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Theorem climcndslem1 15809

Description: Lemma for climcnds 15811: 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 7367	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝑥 + 1) = (0 + 1))
2		0p1e1 12293	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
3	1, 2	eqtrdi 2792	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝑥 + 1) = 1)
4	3	oveq2d 7376	. . . . . . . . 9 ⊢ (𝑥 = 0 → (2↑(𝑥 + 1)) = (2↑1))
5		2cn 12251	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
6		exp1 14024	. . . . . . . . . . 11 ⊢ (2 ∈ ℂ → (2↑1) = 2)
7	5, 6	ax-mp 5	. . . . . . . . . 10 ⊢ (2↑1) = 2
8		df-2 12239	. . . . . . . . . 10 ⊢ 2 = (1 + 1)
9	7, 8	eqtri 2764	. . . . . . . . 9 ⊢ (2↑1) = (1 + 1)
10	4, 9	eqtrdi 2792	. . . . . . . 8 ⊢ (𝑥 = 0 → (2↑(𝑥 + 1)) = (1 + 1))
11	10	oveq1d 7375	. . . . . . 7 ⊢ (𝑥 = 0 → ((2↑(𝑥 + 1)) − 1) = ((1 + 1) − 1))
12		ax-1cn 11091	. . . . . . . 8 ⊢ 1 ∈ ℂ
13	12, 12	pncan3oi 11404	. . . . . . 7 ⊢ ((1 + 1) − 1) = 1
14	11, 13	eqtrdi 2792	. . . . . 6 ⊢ (𝑥 = 0 → ((2↑(𝑥 + 1)) − 1) = 1)
15	14	fveq2d 6835	. . . . 5 ⊢ (𝑥 = 0 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) = (seq1( + , 𝐹)‘1))
16		fveq2 6831	. . . . 5 ⊢ (𝑥 = 0 → (seq0( + , 𝐺)‘𝑥) = (seq0( + , 𝐺)‘0))
17	15, 16	breq12d 5088	. . . 4 ⊢ (𝑥 = 0 → ((seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑥) ↔ (seq1( + , 𝐹)‘1) ≤ (seq0( + , 𝐺)‘0)))
18	17	imbi2d 342	. . 3 ⊢ (𝑥 = 0 → ((𝜑 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑥)) ↔ (𝜑 → (seq1( + , 𝐹)‘1) ≤ (seq0( + , 𝐺)‘0))))
19		oveq1 7367	. . . . . . 7 ⊢ (𝑥 = 𝑗 → (𝑥 + 1) = (𝑗 + 1))
20	19	oveq2d 7376	. . . . . 6 ⊢ (𝑥 = 𝑗 → (2↑(𝑥 + 1)) = (2↑(𝑗 + 1)))
21	20	fvoveq1d 7382	. . . . 5 ⊢ (𝑥 = 𝑗 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) = (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)))
22		fveq2 6831	. . . . 5 ⊢ (𝑥 = 𝑗 → (seq0( + , 𝐺)‘𝑥) = (seq0( + , 𝐺)‘𝑗))
23	21, 22	breq12d 5088	. . . 4 ⊢ (𝑥 = 𝑗 → ((seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑥) ↔ (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑗)))
24	23	imbi2d 342	. . 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 7367	. . . . . . 7 ⊢ (𝑥 = (𝑗 + 1) → (𝑥 + 1) = ((𝑗 + 1) + 1))
26	25	oveq2d 7376	. . . . . 6 ⊢ (𝑥 = (𝑗 + 1) → (2↑(𝑥 + 1)) = (2↑((𝑗 + 1) + 1)))
27	26	fvoveq1d 7382	. . . . 5 ⊢ (𝑥 = (𝑗 + 1) → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) = (seq1( + , 𝐹)‘((2↑((𝑗 + 1) + 1)) − 1)))
28		fveq2 6831	. . . . 5 ⊢ (𝑥 = (𝑗 + 1) → (seq0( + , 𝐺)‘𝑥) = (seq0( + , 𝐺)‘(𝑗 + 1)))
29	27, 28	breq12d 5088	. . . 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 342	. . 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 7367	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑥 + 1) = (𝑁 + 1))
32	31	oveq2d 7376	. . . . . 6 ⊢ (𝑥 = 𝑁 → (2↑(𝑥 + 1)) = (2↑(𝑁 + 1)))
33	32	fvoveq1d 7382	. . . . 5 ⊢ (𝑥 = 𝑁 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) = (seq1( + , 𝐹)‘((2↑(𝑁 + 1)) − 1)))
34		fveq2 6831	. . . . 5 ⊢ (𝑥 = 𝑁 → (seq0( + , 𝐺)‘𝑥) = (seq0( + , 𝐺)‘𝑁))
35	33, 34	breq12d 5088	. . . 4 ⊢ (𝑥 = 𝑁 → ((seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑥) ↔ (seq1( + , 𝐹)‘((2↑(𝑁 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑁)))
36	35	imbi2d 342	. . 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 6831	. . . . . . . 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 3133	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℝ)
41		1nn 12180	. . . . . . . 8 ⊢ 1 ∈ ℕ
42	41	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℕ)
