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Theorem bitsres 16108

Description: Restrict the bits of a number to an upper integer set. (Contributed by Mario Carneiro, 5-Sep-2016.)

**Assertion**
Ref	Expression
bitsres	⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)) = (bits‘((⌊‘(𝐴 / (2↑𝑁))) · (2↑𝑁))))

**Proof of Theorem bitsres**
Step	Hyp	Ref	Expression
1		simpl 482	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → 𝐴 ∈ ℤ)
2		2nn 11976	. . . . . . . 8 ⊢ 2 ∈ ℕ
3	2	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → 2 ∈ ℕ)
4		simpr 484	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℕ₀)
5	3, 4	nnexpcld 13888	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℕ)
6	1, 5	zmodcld 13540	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 mod (2↑𝑁)) ∈ ℕ₀)
7	6	nn0zd 12353	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 mod (2↑𝑁)) ∈ ℤ)
8	7	znegcld 12357	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → -(𝐴 mod (2↑𝑁)) ∈ ℤ)
9		sadadd 16102	. . 3 ⊢ ((-(𝐴 mod (2↑𝑁)) ∈ ℤ ∧ 𝐴 ∈ ℤ) → ((bits‘-(𝐴 mod (2↑𝑁))) sadd (bits‘𝐴)) = (bits‘(-(𝐴 mod (2↑𝑁)) + 𝐴)))
10	8, 1, 9	syl2anc 583	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘-(𝐴 mod (2↑𝑁))) sadd (bits‘𝐴)) = (bits‘(-(𝐴 mod (2↑𝑁)) + 𝐴)))
11		sadadd 16102	. . . . . 6 ⊢ ((-(𝐴 mod (2↑𝑁)) ∈ ℤ ∧ (𝐴 mod (2↑𝑁)) ∈ ℤ) → ((bits‘-(𝐴 mod (2↑𝑁))) sadd (bits‘(𝐴 mod (2↑𝑁)))) = (bits‘(-(𝐴 mod (2↑𝑁)) + (𝐴 mod (2↑𝑁)))))
12	8, 7, 11	syl2anc 583	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘-(𝐴 mod (2↑𝑁))) sadd (bits‘(𝐴 mod (2↑𝑁)))) = (bits‘(-(𝐴 mod (2↑𝑁)) + (𝐴 mod (2↑𝑁)))))
13	8	zcnd 12356	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → -(𝐴 mod (2↑𝑁)) ∈ ℂ)
14	7	zcnd 12356	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 mod (2↑𝑁)) ∈ ℂ)
15	13, 14	addcomd 11107	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-(𝐴 mod (2↑𝑁)) + (𝐴 mod (2↑𝑁))) = ((𝐴 mod (2↑𝑁)) + -(𝐴 mod (2↑𝑁))))
16	14	negidd 11252	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 mod (2↑𝑁)) + -(𝐴 mod (2↑𝑁))) = 0)
17	15, 16	eqtrd 2778	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-(𝐴 mod (2↑𝑁)) + (𝐴 mod (2↑𝑁))) = 0)
18	17	fveq2d 6760	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘(-(𝐴 mod (2↑𝑁)) + (𝐴 mod (2↑𝑁)))) = (bits‘0))
19		0bits 16074	. . . . . 6 ⊢ (bits‘0) = ∅
20	18, 19	eqtrdi 2795	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘(-(𝐴 mod (2↑𝑁)) + (𝐴 mod (2↑𝑁)))) = ∅)
21	12, 20	eqtrd 2778	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘-(𝐴 mod (2↑𝑁))) sadd (bits‘(𝐴 mod (2↑𝑁)))) = ∅)
22	21	oveq1d 7270	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((bits‘-(𝐴 mod (2↑𝑁))) sadd (bits‘(𝐴 mod (2↑𝑁)))) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = (∅ sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))))
23		bitsss 16061	. . . . 5 ⊢ (bits‘-(𝐴 mod (2↑𝑁))) ⊆ ℕ₀
24		bitsss 16061	. . . . 5 ⊢ (bits‘(𝐴 mod (2↑𝑁))) ⊆ ℕ₀
25		inss1 4159	. . . . . 6 ⊢ ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)) ⊆ (bits‘𝐴)
26		bitsss 16061	. . . . . . 7 ⊢ (bits‘𝐴) ⊆ ℕ₀
27	26	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘𝐴) ⊆ ℕ₀)
28	25, 27	sstrid 3928	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)) ⊆ ℕ₀)
29		sadass 16106	. . . . 5 ⊢ (((bits‘-(𝐴 mod (2↑𝑁))) ⊆ ℕ₀ ∧ (bits‘(𝐴 mod (2↑𝑁))) ⊆ ℕ₀ ∧ ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)) ⊆ ℕ₀) → (((bits‘-(𝐴 mod (2↑𝑁))) sadd (bits‘(𝐴 mod (2↑𝑁)))) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = ((bits‘-(𝐴 mod (2↑𝑁))) sadd ((bits‘(𝐴 mod (2↑𝑁))) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)))))
