bitsfzo - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > bitsfzo		Structured version Visualization version GIF version

Theorem bitsfzo 16242

Description: The bits of a number are all less than 𝑀 iff the number is nonnegative and less than 2↑𝑀. (Contributed by Mario Carneiro, 5-Sep-2016.) (Proof shortened by AV, 1-Oct-2020.)

**Assertion**
Ref	Expression
bitsfzo	⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) → (𝑁 ∈ (0..^(2↑𝑀)) ↔ (bits‘𝑁) ⊆ (0..^𝑀)))

Proof of Theorem bitsfzo Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bitsval 16231	. . . 4 ⊢ (𝑚 ∈ (bits‘𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))))
2		simp32 1209	. . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑚 ∈ ℕ₀)
3		nn0uz 12722	. . . . . . 7 ⊢ ℕ₀ = (ℤ_≥‘0)
4	2, 3	eleqtrdi 2847	. . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑚 ∈ (ℤ_≥‘0))
5		simp1r 1197	. . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑀 ∈ ℕ₀)
6	5	nn0zd 12526	. . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑀 ∈ ℤ)
7		2re 12149	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
8	7	a1i 11	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 2 ∈ ℝ)
9	8, 2	reexpcld 13983	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) ∈ ℝ)
10		simp1l 1196	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑁 ∈ ℤ)
11	10	zred 12528	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑁 ∈ ℝ)
12	8, 5	reexpcld 13983	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑀) ∈ ℝ)
13	9	recnd 11105	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) ∈ ℂ)
14	13	mulid2d 11095	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (1 · (2↑𝑚)) = (2↑𝑚))
15		simp33 1210	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))
16		2rp 12837	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ⁺
17	16	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 2 ∈ ℝ⁺)
18	2	nn0zd 12526	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑚 ∈ ℤ)
19	17, 18	rpexpcld 14064	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) ∈ ℝ⁺)
20	11, 19	rerpdivcld 12905	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (𝑁 / (2↑𝑚)) ∈ ℝ)
21		1red 11078	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 1 ∈ ℝ)
22	20, 21	ltnled 11224	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → ((𝑁 / (2↑𝑚)) < 1 ↔ ¬ 1 ≤ (𝑁 / (2↑𝑚))))
23		0p1e1 12197	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
24	23	breq2i 5101	. . . . . . . . . . . . 13 ⊢ ((𝑁 / (2↑𝑚)) < (0 + 1) ↔ (𝑁 / (2↑𝑚)) < 1)
25		elfzole1 13497	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (0..^(2↑𝑀)) → 0 ≤ 𝑁)
26	25	3ad2ant2 1133	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 0 ≤ 𝑁)
27	11, 19, 26	divge0d 12914	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 0 ≤ (𝑁 / (2↑𝑚)))
28		0z 12432	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
29		flbi 13638	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 / (2↑𝑚)) ∈ ℝ ∧ 0 ∈ ℤ) → ((⌊‘(𝑁 / (2↑𝑚))) = 0 ↔ (0 ≤ (𝑁 / (2↑𝑚)) ∧ (𝑁 / (2↑𝑚)) < (0 + 1))))
30	20, 28, 29	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → ((⌊‘(𝑁 / (2↑𝑚))) = 0 ↔ (0 ≤ (𝑁 / (2↑𝑚)) ∧ (𝑁 / (2↑𝑚)) < (0 + 1))))
31		z0even 16176	. . . . . . . . . . . . . . . 16 ⊢ 2 ∥ 0
32		id 22	. . . . . . . . . . . . . . . 16 ⊢ ((⌊‘(𝑁 / (2↑𝑚))) = 0 → (⌊‘(𝑁 / (2↑𝑚))) = 0)
33	31, 32	breqtrrid 5131	. . . . . . . . . . . . . . 15 ⊢ ((⌊‘(𝑁 / (2↑𝑚))) = 0 → 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))
34	30, 33	syl6bir 253	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → ((0 ≤ (𝑁 / (2↑𝑚)) ∧ (𝑁 / (2↑𝑚)) < (0 + 1)) → 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))))
35	27, 34	mpand 692	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → ((𝑁 / (2↑𝑚)) < (0 + 1) → 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))))
36	24, 35	syl5bir 242	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → ((𝑁 / (2↑𝑚)) < 1 → 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))))
37	22, 36	sylbird 259	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (¬ 1 ≤ (𝑁 / (2↑𝑚)) → 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))))
38	15, 37	mt3d 148	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 1 ≤ (𝑁 / (2↑𝑚)))
39	21, 11, 19	lemuldivd 12923	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → ((1 · (2↑𝑚)) ≤ 𝑁 ↔ 1 ≤ (𝑁 / (2↑𝑚))))
40	38, 39	mpbird 256	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (1 · (2↑𝑚)) ≤ 𝑁)
41	14, 40	eqbrtrrd 5117	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) ≤ 𝑁)
42		elfzolt2 13498	. . . . . . . . 9 ⊢ (𝑁 ∈ (0..^(2↑𝑀)) → 𝑁 < (2↑𝑀))
43	42	3ad2ant2 1133	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑁 < (2↑𝑀))
44	9, 11, 12, 41, 43	lelttrd 11235	. . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) < (2↑𝑀))
45		1lt2 12246	. . . . . . . . 9 ⊢ 1 < 2
46	45	a1i 11	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 1 < 2)
47	8, 18, 6, 46	ltexp2d 14070	. . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (𝑚 < 𝑀 ↔ (2↑𝑚) < (2↑𝑀)))
48	44, 47	mpbird 256	. . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑚 < 𝑀)
49		elfzo2 13492	. . . . . 6 ⊢ (𝑚 ∈ (0..^𝑀) ↔ (𝑚 ∈ (ℤ_≥‘0) ∧ 𝑀 ∈ ℤ ∧ 𝑚 < 𝑀))
50	4, 6, 48, 49	syl3anbrc 1342	. . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑚 ∈ (0..^𝑀))