43	38, 40, 42	rspcdva 3563	. . . . . 6 ⊢ (𝜑 → (𝐹‘1) ∈ ℝ)
44	43	leidd 11711	. . . . 5 ⊢ (𝜑 → (𝐹‘1) ≤ (𝐹‘1))
45	43	recnd 11168	. . . . . 6 ⊢ (𝜑 → (𝐹‘1) ∈ ℂ)
46	45	mullidd 11158	. . . . 5 ⊢ (𝜑 → (1 · (𝐹‘1)) = (𝐹‘1))
47	44, 46	breqtrrd 5103	. . . 4 ⊢ (𝜑 → (𝐹‘1) ≤ (1 · (𝐹‘1)))
48		1z 12552	. . . . 5 ⊢ 1 ∈ ℤ
49		eqidd 2742	. . . . 5 ⊢ (𝜑 → (𝐹‘1) = (𝐹‘1))
50	48, 49	seq1i 13972	. . . 4 ⊢ (𝜑 → (seq1( + , 𝐹)‘1) = (𝐹‘1))
51		0z 12530	. . . . 5 ⊢ 0 ∈ ℤ
52		fveq2 6831	. . . . . . 7 ⊢ (𝑛 = 0 → (𝐺‘𝑛) = (𝐺‘0))
53		oveq2 7368	. . . . . . . . 9 ⊢ (𝑛 = 0 → (2↑𝑛) = (2↑0))
54		exp0 14022	. . . . . . . . . 10 ⊢ (2 ∈ ℂ → (2↑0) = 1)
55	5, 54	ax-mp 5	. . . . . . . . 9 ⊢ (2↑0) = 1
56	53, 55	eqtrdi 2792	. . . . . . . 8 ⊢ (𝑛 = 0 → (2↑𝑛) = 1)
57	56	fveq2d 6835	. . . . . . . 8 ⊢ (𝑛 = 0 → (𝐹‘(2↑𝑛)) = (𝐹‘1))
58	56, 57	oveq12d 7378	. . . . . . 7 ⊢ (𝑛 = 0 → ((2↑𝑛) · (𝐹‘(2↑𝑛))) = (1 · (𝐹‘1)))
59	52, 58	eqeq12d 2757	. . . . . 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 3133	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ₀ (𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))
62		0nn0 12447	. . . . . . 7 ⊢ 0 ∈ ℕ₀
63	62	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℕ₀)
64	59, 61, 63	rspcdva 3563	. . . . 5 ⊢ (𝜑 → (𝐺‘0) = (1 · (𝐹‘1)))
65	51, 64	seq1i 13972	. . . 4 ⊢ (𝜑 → (seq0( + , 𝐺)‘0) = (1 · (𝐹‘1)))
66	47, 50, 65	3brtr4d 5107	. . 3 ⊢ (𝜑 → (seq1( + , 𝐹)‘1) ≤ (seq0( + , 𝐺)‘0))
67		fzfid 13930	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1)) ∈ Fin)
68		simpl 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 𝜑)
69		2nn 12249	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ
70		peano2nn0 12472	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ₀ → (𝑗 + 1) ∈ ℕ₀)
71	70	adantl 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝑗 + 1) ∈ ℕ₀)
72		nnexpcl 14031	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℕ ∧ (𝑗 + 1) ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℕ)
73	69, 71, 72	sylancr 594	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℕ)
74		elfzuz 13469	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1)) → 𝑘 ∈ (ℤ_≥‘(2↑(𝑗 + 1))))
75		eluznn 12863	. . . . . . . . . . 11 ⊢ (((2↑(𝑗 + 1)) ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) → 𝑘 ∈ ℕ)
76	73, 74, 75	syl2an 603	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) → 𝑘 ∈ ℕ)
77	68, 76, 39	syl2an2r 692	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) → (𝐹‘𝑘) ∈ ℝ)
78		fveq2 6831	. . . . . . . . . . . 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 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℝ)
81	79, 80, 73	rspcdva 3563	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ)
82	81	adantr 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ)
83		simpr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) → 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1))))
84		simplll 781	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...𝑛)) → 𝜑)
85	73	adantr 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) → (2↑(𝑗 + 1)) ∈ ℕ)
86		elfzuz 13469	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ((2↑(𝑗 + 1))...𝑛) → 𝑘 ∈ (ℤ_≥‘(2↑(𝑗 + 1))))
87	85, 86, 75	syl2an 603	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...𝑛)) → 𝑘 ∈ ℕ)
88	84, 87, 39	syl2anc 591	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...𝑛)) → (𝐹‘𝑘) ∈ ℝ)
89		simplll 781	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...(𝑛 − 1))) → 𝜑)
90		elfzuz 13469	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ((2↑(𝑗 + 1))...(𝑛 − 1)) → 𝑘 ∈ (ℤ_≥‘(2↑(𝑗 + 1))))
