dignn0flhalflem2 - Mathbox for Alexander van der Vekens

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Theorem dignn0flhalflem2 43257

Description: Lemma 2 for dignn0flhalf 43259. (Contributed by AV, 7-Jun-2012.)

**Assertion**
Ref	Expression
dignn0flhalflem2	⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘(𝐴 / (2↑(𝑁 + 1)))) = (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))))

**Proof of Theorem dignn0flhalflem2**
Step	Hyp	Ref	Expression
1		zre 11708	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
2	1	rehalfcld 11605	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → (𝐴 / 2) ∈ ℝ)
3	2	flcld 12894	. . . . . . 7 ⊢ (𝐴 ∈ ℤ → (⌊‘(𝐴 / 2)) ∈ ℤ)
4	3	zred 11810	. . . . . 6 ⊢ (𝐴 ∈ ℤ → (⌊‘(𝐴 / 2)) ∈ ℝ)
5	4	3ad2ant1 1169	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘(𝐴 / 2)) ∈ ℝ)
6		2re 11425	. . . . . . . 8 ⊢ 2 ∈ ℝ
7	6	a1i 11	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 2 ∈ ℝ)
8		id 22	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℕ₀)
9	7, 8	reexpcld 13319	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → (2↑𝑁) ∈ ℝ)
10	9	3ad2ant3 1171	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℝ)
11		2cnd 11429	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → 2 ∈ ℂ)
12		2ne0 11462	. . . . . . 7 ⊢ 2 ≠ 0
13	12	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → 2 ≠ 0)
14		nn0z 11728	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
15	14	3ad2ant3 1171	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℤ)
16	11, 13, 15	expne0d 13308	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ≠ 0)
17	5, 10, 16	redivcld 11179	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → ((⌊‘(𝐴 / 2)) / (2↑𝑁)) ∈ ℝ)
18	17	flcld 12894	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))) ∈ ℤ)
19	1	3ad2ant1 1169	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → 𝐴 ∈ ℝ)
20	6	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → 2 ∈ ℝ)
21		simp3 1174	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℕ₀)
22		1nn0 11636	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
23	22	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → 1 ∈ ℕ₀)
24	21, 23	nn0addcld 11682	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ∈ ℕ₀)
25	20, 24	reexpcld 13319	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2↑(𝑁 + 1)) ∈ ℝ)
26	15	peano2zd 11813	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ∈ ℤ)
27	11, 13, 26	expne0d 13308	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2↑(𝑁 + 1)) ≠ 0)
28	19, 25, 27	redivcld 11179	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (𝐴 / (2↑(𝑁 + 1))) ∈ ℝ)
29	28	flcld 12894	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘(𝐴 / (2↑(𝑁 + 1)))) ∈ ℤ)
30		nn0p1nn 11659	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ)
31		dignn0flhalflem1 43256	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ (𝑁 + 1) ∈ ℕ) → (⌊‘((𝐴 / (2↑(𝑁 + 1))) − 1)) < (⌊‘((𝐴 − 1) / (2↑(𝑁 + 1)))))
32	30, 31	syl3an3 1211	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘((𝐴 / (2↑(𝑁 + 1))) − 1)) < (⌊‘((𝐴 − 1) / (2↑(𝑁 + 1)))))
33		1zzd 11736	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → 1 ∈ ℤ)
34		flsubz 43159	. . . . . 6 ⊢ (((𝐴 / (2↑(𝑁 + 1))) ∈ ℝ ∧ 1 ∈ ℤ) → (⌊‘((𝐴 / (2↑(𝑁 + 1))) − 1)) = ((⌊‘(𝐴 / (2↑(𝑁 + 1)))) − 1))
35	28, 33, 34	syl2anc 581	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘((𝐴 / (2↑(𝑁 + 1))) − 1)) = ((⌊‘(𝐴 / (2↑(𝑁 + 1)))) − 1))
36	35	eqcomd 2831	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → ((⌊‘(𝐴 / (2↑(𝑁 + 1)))) − 1) = (⌊‘((𝐴 / (2↑(𝑁 + 1))) − 1)))
