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Theorem bitsres 16412

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 12230	. . . . . . . 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 14180	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℕ)
6	1, 5	zmodcld 13824	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 mod (2↑𝑁)) ∈ ℕ₀)
7	6	nn0zd 12525	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 mod (2↑𝑁)) ∈ ℤ)
8	7	znegcld 12610	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → -(𝐴 mod (2↑𝑁)) ∈ ℤ)
9		sadadd 16406	. . 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 585	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘-(𝐴 mod (2↑𝑁))) sadd (bits‘𝐴)) = (bits‘(-(𝐴 mod (2↑𝑁)) + 𝐴)))
11		sadadd 16406	. . . . . 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 585	. . . . 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 12609	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → -(𝐴 mod (2↑𝑁)) ∈ ℂ)
14	7	zcnd 12609	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 mod (2↑𝑁)) ∈ ℂ)
15	13, 14	addcomd 11347	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-(𝐴 mod (2↑𝑁)) + (𝐴 mod (2↑𝑁))) = ((𝐴 mod (2↑𝑁)) + -(𝐴 mod (2↑𝑁))))
16	14	negidd 11494	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 mod (2↑𝑁)) + -(𝐴 mod (2↑𝑁))) = 0)
17	15, 16	eqtrd 2772	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-(𝐴 mod (2↑𝑁)) + (𝐴 mod (2↑𝑁))) = 0)
18	17	fveq2d 6846	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘(-(𝐴 mod (2↑𝑁)) + (𝐴 mod (2↑𝑁)))) = (bits‘0))
19		0bits 16378	. . . . . 6 ⊢ (bits‘0) = ∅
20	18, 19	eqtrdi 2788	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘(-(𝐴 mod (2↑𝑁)) + (𝐴 mod (2↑𝑁)))) = ∅)
21	12, 20	eqtrd 2772	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘-(𝐴 mod (2↑𝑁))) sadd (bits‘(𝐴 mod (2↑𝑁)))) = ∅)
22	21	oveq1d 7383	. . 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 16365	. . . . 5 ⊢ (bits‘-(𝐴 mod (2↑𝑁))) ⊆ ℕ₀
24		bitsss 16365	. . . . 5 ⊢ (bits‘(𝐴 mod (2↑𝑁))) ⊆ ℕ₀
25		inss1 4191	. . . . . 6 ⊢ ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)) ⊆ (bits‘𝐴)
26		bitsss 16365	. . . . . . 7 ⊢ (bits‘𝐴) ⊆ ℕ₀
27	26	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘𝐴) ⊆ ℕ₀)
28	25, 27	sstrid 3947	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)) ⊆ ℕ₀)
29		sadass 16410	. . . . 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 1468	. . . 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 16375	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘(𝐴 mod (2↑𝑁))) = ((bits‘𝐴) ∩ (0..^𝑁)))
32	31	oveq1d 7383	. . . . . 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 4191	. . . . . . . . . 10 ⊢ ((bits‘𝐴) ∩ (0..^𝑁)) ⊆ (bits‘𝐴)
34	33, 27	sstrid 3947	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘𝐴) ∩ (0..^𝑁)) ⊆ ℕ₀)
35		fzouzdisj 13623	. . . . . . . . . . . 12 ⊢ ((0..^𝑁) ∩ (ℤ_≥‘𝑁)) = ∅
36	35	ineq2i 4171	. . . . . . . . . . 11 ⊢ ((bits‘𝐴) ∩ ((0..^𝑁) ∩ (ℤ_≥‘𝑁))) = ((bits‘𝐴) ∩ ∅)
37		inindi 4189	. . . . . . . . . . 11 ⊢ ((bits‘𝐴) ∩ ((0..^𝑁) ∩ (ℤ_≥‘𝑁))) = (((bits‘𝐴) ∩ (0..^𝑁)) ∩ ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)))
38		in0 4349	. . . . . . . . . . 11 ⊢ ((bits‘𝐴) ∩ ∅) = ∅
39	36, 37, 38	3eqtr3i 2768	. . . . . . . . . 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 16404	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((bits‘𝐴) ∩ (0..^𝑁)) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = (((bits‘𝐴) ∩ (0..^𝑁)) ∪ ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))))
42		indi 4238	. . . . . . . 8 ⊢ ((bits‘𝐴) ∩ ((0..^𝑁) ∪ (ℤ_≥‘𝑁))) = (((bits‘𝐴) ∩ (0..^𝑁)) ∪ ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)))
43	41, 42	eqtr4di 2790	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((bits‘𝐴) ∩ (0..^𝑁)) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = ((bits‘𝐴) ∩ ((0..^𝑁) ∪ (ℤ_≥‘𝑁))))
44		nn0uz 12801	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