91	85, 90, 75	syl2an 603	. . . . . . . . . . . . 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 591	. . . . . . . . . . . 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 13990	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))) → (𝐹‘𝑛) ≤ (𝐹‘(2↑(𝑗 + 1))))
95	94	ralrimiva 3133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ∀𝑛 ∈ (ℤ_≥‘(2↑(𝑗 + 1)))(𝐹‘𝑛) ≤ (𝐹‘(2↑(𝑗 + 1))))
96		fveq2 6831	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
97	96	breq1d 5085	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → ((𝐹‘𝑛) ≤ (𝐹‘(2↑(𝑗 + 1))) ↔ (𝐹‘𝑘) ≤ (𝐹‘(2↑(𝑗 + 1)))))
98	97	rspccva 3561	. . . . . . . . . 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 603	. . . . . . . . 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 15757	. . . . . . . 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 13930	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (1...((2↑(𝑗 + 1)) − 1)) ∈ Fin)
102		hashcl 14313	. . . . . . . . . . . . 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 12495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (♯‘(1...((2↑(𝑗 + 1)) − 1))) ∈ ℂ)
105	73	nnred 12184	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℝ)
106	105	recnd 11168	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℂ)
107		hashcl 14313	. . . . . . . . . . . . 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 12495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (♯‘((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) ∈ ℂ)
110		2z 12554	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℤ
111		zexpcl 14033	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ ℤ ∧ (𝑗 + 1) ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℤ)
112	110, 71, 111	sylancr 594	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℤ)
113		2re 12250	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℝ
114		1le2 12380	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ≤ 2
115		nn0p1nn 12471	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ ℕ₀ → (𝑗 + 1) ∈ ℕ)
116	115	adantl 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝑗 + 1) ∈ ℕ)
117		nnuz 12822	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℕ = (ℤ_≥‘1)
118	116, 117	eleqtrdi 2851	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝑗 + 1) ∈ (ℤ_≥‘1))
119		leexp2a 14129	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2 ∈ ℝ ∧ 1 ≤ 2 ∧ (𝑗 + 1) ∈ (ℤ_≥‘1)) → (2↑1) ≤ (2↑(𝑗 + 1)))
120	113, 114, 118, 119	mp3an12i 1474	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑1) ≤ (2↑(𝑗 + 1)))
121	7, 120	eqbrtrrid 5111	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 2 ≤ (2↑(𝑗 + 1)))
122	110	eluz1i 12791	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2↑(𝑗 + 1)) ∈ (ℤ_≥‘2) ↔ ((2↑(𝑗 + 1)) ∈ ℤ ∧ 2 ≤ (2↑(𝑗 + 1))))
123	112, 121, 122	sylanbrc 590	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ (ℤ_≥‘2))
124		uz2m1nn 12868	. . . . . . . . . . . . . . . . . 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 12565	. . . . . . . . . . . . . . . . . 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 12472	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 + 1) ∈ ℕ₀ → ((𝑗 + 1) + 1) ∈ ℕ₀)
130	71, 129	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((𝑗 + 1) + 1) ∈ ℕ₀)
131		zexpcl 14033	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2 ∈ ℤ ∧ ((𝑗 + 1) + 1) ∈ ℕ₀) → (2↑((𝑗 + 1) + 1)) ∈ ℤ)
132	110, 130, 131	sylancr 594	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑((𝑗 + 1) + 1)) ∈ ℤ)