37		nnz 11727	. . . . . . . . . 10 ⊢ (((𝐴 − 1) / 2) ∈ ℕ → ((𝐴 − 1) / 2) ∈ ℤ)
38		zob 15457	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → (((𝐴 + 1) / 2) ∈ ℤ ↔ ((𝐴 − 1) / 2) ∈ ℤ))
39	37, 38	syl5ibr 238	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → (((𝐴 − 1) / 2) ∈ ℕ → ((𝐴 + 1) / 2) ∈ ℤ))
40	39	imp 397	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ) → ((𝐴 + 1) / 2) ∈ ℤ)
41		zofldiv2 43172	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 + 1) / 2) ∈ ℤ) → (⌊‘(𝐴 / 2)) = ((𝐴 − 1) / 2))
42	40, 41	syldan 587	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ) → (⌊‘(𝐴 / 2)) = ((𝐴 − 1) / 2))
43	42	3adant3 1168	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘(𝐴 / 2)) = ((𝐴 − 1) / 2))
44	43	fvoveq1d 6927	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))) = (⌊‘(((𝐴 − 1) / 2) / (2↑𝑁))))
45		zcn 11709	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
46		1cnd 10351	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → 1 ∈ ℂ)
47	45, 46	subcld 10713	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → (𝐴 − 1) ∈ ℂ)
48		2rp 12117	. . . . . . . . . 10 ⊢ 2 ∈ ℝ⁺
49	48	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 − 1) / 2) ∈ ℕ → 2 ∈ ℝ⁺)
50	49	rpcnne0d 12165	. . . . . . . 8 ⊢ (((𝐴 − 1) / 2) ∈ ℕ → (2 ∈ ℂ ∧ 2 ≠ 0))
51	48	a1i 11	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → 2 ∈ ℝ⁺)
52	51, 14	rpexpcld 13328	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (2↑𝑁) ∈ ℝ⁺)
53	52	rpcnne0d 12165	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → ((2↑𝑁) ∈ ℂ ∧ (2↑𝑁) ≠ 0))
54		divdiv1 11062	. . . . . . . 8 ⊢ (((𝐴 − 1) ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ ((2↑𝑁) ∈ ℂ ∧ (2↑𝑁) ≠ 0)) → (((𝐴 − 1) / 2) / (2↑𝑁)) = ((𝐴 − 1) / (2 · (2↑𝑁))))
55	47, 50, 53, 54	syl3an 1205	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (((𝐴 − 1) / 2) / (2↑𝑁)) = ((𝐴 − 1) / (2 · (2↑𝑁))))
56	10	recnd 10385	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℂ)
57	11, 56	mulcomd 10378	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2 · (2↑𝑁)) = ((2↑𝑁) · 2))
58	11, 21	expp1d 13303	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2↑(𝑁 + 1)) = ((2↑𝑁) · 2))
59	57, 58	eqtr4d 2864	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2 · (2↑𝑁)) = (2↑(𝑁 + 1)))
60	59	oveq2d 6921	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 − 1) / (2 · (2↑𝑁))) = ((𝐴 − 1) / (2↑(𝑁 + 1))))
61	55, 60	eqtrd 2861	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (((𝐴 − 1) / 2) / (2↑𝑁)) = ((𝐴 − 1) / (2↑(𝑁 + 1))))
62	61	fveq2d 6437	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘(((𝐴 − 1) / 2) / (2↑𝑁))) = (⌊‘((𝐴 − 1) / (2↑(𝑁 + 1)))))
63	44, 62	eqtrd 2861	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))) = (⌊‘((𝐴 − 1) / (2↑(𝑁 + 1)))))
64	32, 36, 63	3brtr4d 4905	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → ((⌊‘(𝐴 / (2↑(𝑁 + 1)))) − 1) < (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))))
65	19	rehalfcld 11605	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (𝐴 / 2) ∈ ℝ)
66	65, 10, 16	redivcld 11179	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 / 2) / (2↑𝑁)) ∈ ℝ)
67		reflcl 12892	. . . . . . 7 ⊢ ((𝐴 / 2) ∈ ℝ → (⌊‘(𝐴 / 2)) ∈ ℝ)
68	65, 67	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘(𝐴 / 2)) ∈ ℝ)
69	48	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → 2 ∈ ℝ⁺)
70	69, 15	rpexpcld 13328	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℝ⁺)
71		flle 12895	. . . . . . 7 ⊢ ((𝐴 / 2) ∈ ℝ → (⌊‘(𝐴 / 2)) ≤ (𝐴 / 2))
