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Theorem bitsfzo 15778

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 15767	. . . 4 ⊢ (𝑚 ∈ (bits‘𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))))
2		simp32 1206	. . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑚 ∈ ℕ₀)
3		nn0uz 12274	. . . . . . 7 ⊢ ℕ₀ = (ℤ_≥‘0)
4	2, 3	eleqtrdi 2923	. . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑚 ∈ (ℤ_≥‘0))
5		simp1r 1194	. . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑀 ∈ ℕ₀)
6	5	nn0zd 12079	. . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑀 ∈ ℤ)
7		2re 11705	. . . . . . . . . 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 13521	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) ∈ ℝ)
10		simp1l 1193	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑁 ∈ ℤ)
11	10	zred 12081	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑁 ∈ ℝ)
12	8, 5	reexpcld 13521	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑀) ∈ ℝ)
13	9	recnd 10663	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) ∈ ℂ)
14	13	mulid2d 10653	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (1 · (2↑𝑚)) = (2↑𝑚))
15		simp33 1207	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))
16		2rp 12388	. . . . . . . . . . . . . . . 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 12079	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑚 ∈ ℤ)
19	17, 18	rpexpcld 13602	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) ∈ ℝ⁺)
20	11, 19	rerpdivcld 12456	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (𝑁 / (2↑𝑚)) ∈ ℝ)
21		1red 10636	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 1 ∈ ℝ)
22	20, 21	ltnled 10781	. . . . . . . . . . . 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 11753	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
24	23	breq2i 5066	. . . . . . . . . . . . 13 ⊢ ((𝑁 / (2↑𝑚)) < (0 + 1) ↔ (𝑁 / (2↑𝑚)) < 1)
25		elfzole1 13040	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (0..^(2↑𝑀)) → 0 ≤ 𝑁)
26	25	3ad2ant2 1130	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 0 ≤ 𝑁)
27	11, 19, 26	divge0d 12465	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 0 ≤ (𝑁 / (2↑𝑚)))
28		0z 11986	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
29		flbi 13180	. . . . . . . . . . . . . . . 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 588	. . . . . . . . . . . . . . 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 15710	. . . . . . . . . . . . . . . 16 ⊢ 2 ∥ 0
32		id 22	. . . . . . . . . . . . . . . 16 ⊢ ((⌊‘(𝑁 / (2↑𝑚))) = 0 → (⌊‘(𝑁 / (2↑𝑚))) = 0)
33	31, 32	breqtrrid 5096	. . . . . . . . . . . . . . 15 ⊢ ((⌊‘(𝑁 / (2↑𝑚))) = 0 → 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))
34	30, 33	syl6bir 256	. . . . . . . . . . . . . 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 693	. . . . . . . . . . . . 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 245	. . . . . . . . . . . 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 262	. . . . . . . . . . 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 150	. . . . . . . . . 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 12474	. . . . . . . . . 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 259	. . . . . . . . 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 5082	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) ≤ 𝑁)
42		elfzolt2 13041	. . . . . . . . 9 ⊢ (𝑁 ∈ (0..^(2↑𝑀)) → 𝑁 < (2↑𝑀))
43	42	3ad2ant2 1130	. . . . . . . 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 10792	. . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) < (2↑𝑀))
45		1lt2 11802	. . . . . . . . 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 13608	. . . . . . 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 259	. . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑚 < 𝑀)
49		elfzo2 13035	. . . . . 6 ⊢ (𝑚 ∈ (0..^𝑀) ↔ (𝑚 ∈ (ℤ_≥‘0) ∧ 𝑀 ∈ ℤ ∧ 𝑚 < 𝑀))
50	4, 6, 48, 49	syl3anbrc 1339	. . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑚 ∈ (0..^𝑀))
51	50	3expia 1117	. . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀))) → ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))) → 𝑚 ∈ (0..^𝑀)))
52	1, 51	syl5bi 244	. . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀))) → (𝑚 ∈ (bits‘𝑁) → 𝑚 ∈ (0..^𝑀)))
53	52	ssrdv 3972	. 2 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀))) → (bits‘𝑁) ⊆ (0..^𝑀))
54		simpr 487	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ∈ ℕ)
55	54	nnred 11647	. . . . . . 7 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ∈ ℝ)
56		simpllr 774	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑀 ∈ ℕ₀)
57	56	nn0red 11950	. . . . . . 7 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑀 ∈ ℝ)
58		max2 12574	. . . . . . 7 ⊢ ((-𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → 𝑀 ≤ if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))
59	55, 57, 58	syl2anc 586	. . . . . 6 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑀 ≤ if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))
60		simplr 767	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (bits‘𝑁) ⊆ (0..^𝑀))
61		n2dvdsm1 15713	. . . . . . . . . . 11 ⊢ ¬ 2 ∥ -1
62		simplll 773	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑁 ∈ ℤ)
63	62	zred 12081	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑁 ∈ ℝ)
64		2nn 11704	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ
65	64	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 2 ∈ ℕ)
66	54	nnnn0d 11949	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ∈ ℕ₀)
67	56, 66	ifcld 4511	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) ∈ ℕ₀)
68	65, 67	nnexpcld 13600	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)) ∈ ℕ)
69	63, 68	nndivred 11685	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))) ∈ ℝ)
70		1red 10636	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 1 ∈ ℝ)
71	62	zcnd 12082	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑁 ∈ ℂ)
72	68	nncnd 11648	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)) ∈ ℂ)
73		2cnd 11709	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 2 ∈ ℂ)
74		2ne0 11735	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ≠ 0
75	74	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 2 ≠ 0)
76	67	nn0zd 12079	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) ∈ ℤ)