133		peano2zm 12565	. . . . . . . . . . . . . . . . . 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 12628	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) ∈ ℝ)
136	132	zred 12628	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑((𝑗 + 1) + 1)) ∈ ℝ)
137		1red 11140	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 1 ∈ ℝ)
138	71	nn0zd 12544	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝑗 + 1) ∈ ℤ)
139		uzid 12798	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 + 1) ∈ ℤ → (𝑗 + 1) ∈ (ℤ_≥‘(𝑗 + 1)))
140		peano2uz 12846	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 + 1) ∈ (ℤ_≥‘(𝑗 + 1)) → ((𝑗 + 1) + 1) ∈ (ℤ_≥‘(𝑗 + 1)))
141		leexp2a 14129	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ ℝ ∧ 1 ≤ 2 ∧ ((𝑗 + 1) + 1) ∈ (ℤ_≥‘(𝑗 + 1))) → (2↑(𝑗 + 1)) ≤ (2↑((𝑗 + 1) + 1)))
142	113, 114, 141	mp3an12 1460	. . . . . . . . . . . . . . . . . . 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 11761	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) − 1) ≤ ((2↑((𝑗 + 1) + 1)) − 1))
145		eluz2 12789	. . . . . . . . . . . . . . . . 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 1351	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑((𝑗 + 1) + 1)) − 1) ∈ (ℤ_≥‘((2↑(𝑗 + 1)) − 1)))
147		elfzuzb 13467	. . . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) − 1) ∈ (1...((2↑((𝑗 + 1) + 1)) − 1)))
149		fzsplit 13499	. . . . . . . . . . . . . . 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 11397	. . . . . . . . . . . . . . . . 17 ⊢ (((2↑(𝑗 + 1)) ∈ ℂ ∧ 1 ∈ ℂ) → (((2↑(𝑗 + 1)) − 1) + 1) = (2↑(𝑗 + 1)))
152	106, 12, 151	sylancl 593	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (((2↑(𝑗 + 1)) − 1) + 1) = (2↑(𝑗 + 1)))
153	152	oveq1d 7375	. . . . . . . . . . . . . . 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 4101	. . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . 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 6835	. . . . . . . . . . . 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 14025	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℂ ∧ (𝑗 + 1) ∈ ℕ₀) → (2↑((𝑗 + 1) + 1)) = ((2↑(𝑗 + 1)) · 2))
158	5, 71, 157	sylancr 594	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑((𝑗 + 1) + 1)) = ((2↑(𝑗 + 1)) · 2))
159	106	times2d 12416	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) · 2) = ((2↑(𝑗 + 1)) + (2↑(𝑗 + 1))))
160	158, 159	eqtrd 2776	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑((𝑗 + 1) + 1)) = ((2↑(𝑗 + 1)) + (2↑(𝑗 + 1))))
161	160	oveq1d 7375	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑((𝑗 + 1) + 1)) − 1) = (((2↑(𝑗 + 1)) + (2↑(𝑗 + 1))) − 1))
162		1cnd 11134	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 1 ∈ ℂ)
163	106, 106, 162	addsubd 11521	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (((2↑(𝑗 + 1)) + (2↑(𝑗 + 1))) − 1) = (((2↑(𝑗 + 1)) − 1) + (2↑(𝑗 + 1))))
164	161, 163	eqtrd 2776	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑((𝑗 + 1) + 1)) − 1) = (((2↑(𝑗 + 1)) − 1) + (2↑(𝑗 + 1))))
165		uztrn 12801	. . . . . . . . . . . . . . . . 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 591	. . . . . . . . . . . . . . . 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 12493	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑((𝑗 + 1) + 1)) − 1) ∈ ℕ₀)
169		hashfz1 14303	. . . . . . . . . . . . . 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 12493	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) − 1) ∈ ℕ₀)
172		hashfz1 14303	. . . . . . . . . . . . . . 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 7375	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((♯‘(1...((2↑(𝑗 + 1)) − 1))) + (2↑(𝑗 + 1))) = (((2↑(𝑗 + 1)) − 1) + (2↑(𝑗 + 1))))
