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Theorem bitsfzo 15774

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 15763	. . . 4 ⊢ (𝑚 ∈ (bits‘𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))))
2		simp32 1207	. . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑚 ∈ ℕ₀)
3		nn0uz 12268	. . . . . . 7 ⊢ ℕ₀ = (ℤ_≥‘0)
4	2, 3	eleqtrdi 2900	. . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑚 ∈ (ℤ_≥‘0))
5		simp1r 1195	. . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑀 ∈ ℕ₀)
6	5	nn0zd 12073	. . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑀 ∈ ℤ)
7		2re 11699	. . . . . . . . . 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 13523	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) ∈ ℝ)
10		simp1l 1194	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑁 ∈ ℤ)
11	10	zred 12075	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑁 ∈ ℝ)
12	8, 5	reexpcld 13523	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑀) ∈ ℝ)
13	9	recnd 10658	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) ∈ ℂ)
14	13	mulid2d 10648	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (1 · (2↑𝑚)) = (2↑𝑚))
15		simp33 1208	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))
16		2rp 12382	. . . . . . . . . . . . . . . 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 12073	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑚 ∈ ℤ)
19	17, 18	rpexpcld 13604	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) ∈ ℝ⁺)
20	11, 19	rerpdivcld 12450	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (𝑁 / (2↑𝑚)) ∈ ℝ)
21		1red 10631	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 1 ∈ ℝ)
22	20, 21	ltnled 10776	. . . . . . . . . . . 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 11747	. . . . . . . . . . . . . 14 ⊢ (0 + 1) = 1
24	23	breq2i 5038	. . . . . . . . . . . . 13 ⊢ ((𝑁 / (2↑𝑚)) < (0 + 1) ↔ (𝑁 / (2↑𝑚)) < 1)
25		elfzole1 13041	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (0..^(2↑𝑀)) → 0 ≤ 𝑁)
26	25	3ad2ant2 1131	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 0 ≤ 𝑁)
27	11, 19, 26	divge0d 12459	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 0 ≤ (𝑁 / (2↑𝑚)))
28		0z 11980	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
29		flbi 13181	. . . . . . . . . . . . . . . 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 589	. . . . . . . . . . . . . . 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 15708	. . . . . . . . . . . . . . . 16 ⊢ 2 ∥ 0
32		id 22	. . . . . . . . . . . . . . . 16 ⊢ ((⌊‘(𝑁 / (2↑𝑚))) = 0 → (⌊‘(𝑁 / (2↑𝑚))) = 0)
33	31, 32	breqtrrid 5068	. . . . . . . . . . . . . . 15 ⊢ ((⌊‘(𝑁 / (2↑𝑚))) = 0 → 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))
34	30, 33	syl6bir 257	. . . . . . . . . . . . . 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 694	. . . . . . . . . . . . 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 246	. . . . . . . . . . . 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 263	. . . . . . . . . . 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 12468	. . . . . . . . . 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 260	. . . . . . . . 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 5054	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) ≤ 𝑁)
42		elfzolt2 13042	. . . . . . . . 9 ⊢ (𝑁 ∈ (0..^(2↑𝑀)) → 𝑁 < (2↑𝑀))
43	42	3ad2ant2 1131	. . . . . . . 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 10787	. . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) < (2↑𝑀))
45		1lt2 11796	. . . . . . . . 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 13610	. . . . . . 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 260	. . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑚 < 𝑀)
49		elfzo2 13036	. . . . . 6 ⊢ (𝑚 ∈ (0..^𝑀) ↔ (𝑚 ∈ (ℤ_≥‘0) ∧ 𝑀 ∈ ℤ ∧ 𝑚 < 𝑀))
50	4, 6, 48, 49	syl3anbrc 1340	. . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑚 ∈ (0..^𝑀))
51	50	3expia 1118	. . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀))) → ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))) → 𝑚 ∈ (0..^𝑀)))
52	1, 51	syl5bi 245	. . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀))) → (𝑚 ∈ (bits‘𝑁) → 𝑚 ∈ (0..^𝑀)))
53	52	ssrdv 3921	. 2 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀))) → (bits‘𝑁) ⊆ (0..^𝑀))
54		simpr 488	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ∈ ℕ)
55	54	nnred 11640	. . . . . . 7 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ∈ ℝ)
56		simpllr 775	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑀 ∈ ℕ₀)
57	56	nn0red 11944	. . . . . . 7 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑀 ∈ ℝ)
58		max2 12568	. . . . . . 7 ⊢ ((-𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → 𝑀 ≤ if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))
59	55, 57, 58	syl2anc 587	. . . . . 6 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑀 ≤ if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))
60		simplr 768	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (bits‘𝑁) ⊆ (0..^𝑀))
61		n2dvdsm1 15710	. . . . . . . . . . 11 ⊢ ¬ 2 ∥ -1
62		simplll 774	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑁 ∈ ℤ)
63	62	zred 12075	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑁 ∈ ℝ)
64		2nn 11698	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ
65	64	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 2 ∈ ℕ)
66	54	nnnn0d 11943	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ∈ ℕ₀)
67	56, 66	ifcld 4470	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) ∈ ℕ₀)
68	65, 67	nnexpcld 13602	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)) ∈ ℕ)
69	63, 68	nndivred 11679	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))) ∈ ℝ)
70		1red 10631	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 1 ∈ ℝ)
71	62	zcnd 12076	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑁 ∈ ℂ)
72	68	nncnd 11641	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)) ∈ ℂ)
73		2cnd 11703	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 2 ∈ ℂ)
74		2ne0 11729	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ≠ 0