72	65, 71	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘(𝐴 / 2)) ≤ (𝐴 / 2))
73	68, 65, 70, 72	lediv1dd 12214	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → ((⌊‘(𝐴 / 2)) / (2↑𝑁)) ≤ ((𝐴 / 2) / (2↑𝑁)))
74		flwordi 12908	. . . . 5 ⊢ ((((⌊‘(𝐴 / 2)) / (2↑𝑁)) ∈ ℝ ∧ ((𝐴 / 2) / (2↑𝑁)) ∈ ℝ ∧ ((⌊‘(𝐴 / 2)) / (2↑𝑁)) ≤ ((𝐴 / 2) / (2↑𝑁))) → (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))) ≤ (⌊‘((𝐴 / 2) / (2↑𝑁))))
75	17, 66, 73, 74	syl3anc 1496	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))) ≤ (⌊‘((𝐴 / 2) / (2↑𝑁))))
76		divdiv1 11062	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ ((2↑𝑁) ∈ ℂ ∧ (2↑𝑁) ≠ 0)) → ((𝐴 / 2) / (2↑𝑁)) = (𝐴 / (2 · (2↑𝑁))))
77	45, 50, 53, 76	syl3an 1205	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 / 2) / (2↑𝑁)) = (𝐴 / (2 · (2↑𝑁))))
78	52	rpcnd 12158	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → (2↑𝑁) ∈ ℂ)
79	78	3ad2ant3 1171	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℂ)
80	11, 79	mulcomd 10378	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2 · (2↑𝑁)) = ((2↑𝑁) · 2))
81	11, 13, 15	expp1zd 13311	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2↑(𝑁 + 1)) = ((2↑𝑁) · 2))
82	80, 81	eqtr4d 2864	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2 · (2↑𝑁)) = (2↑(𝑁 + 1)))
83	82	oveq2d 6921	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (𝐴 / (2 · (2↑𝑁))) = (𝐴 / (2↑(𝑁 + 1))))
84	77, 83	eqtrd 2861	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 / 2) / (2↑𝑁)) = (𝐴 / (2↑(𝑁 + 1))))
85	84	eqcomd 2831	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (𝐴 / (2↑(𝑁 + 1))) = ((𝐴 / 2) / (2↑𝑁)))
86	85	fveq2d 6437	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘(𝐴 / (2↑(𝑁 + 1)))) = (⌊‘((𝐴 / 2) / (2↑𝑁))))
87	75, 86	breqtrrd 4901	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))) ≤ (⌊‘(𝐴 / (2↑(𝑁 + 1)))))
88		zgtp1leeq 43158	. . . 4 ⊢ (((⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))) ∈ ℤ ∧ (⌊‘(𝐴 / (2↑(𝑁 + 1)))) ∈ ℤ) → ((((⌊‘(𝐴 / (2↑(𝑁 + 1)))) − 1) < (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))) ∧ (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))) ≤ (⌊‘(𝐴 / (2↑(𝑁 + 1))))) → (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))) = (⌊‘(𝐴 / (2↑(𝑁 + 1))))))
89	88	imp 397	. . 3 ⊢ ((((⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))) ∈ ℤ ∧ (⌊‘(𝐴 / (2↑(𝑁 + 1)))) ∈ ℤ) ∧ (((⌊‘(𝐴 / (2↑(𝑁 + 1)))) − 1) < (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))) ∧ (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))) ≤ (⌊‘(𝐴 / (2↑(𝑁 + 1)))))) → (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))) = (⌊‘(𝐴 / (2↑(𝑁 + 1)))))
90	18, 29, 64, 87, 89	syl22anc 874	. 2 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))) = (⌊‘(𝐴 / (2↑(𝑁 + 1)))))
91	90	eqcomd 2831	1 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘(𝐴 / (2↑(𝑁 + 1)))) = (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ≠ wne 2999 class class class wbr 4873 ‘cfv 6123 (class class class)co 6905 ℂcc 10250 ℝcr 10251 0cc0 10252 1c1 10253 + caddc 10255 · cmul 10257 < clt 10391 ≤ cle 10392 − cmin 10585 / cdiv 11009 ℕcn 11350 2c2 11406 ℕ₀cn0 11618 ℤcz 11704 ℝ⁺crp 12112 ⌊cfl 12886 ↑cexp 13154

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-sup 8617 df-inf 8618 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-n0 11619 df-z 11705 df-uz 11969 df-rp 12113 df-fz 12620 df-fzo 12761 df-fl 12888 df-mod 12964 df-seq 13096 df-exp 13155

This theorem is referenced by: dignn0flhalf 43259

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