175	164, 170, 174	3eqtr4d 2786	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (♯‘(1...((2↑((𝑗 + 1) + 1)) − 1))) = ((♯‘(1...((2↑(𝑗 + 1)) − 1))) + (2↑(𝑗 + 1))))
176	105	ltm1d 12083	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((2↑(𝑗 + 1)) − 1) < (2↑(𝑗 + 1)))
177		fzdisj 13500	. . . . . . . . . . . . . 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 14339	. . . . . . . . . . . . 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 1380	. . . . . . . . . . . 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 2784	. . . . . . . . . . 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 11346	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (2↑(𝑗 + 1)) = (♯‘((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))))
183	182	oveq1d 7375	. . . . . . . . 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 6831	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑗 + 1) → (𝐺‘𝑛) = (𝐺‘(𝑗 + 1)))
185		oveq2 7368	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑗 + 1) → (2↑𝑛) = (2↑(𝑗 + 1)))
186	185	fveq2d 6835	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑗 + 1) → (𝐹‘(2↑𝑛)) = (𝐹‘(2↑(𝑗 + 1))))
187	185, 186	oveq12d 7378	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑗 + 1) → ((2↑𝑛) · (𝐹‘(2↑𝑛))) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))))
188	184, 187	eqeq12d 2757	. . . . . . . . . 10 ⊢ (𝑛 = (𝑗 + 1) → ((𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))) ↔ (𝐺‘(𝑗 + 1)) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1))))))
189	61	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ∀𝑛 ∈ ℕ₀ (𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))))
190	188, 189, 71	rspcdva 3563	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐺‘(𝑗 + 1)) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))))
191	81	recnd 11168	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐹‘(2↑(𝑗 + 1))) ∈ ℂ)
192		fsumconst 15747	. . . . . . . . . 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 591	. . . . . . . . 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 2786	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐺‘(𝑗 + 1)) = Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘(2↑(𝑗 + 1))))
195	100, 194	breqtrrd 5103	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘) ≤ (𝐺‘(𝑗 + 1)))
196		elfznn 13502	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1)) → 𝑘 ∈ ℕ)
197	68, 196, 39	syl2an 603	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))) → (𝐹‘𝑘) ∈ ℝ)
198	101, 197	fsumrecl 15691	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) ∈ ℝ)
199	67, 77	fsumrecl 15691	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘) ∈ ℝ)
200		nn0uz 12821	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
201		0zd 12531	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℤ)
202		simpr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ ℕ₀)
203		nnexpcl 14031	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
204	69, 202, 203	sylancr 594	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
205	204	nnred 12184	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℝ)
206		fveq2 6831	. . . . . . . . . . . . . 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 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℝ)
209	207, 208, 204	rspcdva 3563	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐹‘(2↑𝑛)) ∈ ℝ)
210	205, 209	remulcld 11170	. . . . . . . . . . 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 13988	. . . . . . . . 9 ⊢ (𝜑 → seq0( + , 𝐺):ℕ₀⟶ℝ)