75	74	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 2 ≠ 0)
76	67	nn0zd 12073	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) ∈ ℤ)
77	73, 75, 76	expne0d 13512	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)) ≠ 0)
78	71, 72, 77	divnegd 11418	. . . . . . . . . . . . . . 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 11944	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) ∈ ℝ)
80	68	nnred 11640	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)) ∈ ℝ)
81		max1 12566	. . . . . . . . . . . . . . . . . . 19 ⊢ ((-𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → -𝑁 ≤ if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))
82	55, 57, 81	syl2anc 587	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ≤ if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))
83		2z 12002	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℤ
84		uzid 12246	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
85	83, 84	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ (ℤ_≥‘2)
86		bernneq3 13588	. . . . . . . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . . . . . . 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 10777	. . . . . . . . . . . . . . . . . 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 10786	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ≤ (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)))
90	72	mulid1d 10647	. . . . . . . . . . . . . . . . 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 5058	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ≤ ((2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)) · 1))
92	68	nnrpd 12417	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)) ∈ ℝ⁺)
93	55, 70, 92	ledivmuld 12472	. . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (-𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))) ≤ 1)
95	78, 94	eqbrtrd 5052	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -(𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))) ≤ 1)
96	69, 70, 95	lenegcon1d 11211	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -1 ≤ (𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))))
97	54	nngt0d 11674	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 0 < -𝑁)
98	68	nngt0d 11674	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 0 < (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)))
99	55, 80, 97, 98	divgt0d 11564	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 0 < (-𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))))
100	99, 78	breqtrrd 5058	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 0 < -(𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))))
101	69	lt0neg1d 11198	. . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))) < 0)
103		ax-1cn 10584	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
104		neg1cn 11739	. . . . . . . . . . . . . . 15 ⊢ -1 ∈ ℂ
105		1pneg1e0 11744	. . . . . . . . . . . . . . 15 ⊢ (1 + -1) = 0
106	103, 104, 105	addcomli 10821	. . . . . . . . . . . . . 14 ⊢ (-1 + 1) = 0
107	102, 106	breqtrrdi 5072	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))) < (-1 + 1))
108		neg1z 12006	. . . . . . . . . . . . . 14 ⊢ -1 ∈ ℤ
109		flbi 13181	. . . . . . . . . . . . . 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 589	. . . . . . . . . . . . 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 712	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (⌊‘(𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)))) = -1)
112	111	breq2d 5042	. . . . . . . . . . 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 330	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → ¬ 2 ∥ (⌊‘(𝑁 / (2↑if(-𝑁 ≤ 𝑀, 𝑀, -𝑁)))))
114		bitsval2 15764	. . . . . . . . . . 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 587	. . . . . . . . . 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 260	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) ∈ (bits‘𝑁))
117	60, 116	sseldd 3916	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → if(-𝑁 ≤ 𝑀, 𝑀, -𝑁) ∈ (0..^𝑀))
118		elfzolt2 13042	. . . . . . . 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 10776	. . . . . . 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 235	. . . . . 6 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → ¬ 𝑀 ≤ if(-𝑁 ≤ 𝑀, 𝑀, -𝑁))
122	59, 121	pm2.65da 816	. . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → ¬ -𝑁 ∈ ℕ)
123	122	intnand 492	. . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → ¬ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))
124		simpll 766	. . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → 𝑁 ∈ ℤ)
125		elznn0nn 11983	. . . . . 6 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ₀ ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)))
126	124, 125	sylib 221	. . . . 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 150	. . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → 𝑁 ∈ ℕ₀)
129		simplr 768	. . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → 𝑀 ∈ ℕ₀)
130		simpr 488	. . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → (bits‘𝑁) ⊆ (0..^𝑀))
131		eqid 2798	. . 3 ⊢ inf({𝑛 ∈ ℕ₀ ∣ 𝑁 < (2↑𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ₀ ∣ 𝑁 < (2↑𝑛)}, ℝ, < )
132	128, 129, 130, 131	bitsfzolem 15773	. 2 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → 𝑁 ∈ (0..^(2↑𝑀)))
133	53, 132	impbida 800	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 209 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 {crab 3110 ⊆ wss 3881 ifcif 4425 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 infcinf 8889 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 < clt 10664 ≤ cle 10665 -cneg 10860 / cdiv 11286 ℕcn 11625 2c2 11680 ℕ₀cn0 11885 ℤcz 11969 ℤ_≥cuz 12231 ℝ⁺crp 12377 ..^cfzo 13028 ⌊cfl 13155 ↑cexp 13425 ∥ cdvds 15599 bitscbits 15758

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-seq 13365 df-exp 13426 df-dvds 15600 df-bits 15761

This theorem is referenced by: bitsfi 15776 0bits 15778 bitsinv1 15781 sadcaddlem 15796 sadaddlem 15805 sadasslem 15809 sadeq 15811

W3C validator