51	50	3expia 1120	. . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀))) → ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))) → 𝑚 ∈ (0..^𝑀)))
52	1, 51	biimtrid 241	. . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀))) → (𝑚 ∈ (bits‘𝑁) → 𝑚 ∈ (0..^𝑀)))
53	52	ssrdv 3938	. 2 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀))) → (bits‘𝑁) ⊆ (0..^𝑀))
54		simpr 485	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ∈ ℕ)
55	54	nnred 12090	. . . . . . 7 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ∈ ℝ)
56		simpllr 773	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑀 ∈ ℕ₀)
57	56	nn0red 12396	. . . . . . 7 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑀 ∈ ℝ)
58		max2 13023	. . . . . . 7 ⊢ ((-𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → 𝑀 ≤ if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))
59	55, 57, 58	syl2anc 584	. . . . . 6 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑀 ≤ if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))
60		simplr 766	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (bits‘𝑁) ⊆ (0..^𝑀))
61		n2dvdsm1 16178	. . . . . . . . . . 11 ⊢ ¬ 2 ∥ -1
62		simplll 772	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑁 ∈ ℤ)
63	62	zred 12528	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑁 ∈ ℝ)
64		2nn 12148	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ
65	64	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 2 ∈ ℕ)
66	54	nnnn0d 12395	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ∈ ℕ₀)
67	56, 66	ifcld 4520	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) ∈ ℕ₀)
68	65, 67	nnexpcld 14062	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)) ∈ ℕ)
69	63, 68	nndivred 12129	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))) ∈ ℝ)
70		1red 11078	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 1 ∈ ℝ)
71	62	zcnd 12529	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑁 ∈ ℂ)
72	68	nncnd 12091	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)) ∈ ℂ)
73		2cnd 12153	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 2 ∈ ℂ)
74		2ne0 12179	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ≠ 0
75	74	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 2 ≠ 0)
76	67	nn0zd 12526	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) ∈ ℤ)
77	73, 75, 76	expne0d 13972	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)) ≠ 0)
78	71, 72, 77	divnegd 11866	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -(𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))) = (-𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))))
79	67	nn0red 12396	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) ∈ ℝ)
80	68	nnred 12090	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)) ∈ ℝ)
81		max1 13021	. . . . . . . . . . . . . . . . . . 19 ⊢ ((-𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → -𝑁 ≤ if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))
82	55, 57, 81	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ≤ if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))
83		2z 12454	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℤ
84		uzid 12699	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
85	83, 84	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ (ℤ_≥‘2)
86		bernneq3 14048	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) ∈ ℕ₀) → if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) < (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)))
87	85, 67, 86	sylancr 587	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) < (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)))
88	79, 80, 87	ltled 11225	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) ≤ (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)))
89	55, 79, 80, 82, 88	letrd 11234	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ≤ (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)))
90	72	mulid1d 11094	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → ((2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)) · 1) = (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)))
91	89, 90	breqtrrd 5121	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ≤ ((2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)) · 1))
92	68	nnrpd 12872	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)) ∈ ℝ⁺)
93	55, 70, 92	ledivmuld 12927	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → ((-𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))) ≤ 1 ↔ -𝑁 ≤ ((2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)) · 1)))
94	91, 93	mpbird 256	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (-𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))) ≤ 1)
95	78, 94	eqbrtrd 5115	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -(𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))) ≤ 1)
96	69, 70, 95	lenegcon1d 11659	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -1 ≤ (𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))))
97	54	nngt0d 12124	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 0 < -𝑁)