30	23, 24, 28, 29	mp3an12i 1463	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((bits‘-(𝐴 mod (2↑𝑁))) sadd (bits‘(𝐴 mod (2↑𝑁)))) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = ((bits‘-(𝐴 mod (2↑𝑁))) sadd ((bits‘(𝐴 mod (2↑𝑁))) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)))))
31		bitsmod 16071	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘(𝐴 mod (2↑𝑁))) = ((bits‘𝐴) ∩ (0..^𝑁)))
32	31	oveq1d 7270	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘(𝐴 mod (2↑𝑁))) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = (((bits‘𝐴) ∩ (0..^𝑁)) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))))
33		inss1 4159	. . . . . . . . . 10 ⊢ ((bits‘𝐴) ∩ (0..^𝑁)) ⊆ (bits‘𝐴)
34	33, 27	sstrid 3928	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘𝐴) ∩ (0..^𝑁)) ⊆ ℕ₀)
35		fzouzdisj 13351	. . . . . . . . . . . 12 ⊢ ((0..^𝑁) ∩ (ℤ_≥‘𝑁)) = ∅
36	35	ineq2i 4140	. . . . . . . . . . 11 ⊢ ((bits‘𝐴) ∩ ((0..^𝑁) ∩ (ℤ_≥‘𝑁))) = ((bits‘𝐴) ∩ ∅)
37		inindi 4157	. . . . . . . . . . 11 ⊢ ((bits‘𝐴) ∩ ((0..^𝑁) ∩ (ℤ_≥‘𝑁))) = (((bits‘𝐴) ∩ (0..^𝑁)) ∩ ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)))
38		in0 4322	. . . . . . . . . . 11 ⊢ ((bits‘𝐴) ∩ ∅) = ∅
39	36, 37, 38	3eqtr3i 2774	. . . . . . . . . 10 ⊢ (((bits‘𝐴) ∩ (0..^𝑁)) ∩ ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = ∅
40	39	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((bits‘𝐴) ∩ (0..^𝑁)) ∩ ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = ∅)
41	34, 28, 40	saddisj 16100	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((bits‘𝐴) ∩ (0..^𝑁)) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = (((bits‘𝐴) ∩ (0..^𝑁)) ∪ ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))))
42		indi 4204	. . . . . . . 8 ⊢ ((bits‘𝐴) ∩ ((0..^𝑁) ∪ (ℤ_≥‘𝑁))) = (((bits‘𝐴) ∩ (0..^𝑁)) ∪ ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)))
43	41, 42	eqtr4di 2797	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((bits‘𝐴) ∩ (0..^𝑁)) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = ((bits‘𝐴) ∩ ((0..^𝑁) ∪ (ℤ_≥‘𝑁))))
44		nn0uz 12549	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
45	4, 44	eleqtrdi 2849	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ (ℤ_≥‘0))
46		fzouzsplit 13350	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘0) → (ℤ_≥‘0) = ((0..^𝑁) ∪ (ℤ_≥‘𝑁)))
47	45, 46	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (ℤ_≥‘0) = ((0..^𝑁) ∪ (ℤ_≥‘𝑁)))
48	44, 47	eqtrid 2790	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ℕ₀ = ((0..^𝑁) ∪ (ℤ_≥‘𝑁)))
49	26, 48	sseqtrid 3969	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘𝐴) ⊆ ((0..^𝑁) ∪ (ℤ_≥‘𝑁)))
50		df-ss 3900	. . . . . . . 8 ⊢ ((bits‘𝐴) ⊆ ((0..^𝑁) ∪ (ℤ_≥‘𝑁)) ↔ ((bits‘𝐴) ∩ ((0..^𝑁) ∪ (ℤ_≥‘𝑁))) = (bits‘𝐴))
51	49, 50	sylib 217	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘𝐴) ∩ ((0..^𝑁) ∪ (ℤ_≥‘𝑁))) = (bits‘𝐴))
52	43, 51	eqtrd 2778	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((bits‘𝐴) ∩ (0..^𝑁)) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = (bits‘𝐴))
53	32, 52	eqtrd 2778	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘(𝐴 mod (2↑𝑁))) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = (bits‘𝐴))
