dignn0flhalflem2 - Mathbox for Alexander van der Vekens

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Theorem dignn0flhalflem2 45030

Description: Lemma 2 for dignn0flhalf 45032. (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 11973	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
2	1	rehalfcld 11872	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → (𝐴 / 2) ∈ ℝ)
3	2	flcld 13163	. . . . . . 7 ⊢ (𝐴 ∈ ℤ → (⌊‘(𝐴 / 2)) ∈ ℤ)
4	3	zred 12075	. . . . . 6 ⊢ (𝐴 ∈ ℤ → (⌊‘(𝐴 / 2)) ∈ ℝ)
5	4	3ad2ant1 1130	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘(𝐴 / 2)) ∈ ℝ)
6		2re 11699	. . . . . . . 8 ⊢ 2 ∈ ℝ
7	6	a1i 11	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 2 ∈ ℝ)
8		id 22	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℕ₀)
9	7, 8	reexpcld 13523	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → (2↑𝑁) ∈ ℝ)
10	9	3ad2ant3 1132	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℝ)
11		2cnd 11703	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → 2 ∈ ℂ)
12		2ne0 11729	. . . . . . 7 ⊢ 2 ≠ 0
13	12	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → 2 ≠ 0)
14		nn0z 11993	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
15	14	3ad2ant3 1132	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℤ)
16	11, 13, 15	expne0d 13512	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ≠ 0)
17	5, 10, 16	redivcld 11457	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → ((⌊‘(𝐴 / 2)) / (2↑𝑁)) ∈ ℝ)
18	17	flcld 13163	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))) ∈ ℤ)
19	1	3ad2ant1 1130	. . . . 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 1135	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℕ₀)
22		1nn0 11901	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
23	22	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → 1 ∈ ℕ₀)
24	21, 23	nn0addcld 11947	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ∈ ℕ₀)
25	20, 24	reexpcld 13523	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2↑(𝑁 + 1)) ∈ ℝ)
26	15	peano2zd 12078	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ∈ ℤ)
27	11, 13, 26	expne0d 13512	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2↑(𝑁 + 1)) ≠ 0)
28	19, 25, 27	redivcld 11457	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (𝐴 / (2↑(𝑁 + 1))) ∈ ℝ)
29	28	flcld 13163	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘(𝐴 / (2↑(𝑁 + 1)))) ∈ ℤ)
30		nn0p1nn 11924	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ)
31		dignn0flhalflem1 45029	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ (𝑁 + 1) ∈ ℕ) → (⌊‘((𝐴 / (2↑(𝑁 + 1))) − 1)) < (⌊‘((𝐴 − 1) / (2↑(𝑁 + 1)))))
32	30, 31	syl3an3 1162	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘((𝐴 / (2↑(𝑁 + 1))) − 1)) < (⌊‘((𝐴 − 1) / (2↑(𝑁 + 1)))))
33		1zzd 12001	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → 1 ∈ ℤ)
34		flsubz 44931	. . . . . 6 ⊢ (((𝐴 / (2↑(𝑁 + 1))) ∈ ℝ ∧ 1 ∈ ℤ) → (⌊‘((𝐴 / (2↑(𝑁 + 1))) − 1)) = ((⌊‘(𝐴 / (2↑(𝑁 + 1)))) − 1))
35	28, 33, 34	syl2anc 587	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘((𝐴 / (2↑(𝑁 + 1))) − 1)) = ((⌊‘(𝐴 / (2↑(𝑁 + 1)))) − 1))
36	35	eqcomd 2804	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → ((⌊‘(𝐴 / (2↑(𝑁 + 1)))) − 1) = (⌊‘((𝐴 / (2↑(𝑁 + 1))) − 1)))
37		nnz 11992	. . . . . . . . . 10 ⊢ (((𝐴 − 1) / 2) ∈ ℕ → ((𝐴 − 1) / 2) ∈ ℤ)
38		zob 15700	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → (((𝐴 + 1) / 2) ∈ ℤ ↔ ((𝐴 − 1) / 2) ∈ ℤ))
39	37, 38	syl5ibr 249	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → (((𝐴 − 1) / 2) ∈ ℕ → ((𝐴 + 1) / 2) ∈ ℤ))
40	39	imp 410	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ) → ((𝐴 + 1) / 2) ∈ ℤ)
41		zofldiv2 44945	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 + 1) / 2) ∈ ℤ) → (⌊‘(𝐴 / 2)) = ((𝐴 − 1) / 2))
42	40, 41	syldan 594	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ) → (⌊‘(𝐴 / 2)) = ((𝐴 − 1) / 2))
43	42	3adant3 1129	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘(𝐴 / 2)) = ((𝐴 − 1) / 2))
44	43	fvoveq1d 7157	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))) = (⌊‘(((𝐴 − 1) / 2) / (2↑𝑁))))
45		zcn 11974	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
46		1cnd 10625	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → 1 ∈ ℂ)
47	45, 46	subcld 10986	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → (𝐴 − 1) ∈ ℂ)