98	68	nngt0d 12124	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 0 < (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)))
99	55, 80, 97, 98	divgt0d 12012	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 0 < (-𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))))
100	99, 78	breqtrrd 5121	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 0 < -(𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))))
101	69	lt0neg1d 11646	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → ((𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))) < 0 ↔ 0 < -(𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)))))
102	100, 101	mpbird 256	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))) < 0)
103		ax-1cn 11031	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
104		neg1cn 12189	. . . . . . . . . . . . . . 15 ⊢ -1 ∈ ℂ
105		1pneg1e0 12194	. . . . . . . . . . . . . . 15 ⊢ (1 + -1) = 0
106	103, 104, 105	addcomli 11269	. . . . . . . . . . . . . 14 ⊢ (-1 + 1) = 0
107	102, 106	breqtrrdi 5135	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))) < (-1 + 1))
108		neg1z 12458	. . . . . . . . . . . . . 14 ⊢ -1 ∈ ℤ
109		flbi 13638	. . . . . . . . . . . . . 14 ⊢ (((𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))) ∈ ℝ ∧ -1 ∈ ℤ) → ((⌊‘(𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)))) = -1 ↔ (-1 ≤ (𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))) ∧ (𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))) < (-1 + 1))))
110	69, 108, 109	sylancl 586	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → ((⌊‘(𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)))) = -1 ↔ (-1 ≤ (𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))) ∧ (𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))) < (-1 + 1))))
111	96, 107, 110	mpbir2and 710	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (⌊‘(𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)))) = -1)
112	111	breq2d 5105	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2 ∥ (⌊‘(𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)))) ↔ 2 ∥ -1))
113	61, 112	mtbiri 326	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → ¬ 2 ∥ (⌊‘(𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)))))
114		bitsval2 16232	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) ∈ ℕ₀) → (if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) ∈ (bits‘𝑁) ↔ ¬ 2 ∥ (⌊‘(𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))))))
115	62, 67, 114	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) ∈ (bits‘𝑁) ↔ ¬ 2 ∥ (⌊‘(𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))))))
116	113, 115	mpbird 256	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) ∈ (bits‘𝑁))
117	60, 116	sseldd 3933	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) ∈ (0..^𝑀))
118		elfzolt2 13498	. . . . . . . 8 ⊢ (if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) ∈ (0..^𝑀) → if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) < 𝑀)
119	117, 118	syl 17	. . . . . . 7 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) < 𝑀)
120	79, 57	ltnled 11224	. . . . . . 7 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) < 𝑀 ↔ ¬ 𝑀 ≤ if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)))
121	119, 120	mpbid 231	. . . . . 6 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → ¬ 𝑀 ≤ if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))
122	59, 121	pm2.65da 814	. . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → ¬ -𝑁 ∈ ℕ)
123	122	intnand 489	. . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → ¬ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))
124		simpll 764	. . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → 𝑁 ∈ ℤ)
125		elznn0nn 12435	. . . . . 6 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ₀ ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)))
126	124, 125	sylib 217	. . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → (𝑁 ∈ ℕ₀ ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)))
127	126	ord 861	. . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → (¬ 𝑁 ∈ ℕ₀ → (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)))
128	123, 127	mt3d 148	. . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → 𝑁 ∈ ℕ₀)
129		simplr 766	. . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → 𝑀 ∈ ℕ₀)
130		simpr 485	. . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → (bits‘𝑁) ⊆ (0..^𝑀))
131		eqid 2736	. . 3 ⊢ inf({𝑛 ∈ ℕ₀ ∣ 𝑁 < (2↑𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ₀ ∣ 𝑁 < (2↑𝑛)}, ℝ, < )
132	128, 129, 130, 131	bitsfzolem 16241	. 2 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → 𝑁 ∈ (0..^(2↑𝑀)))
133	53, 132	impbida 798	1 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) → (𝑁 ∈ (0..^(2↑𝑀)) ↔ (bits‘𝑁) ⊆ (0..^𝑀)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2940 {crab 3403 ⊆ wss 3898 ifcif 4474 class class class wbr 5093 ‘cfv 6480 (class class class)co 7338 infcinf 9299 ℝcr 10972 0cc0 10973 1c1 10974 + caddc 10976 · cmul 10978 < clt 11111 ≤ cle 11112 -cneg 11308 / cdiv 11734 ℕcn 12075 2c2 12130 ℕ₀cn0 12335 ℤcz 12421 ℤ_≥cuz 12684 ℝ⁺crp 12832 ..^cfzo 13484 ⌊cfl 13612 ↑cexp 13884 ∥ cdvds 16063 bitscbits 16226