45	4, 44	eleqtrdi 2847	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ (ℤ_≥‘0))
46		fzouzsplit 13622	. . . . . . . . . . 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 2784	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ℕ₀ = ((0..^𝑁) ∪ (ℤ_≥‘𝑁)))
49	26, 48	sseqtrid 3978	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘𝐴) ⊆ ((0..^𝑁) ∪ (ℤ_≥‘𝑁)))
50		dfss2 3921	. . . . . . . 8 ⊢ ((bits‘𝐴) ⊆ ((0..^𝑁) ∪ (ℤ_≥‘𝑁)) ↔ ((bits‘𝐴) ∩ ((0..^𝑁) ∪ (ℤ_≥‘𝑁))) = (bits‘𝐴))
51	49, 50	sylib 218	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘𝐴) ∩ ((0..^𝑁) ∪ (ℤ_≥‘𝑁))) = (bits‘𝐴))
52	43, 51	eqtrd 2772	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((bits‘𝐴) ∩ (0..^𝑁)) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = (bits‘𝐴))
53	32, 52	eqtrd 2772	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘(𝐴 mod (2↑𝑁))) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = (bits‘𝐴))
54	53	oveq2d 7384	. . . 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 2772	. . 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 16408	. . . 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 2780	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘-(𝐴 mod (2↑𝑁))) sadd (bits‘𝐴)) = ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)))
59	1	zcnd 12609	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → 𝐴 ∈ ℂ)
60	13, 59	addcomd 11347	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-(𝐴 mod (2↑𝑁)) + 𝐴) = (𝐴 + -(𝐴 mod (2↑𝑁))))
61	59, 14	negsubd 11510	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 + -(𝐴 mod (2↑𝑁))) = (𝐴 − (𝐴 mod (2↑𝑁))))
62	59, 14	subcld 11504	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 − (𝐴 mod (2↑𝑁))) ∈ ℂ)
63	5	nncnd 12173	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℂ)
64	5	nnne0d 12207	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ≠ 0)
65	62, 63, 64	divcan1d 11930	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)) · (2↑𝑁)) = (𝐴 − (𝐴 mod (2↑𝑁))))
66	1	zred 12608	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → 𝐴 ∈ ℝ)
67	5	nnrpd 12959	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℝ⁺)
68		moddiffl 13814	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (2↑𝑁) ∈ ℝ⁺) → ((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)) = (⌊‘(𝐴 / (2↑𝑁))))
69	66, 67, 68	syl2anc 585	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)) = (⌊‘(𝐴 / (2↑𝑁))))
70	69	oveq1d 7383	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)) · (2↑𝑁)) = ((⌊‘(𝐴 / (2↑𝑁))) · (2↑𝑁)))
71	61, 65, 70	3eqtr2d 2778	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 + -(𝐴 mod (2↑𝑁))) = ((⌊‘(𝐴 / (2↑𝑁))) · (2↑𝑁)))
72	60, 71	eqtrd 2772	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-(𝐴 mod (2↑𝑁)) + 𝐴) = ((⌊‘(𝐴 / (2↑𝑁))) · (2↑𝑁)))
73	72	fveq2d 6846	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘(-(𝐴 mod (2↑𝑁)) + 𝐴)) = (bits‘((⌊‘(𝐴 / (2↑𝑁))) · (2↑𝑁))))
74	10, 58, 73	3eqtr3d 2780	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 1542 ∈ wcel 2114 ∪ cun 3901 ∩ cin 3902 ⊆ wss 3903 ∅c0 4287 ‘cfv 6500 (class class class)co 7368 ℝcr 11037 0cc0 11038 + caddc 11041 · cmul 11043 − cmin 11376 -cneg 11377 / cdiv 11806 ℕcn 12157 2c2 12212 ℕ₀cn0 12413 ℤcz 12500 ℤ_≥cuz 12763 ℝ⁺crp 12917 ..^cfzo 13582 ⌊cfl 13722 mod cmo 13801 ↑cexp 13996 bitscbits 16358 sadd csad 16359

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-xor 1514 df-tru 1545 df-fal 1555 df-had 1596 df-cad 1609 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-disj 5068 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-er 8645 df-map 8777 df-pm 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-inf 9358 df-oi 9427 df-dju 9825 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-n0 12414 df-xnn0 12487 df-z 12501 df-uz 12764 df-rp 12918 df-fz 13436 df-fzo 13583 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-sum 15622 df-dvds 16192 df-bits 16361 df-sad 16390

This theorem is referenced by: bitsuz 16413

W3C validator