48		2rp 12382	. . . . . . . . . 10 ⊢ 2 ∈ ℝ⁺
49	48	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 − 1) / 2) ∈ ℕ → 2 ∈ ℝ⁺)
50	49	rpcnne0d 12428	. . . . . . . 8 ⊢ (((𝐴 − 1) / 2) ∈ ℕ → (2 ∈ ℂ ∧ 2 ≠ 0))
51	48	a1i 11	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → 2 ∈ ℝ⁺)
52	51, 14	rpexpcld 13604	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (2↑𝑁) ∈ ℝ⁺)
53	52	rpcnne0d 12428	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → ((2↑𝑁) ∈ ℂ ∧ (2↑𝑁) ≠ 0))
54		divdiv1 11340	. . . . . . . 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 1157	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (((𝐴 − 1) / 2) / (2↑𝑁)) = ((𝐴 − 1) / (2 · (2↑𝑁))))
56	10	recnd 10658	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℂ)
57	11, 56	mulcomd 10651	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2 · (2↑𝑁)) = ((2↑𝑁) · 2))
58	11, 21	expp1d 13507	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2↑(𝑁 + 1)) = ((2↑𝑁) · 2))
59	57, 58	eqtr4d 2836	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2 · (2↑𝑁)) = (2↑(𝑁 + 1)))
60	59	oveq2d 7151	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 − 1) / (2 · (2↑𝑁))) = ((𝐴 − 1) / (2↑(𝑁 + 1))))
61	55, 60	eqtrd 2833	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (((𝐴 − 1) / 2) / (2↑𝑁)) = ((𝐴 − 1) / (2↑(𝑁 + 1))))
62	61	fveq2d 6649	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘(((𝐴 − 1) / 2) / (2↑𝑁))) = (⌊‘((𝐴 − 1) / (2↑(𝑁 + 1)))))
63	44, 62	eqtrd 2833	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))) = (⌊‘((𝐴 − 1) / (2↑(𝑁 + 1)))))
64	32, 36, 63	3brtr4d 5062	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → ((⌊‘(𝐴 / (2↑(𝑁 + 1)))) − 1) < (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))))
65	19	rehalfcld 11872	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (𝐴 / 2) ∈ ℝ)
66	65, 10, 16	redivcld 11457	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 / 2) / (2↑𝑁)) ∈ ℝ)
67		reflcl 13161	. . . . . . 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 13604	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℝ⁺)
71		flle 13164	. . . . . . 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 12477	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → ((⌊‘(𝐴 / 2)) / (2↑𝑁)) ≤ ((𝐴 / 2) / (2↑𝑁)))
74		flwordi 13177	. . . . 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 1368	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))) ≤ (⌊‘((𝐴 / 2) / (2↑𝑁))))
76		divdiv1 11340	. . . . . . . 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 1157	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 / 2) / (2↑𝑁)) = (𝐴 / (2 · (2↑𝑁))))
78	52	rpcnd 12421	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → (2↑𝑁) ∈ ℂ)
79	78	3ad2ant3 1132	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℂ)
80	11, 79	mulcomd 10651	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2 · (2↑𝑁)) = ((2↑𝑁) · 2))
81	11, 13, 15	expp1zd 13515	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2↑(𝑁 + 1)) = ((2↑𝑁) · 2))
82	80, 81	eqtr4d 2836	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2 · (2↑𝑁)) = (2↑(𝑁 + 1)))
83	82	oveq2d 7151	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (𝐴 / (2 · (2↑𝑁))) = (𝐴 / (2↑(𝑁 + 1))))
84	77, 83	eqtrd 2833	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 / 2) / (2↑𝑁)) = (𝐴 / (2↑(𝑁 + 1))))
85	84	eqcomd 2804	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (𝐴 / (2↑(𝑁 + 1))) = ((𝐴 / 2) / (2↑𝑁)))
86	85	fveq2d 6649	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘(𝐴 / (2↑(𝑁 + 1)))) = (⌊‘((𝐴 / 2) / (2↑𝑁))))
87	75, 86	breqtrrd 5058	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))) ≤ (⌊‘(𝐴 / (2↑(𝑁 + 1)))))
88		zgtp1leeq 44930	. . . 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 410	. . 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 837	. 2 ⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (⌊‘((⌊‘(𝐴 / 2)) / (2↑𝑁))) = (⌊‘(𝐴 / (2↑(𝑁 + 1)))))
91	90	eqcomd 2804	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 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 < clt 10664 ≤ cle 10665 − cmin 10859 / cdiv 11286 ℕcn 11625 2c2 11680 ℕ₀cn0 11885 ℤcz 11969 ℝ⁺crp 12377 ⌊cfl 13155 ↑cexp 13425

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426

This theorem is referenced by: dignn0flhalf 45032

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