54	53	oveq2d 7271	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘-(𝐴 mod (2↑𝑁))) sadd ((bits‘(𝐴 mod (2↑𝑁))) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)))) = ((bits‘-(𝐴 mod (2↑𝑁))) sadd (bits‘𝐴)))
55	30, 54	eqtrd 2778	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((bits‘-(𝐴 mod (2↑𝑁))) sadd (bits‘(𝐴 mod (2↑𝑁)))) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = ((bits‘-(𝐴 mod (2↑𝑁))) sadd (bits‘𝐴)))
56		sadid2 16104	. . . 4 ⊢ (((bits‘𝐴) ∩ (ℤ_≥‘𝑁)) ⊆ ℕ₀ → (∅ sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)))
57	28, 56	syl 17	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (∅ sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)))
58	22, 55, 57	3eqtr3d 2786	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘-(𝐴 mod (2↑𝑁))) sadd (bits‘𝐴)) = ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)))
59	1	zcnd 12356	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → 𝐴 ∈ ℂ)
60	13, 59	addcomd 11107	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-(𝐴 mod (2↑𝑁)) + 𝐴) = (𝐴 + -(𝐴 mod (2↑𝑁))))
61	59, 14	negsubd 11268	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 + -(𝐴 mod (2↑𝑁))) = (𝐴 − (𝐴 mod (2↑𝑁))))
62	59, 14	subcld 11262	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 − (𝐴 mod (2↑𝑁))) ∈ ℂ)
63	5	nncnd 11919	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℂ)
64	5	nnne0d 11953	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ≠ 0)
65	62, 63, 64	divcan1d 11682	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)) · (2↑𝑁)) = (𝐴 − (𝐴 mod (2↑𝑁))))
66	1	zred 12355	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → 𝐴 ∈ ℝ)
67	5	nnrpd 12699	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℝ⁺)
68		moddiffl 13530	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (2↑𝑁) ∈ ℝ⁺) → ((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)) = (⌊‘(𝐴 / (2↑𝑁))))
69	66, 67, 68	syl2anc 583	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)) = (⌊‘(𝐴 / (2↑𝑁))))
70	69	oveq1d 7270	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)) · (2↑𝑁)) = ((⌊‘(𝐴 / (2↑𝑁))) · (2↑𝑁)))
71	61, 65, 70	3eqtr2d 2784	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 + -(𝐴 mod (2↑𝑁))) = ((⌊‘(𝐴 / (2↑𝑁))) · (2↑𝑁)))
72	60, 71	eqtrd 2778	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-(𝐴 mod (2↑𝑁)) + 𝐴) = ((⌊‘(𝐴 / (2↑𝑁))) · (2↑𝑁)))
73	72	fveq2d 6760	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘(-(𝐴 mod (2↑𝑁)) + 𝐴)) = (bits‘((⌊‘(𝐴 / (2↑𝑁))) · (2↑𝑁))))
74	10, 58, 73	3eqtr3d 2786	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)) = (bits‘((⌊‘(𝐴 / (2↑𝑁))) · (2↑𝑁))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∪ cun 3881 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 ‘cfv 6418 (class class class)co 7255 ℝcr 10801 0cc0 10802 + caddc 10805 · cmul 10807 − cmin 11135 -cneg 11136 / cdiv 11562 ℕcn 11903 2c2 11958 ℕ₀cn0 12163 ℤcz 12249 ℤ_≥cuz 12511 ℝ⁺crp 12659 ..^cfzo 13311 ⌊cfl 13438 mod cmo 13517 ↑cexp 13710 bitscbits 16054 sadd csad 16055

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-xor 1504 df-tru 1542 df-fal 1552 df-had 1596 df-cad 1610 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-disj 5036 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-er 8456 df-map 8575 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-xnn0 12236 df-z 12250 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-sum 15326 df-dvds 15892 df-bits 16057 df-sad 16086

This theorem is referenced by: bitsuz 16109

W3C validator