77	73, 75, 76	expne0d 13510	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)) ≠ 0)
78	71, 72, 77	divnegd 11423	. . . . . . . . . . . . . . 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 11950	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) ∈ ℝ)
80	68	nnred 11647	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)) ∈ ℝ)
81		max1 12572	. . . . . . . . . . . . . . . . . . 19 ⊢ ((-𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → -𝑁 ≤ if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))
82	55, 57, 81	syl2anc 586	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ≤ if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))
83		2z 12008	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℤ
84		uzid 12252	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
85	83, 84	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ (ℤ_≥‘2)
86		bernneq3 13586	. . . . . . . . . . . . . . . . . . . 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 589	. . . . . . . . . . . . . . . . . . 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 10782	. . . . . . . . . . . . . . . . . 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 10791	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ≤ (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)))
90	72	mulid1d 10652	. . . . . . . . . . . . . . . . 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 5086	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ≤ ((2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)) · 1))
92	68	nnrpd 12423	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)) ∈ ℝ⁺)
93	55, 70, 92	ledivmuld 12478	. . . . . . . . . . . . . . . 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 259	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (-𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))) ≤ 1)
95	78, 94	eqbrtrd 5080	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -(𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))) ≤ 1)
96	69, 70, 95	lenegcon1d 11216	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -1 ≤ (𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))))
97	54	nngt0d 11680	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 0 < -𝑁)
98	68	nngt0d 11680	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 0 < (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)))
99	55, 80, 97, 98	divgt0d 11569	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 0 < (-𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))))
100	99, 78	breqtrrd 5086	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 0 < -(𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))))
101	69	lt0neg1d 11203	. . . . . . . . . . . . . . 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 259	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))) < 0)
103		ax-1cn 10589	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
104		neg1cn 11745	. . . . . . . . . . . . . . 15 ⊢ -1 ∈ ℂ
105		1pneg1e0 11750	. . . . . . . . . . . . . . 15 ⊢ (1 + -1) = 0
106	103, 104, 105	addcomli 10826	. . . . . . . . . . . . . 14 ⊢ (-1 + 1) = 0
107	102, 106	breqtrrdi 5100	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))) < (-1 + 1))
108		neg1z 12012	. . . . . . . . . . . . . 14 ⊢ -1 ∈ ℤ
109		flbi 13180	. . . . . . . . . . . . . 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 588	. . . . . . . . . . . . 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 711	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (⌊‘(𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)))) = -1)
112	111	breq2d 5070	. . . . . . . . . . 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 329	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → ¬ 2 ∥ (⌊‘(𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)))))
114		bitsval2 15768	. . . . . . . . . . 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 586	. . . . . . . . . 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 259	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) ∈ (bits‘𝑁))
117	60, 116	sseldd 3967	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) ∈ (0..^𝑀))
118		elfzolt2 13041	. . . . . . . 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 10781	. . . . . . 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 234	. . . . . 6 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → ¬ 𝑀 ≤ if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))
122	59, 121	pm2.65da 815	. . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → ¬ -𝑁 ∈ ℕ)
123	122	intnand 491	. . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → ¬ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))
124		simpll 765	. . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → 𝑁 ∈ ℤ)
125		elznn0nn 11989	. . . . . 6 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ₀ ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)))
126	124, 125	sylib 220	. . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → (𝑁 ∈ ℕ₀ ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)))
127	126	ord 860	. . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → (¬ 𝑁 ∈ ℕ₀ → (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)))
128	123, 127	mt3d 150	. . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → 𝑁 ∈ ℕ₀)
129		simplr 767	. . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → 𝑀 ∈ ℕ₀)
130		simpr 487	. . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → (bits‘𝑁) ⊆ (0..^𝑀))
131		eqid 2821	. . 3 ⊢ inf({𝑛 ∈ ℕ₀ ∣ 𝑁 < (2↑𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ₀ ∣ 𝑁 < (2↑𝑛)}, ℝ, < )
132	128, 129, 130, 131	bitsfzolem 15777	. 2 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → 𝑁 ∈ (0..^(2↑𝑀)))
133	53, 132	impbida 799	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 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 {crab 3142 ⊆ wss 3935 ifcif 4466 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 infcinf 8899 ℝcr 10530 0cc0 10531 1c1 10532 + caddc 10534 · cmul 10536 < clt 10669 ≤ cle 10670 -cneg 10865 / cdiv 11291 ℕcn 11632 2c2 11686 ℕ₀cn0 11891 ℤcz 11975 ℤ_≥cuz 12237 ℝ⁺crp 12383 ..^cfzo 13027 ⌊cfl 13154 ↑cexp 13423 ∥ cdvds 15601 bitscbits 15762

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-dvds 15602 df-bits 15765

This theorem is referenced by: bitsfi 15780 0bits 15782 bitsinv1 15785 sadcaddlem 15800 sadaddlem 15809 sadasslem 15813 sadeq 15815

W3C validator