2↑(𝑗 + 1)) + (2↑(𝑗 + 1))))
161	160	oveq1d 7446	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑((𝑗 + 1) + 1)) − 1) = (((2↑(𝑗 + 1)) + (2↑(𝑗 + 1))) − 1))
162		1cnd 11256	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 1 ∈ ℂ)
163	106, 106, 162	addsubd 11641	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (((2↑(𝑗 + 1)) + (2↑(𝑗 + 1))) − 1) = (((2↑(𝑗 + 1)) − 1) + (2↑(𝑗 + 1))))
164	161, 163	eqtrd 2777	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑((𝑗 + 1) + 1)) − 1) = (((2↑(𝑗 + 1)) − 1) + (2↑(𝑗 + 1))))
165		uztrn 12896	. . . . . . . . . . . . . . . . 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 2852	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑((𝑗 + 1) + 1)) − 1) ∈ ℕ)
168	167	nnnn0d 12587	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑((𝑗 + 1) + 1)) − 1) ∈ ℕ₀)
169		hashfz1 14385	. . . . . . . . . . . . . 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 12587	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) − 1) ∈ ℕ₀)
172		hashfz1 14385	. . . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((♯‘(1...((2↑(𝑗 + 1)) − 1))) + (2↑(𝑗 + 1))) = (((2↑(𝑗 + 1)) − 1) + (2↑(𝑗 + 1))))
175	164, 170, 174	3eqtr4d 2787	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (♯‘(1...((2↑((𝑗 + 1) + 1)) − 1))) = ((♯‘(1...((2↑(𝑗 + 1)) − 1))) + (2↑(𝑗 + 1))))
176	105	ltm1d 12200	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) − 1) < (2↑(𝑗 + 1)))
177		fzdisj 13591	. . . . . . . . . . . . . 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 14421	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 2785	. . . . . . . . . . 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 11466	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) = (♯‘((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))))
183	182	oveq1d 7446	. . . . . . . . 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 6906	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑗 + 1) → (𝐺‘𝑛) = (𝐺‘(𝑗 + 1)))
185		oveq2 7439	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑗 + 1) → (2↑𝑛) = (2↑(𝑗 + 1)))
186	185	fveq2d 6910	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑗 + 1) → (𝐹‘(2↑𝑛)) = (𝐹‘(2↑(𝑗 + 1))))
187	185, 186	oveq12d 7449	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑗 + 1) → ((2↑𝑛) · (𝐹‘(2↑𝑛))) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))))
188	184, 187	eqeq12d 2753	. . . . . . . . . 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 3623	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐺‘(𝑗 + 1)) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))))
191	81	recnd 11289	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℂ)
192		fsumconst 15826	. . . . . . . . . 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 2787	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐺‘(𝑗 + 1)) = Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘(2↑(𝑗 + 1))))
195	100, 194	breqtrrd 5171	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘) ≤ (𝐺‘(𝑗 + 1)))
196		elfznn 13593	. . . . . . . . . 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 15770	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) ∈ ℝ)
199	67, 77	fsumrecl 15770	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘) ∈ ℝ)
200		nn0uz 12920	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
201		0zd 12625	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℤ)
202		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ ℕ₀)
203		nnexpcl 14115	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
204	69, 202, 203	sylancr 587	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
205	204	nnred 12281	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℝ)
206		fveq2 6906	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (2↑𝑛) → (𝐹‘𝑘) = (𝐹‘(2↑𝑛)))
207	206	eleq1d 2826	. . . . . . . . . . . . 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 3623	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐹‘(2↑𝑛)) ∈ ℝ)
210	205, 209	remulcld 11291	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((2↑𝑛) · (𝐹‘(2↑𝑛))) ∈ ℝ)
211	60, 210	eqeltrd 2841	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) ∈ ℝ)
212	200, 201, 211	serfre 14072	. . . . . . . . 9 ⊢ (𝜑 → seq0( + , 𝐺):ℕ₀⟶ℝ)
213	212	ffvelcdmda 7104	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (seq0( + , 𝐺)‘𝑗) ∈ ℝ)
214	135, 81	remulcld 11291	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))) ∈ ℝ)
215	190, 214	eqeltrd 2841	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐺‘(𝑗 + 1)) ∈ ℝ)
216		le2add 11745	. . . . . . . 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 839	. . . . . . 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 694	. . . . . 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 2738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))) → (𝐹‘𝑘) = (𝐹‘𝑘))
220	39	recnd 11289	. . . . . . . . . 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 15766	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) = (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)))
223	222	eqcomd 2743	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)) = Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘))
224	223	breq1d 5153	. . . . . 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 2738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ (1...((2↑((𝑗 + 1) + 1)) − 1))) → (𝐹‘𝑘) = (𝐹‘𝑘))
226		elfznn 13593	. . . . . . . . . 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 15766	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → Σ𝑘 ∈ (1...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘) = (seq1( + , 𝐹)‘((2↑((𝑗 + 1) + 1)) − 1)))
229		fzfid 14014	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (1...((2↑((𝑗 + 1) + 1)) − 1)) ∈ Fin)
230	178, 155, 229, 227	fsumsplit 15777	. . . . . . . 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 2779	. . . . . . 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 2851	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 𝑗 ∈ (ℤ_≥‘0))
234		seqp1 14057	. . . . . . . 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 5156	. . . . . 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 12713	. 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 1540 ∈ wcel 2108 ∀wral 3061 ∪ cun 3949 ∩ cin 3950 ∅c0 4333 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 ≤ cle 11296 − cmin 11492 ℕcn 12266 2c2 12321 ℕ₀cn0 12526 ℤcz 12613 ℤ_≥cuz 12878 ...cfz 13547 seqcseq 14042 ↑cexp 14102 ♯chash 14369 Σcsu 15722

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-ico 13393 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723

This theorem is referenced by: climcnds 15887

W3C validator