213	212	ffvelcdmda 7029	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (seq0( + , 𝐺)‘𝑗) ∈ ℝ)
214	135, 81	remulcld 11170	. . . . . . . . 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 11627	. . . . . . . 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 845	. . . . . . 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 701	. . . . . 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 2742	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))) → (𝐹‘𝑘) = (𝐹‘𝑘))
220	39	recnd 11168	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
221	68, 196, 220	syl2an 603	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))) → (𝐹‘𝑘) ∈ ℂ)
222	219, 126, 221	fsumser 15687	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) = (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)))
223	222	eqcomd 2747	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)) = Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘))
224	223	breq1d 5085	. . . . . 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 2742	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ (1...((2↑((𝑗 + 1) + 1)) − 1))) → (𝐹‘𝑘) = (𝐹‘𝑘))
226		elfznn 13502	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...((2↑((𝑗 + 1) + 1)) − 1)) → 𝑘 ∈ ℕ)
227	68, 226, 220	syl2an 603	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ₀) ∧ 𝑘 ∈ (1...((2↑((𝑗 + 1) + 1)) − 1))) → (𝐹‘𝑘) ∈ ℂ)
228	225, 166, 227	fsumser 15687	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → Σ𝑘 ∈ (1...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘) = (seq1( + , 𝐹)‘((2↑((𝑗 + 1) + 1)) − 1)))
229		fzfid 13930	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (1...((2↑((𝑗 + 1) + 1)) − 1)) ∈ Fin)
230	178, 155, 229, 227	fsumsplit 15698	. . . . . . . 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 2778	. . . . . . 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 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 𝑗 ∈ ℕ₀)
233	232, 200	eleqtrdi 2851	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 𝑗 ∈ (ℤ_≥‘0))
234		seqp1 13973	. . . . . . . 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 5088	. . . . . 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 296	. . . . 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 415	. . . 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 12619	. 2 ⊢ (𝑁 ∈ ℕ₀ → (𝜑 → (seq1( + , 𝐹)‘((2↑(𝑁 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑁)))
241	240	impcom 409	1 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ₀) → (seq1( + , 𝐹)‘((2↑(𝑁 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1548 ∈ wcel 2121 ∀wral 3055 ∪ cun 3883 ∩ cin 3884 ∅c0 4264 class class class wbr 5075 ‘cfv 6489 (class class class)co 7360 Fincfn 8887 ℂcc 11031 ℝcr 11032 0cc0 11033 1c1 11034 + caddc 11036 · cmul 11038 < clt 11174 ≤ cle 11175 − cmin 11372 ℕcn 12169 2c2 12231 ℕ₀cn0 12432 ℤcz 12519 ℤ_≥cuz 12783 ...cfz 13456 seqcseq 13958 ↑cexp 14018 ♯chash 14287 Σcsu 15643

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-inf2 9557 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-oadd 8403 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-oi 9419 df-dju 9820 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-n0 12433 df-z 12520 df-uz 12784 df-rp 12938 df-ico 13299 df-fz 13457 df-fzo 13604 df-seq 13959 df-exp 14019 df-hash 14288 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-clim 15445 df-sum 15644

This theorem is referenced by: climcnds 15811

W3C validator