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5244 ax-nul 5251 ax-pow 5309 ax-pr 5373 ax-un 7651 ax-cnex 11029 ax-resscn 11030 ax-1cn 11031 ax-icn 11032 ax-addcl 11033 ax-addrcl 11034 ax-mulcl 11035 ax-mulrcl 11036 ax-mulcom 11037 ax-addass 11038 ax-mulass 11039 ax-distr 11040 ax-i2m1 11041 ax-1ne0 11042 ax-1rid 11043 ax-rnegex 11044 ax-rrecex 11045 ax-cnre 11046 ax-pre-lttri 11047 ax-pre-lttrn 11048 ax-pre-ltadd 11049 ax-pre-mulgt0 11050 ax-pre-sup 11051

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4271 df-if 4475 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4854 df-iun 4944 df-br 5094 df-opab 5156 df-mpt 5177 df-tr 5211 df-id 5519 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5576 df-we 5578 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 6239 df-ord 6306 df-on 6307 df-lim 6308 df-suc 6309 df-iota 6432 df-fun 6482 df-fn 6483 df-f 6484 df-f1 6485 df-fo 6486 df-f1o 6487 df-fv 6488 df-riota 7294 df-ov 7341 df-oprab 7342 df-mpo 7343 df-om 7782 df-1st 7900 df-2nd 7901 df-frecs 8168 df-wrecs 8199 df-recs 8273 df-rdg 8312 df-er 8570 df-en 8806 df-dom 8807 df-sdom 8808 df-sup 9300 df-inf 9301 df-pnf 11113 df-mnf 11114 df-xr 11115 df-ltxr 11116 df-le 11117 df-sub 11309 df-neg 11310 df-div 11735 df-nn 12076 df-2 12138 df-n0 12336 df-z 12422 df-uz 12685 df-rp 12833 df-fz 13342 df-fzo 13485 df-fl 13614 df-seq 13824 df-exp 13885 df-dvds 16064 df-bits 16229

This theorem is referenced by: bitsfi 16244 0bits 16246 bitsinv1 16249 sadcaddlem 16264 sadaddlem 16273 sadasslem 16277 sadeq 16279

W3C validator