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Theorem bitsres 16404

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 12222	. . . . . . . 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 14172	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℕ)
6	1, 5	zmodcld 13816	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 mod (2↑𝑁)) ∈ ℕ₀)
7	6	nn0zd 12517	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 mod (2↑𝑁)) ∈ ℤ)
8	7	znegcld 12602	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → -(𝐴 mod (2↑𝑁)) ∈ ℤ)
9		sadadd 16398	. . 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 16398	. . . . . 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 12601	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → -(𝐴 mod (2↑𝑁)) ∈ ℂ)
14	7	zcnd 12601	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 mod (2↑𝑁)) ∈ ℂ)
15	13, 14	addcomd 11339	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-(𝐴 mod (2↑𝑁)) + (𝐴 mod (2↑𝑁))) = ((𝐴 mod (2↑𝑁)) + -(𝐴 mod (2↑𝑁))))
16	14	negidd 11486	. . . . . . . 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 6839	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘(-(𝐴 mod (2↑𝑁)) + (𝐴 mod (2↑𝑁)))) = (bits‘0))
19		0bits 16370	. . . . . 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 7375	. . 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 16357	. . . . 5 ⊢ (bits‘-(𝐴 mod (2↑𝑁))) ⊆ ℕ₀
24		bitsss 16357	. . . . 5 ⊢ (bits‘(𝐴 mod (2↑𝑁))) ⊆ ℕ₀
25		inss1 4190	. . . . . 6 ⊢ ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)) ⊆ (bits‘𝐴)
26		bitsss 16357	. . . . . . 7 ⊢ (bits‘𝐴) ⊆ ℕ₀
27	26	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘𝐴) ⊆ ℕ₀)
28	25, 27	sstrid 3946	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)) ⊆ ℕ₀)
29		sadass 16402	. . . . 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 16367	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘(𝐴 mod (2↑𝑁))) = ((bits‘𝐴) ∩ (0..^𝑁)))
32	31	oveq1d 7375	. . . . . 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 4190	. . . . . . . . . 10 ⊢ ((bits‘𝐴) ∩ (0..^𝑁)) ⊆ (bits‘𝐴)
34	33, 27	sstrid 3946	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘𝐴) ∩ (0..^𝑁)) ⊆ ℕ₀)
35		fzouzdisj 13615	. . . . . . . . . . . 12 ⊢ ((0..^𝑁) ∩ (ℤ_≥‘𝑁)) = ∅
36	35	ineq2i 4170	. . . . . . . . . . 11 ⊢ ((bits‘𝐴) ∩ ((0..^𝑁) ∩ (ℤ_≥‘𝑁))) = ((bits‘𝐴) ∩ ∅)
37		inindi 4188	. . . . . . . . . . 11 ⊢ ((bits‘𝐴) ∩ ((0..^𝑁) ∩ (ℤ_≥‘𝑁))) = (((bits‘𝐴) ∩ (0..^𝑁)) ∩ ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)))
38		in0 4348	. . . . . . . . . . 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 16396	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((bits‘𝐴) ∩ (0..^𝑁)) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = (((bits‘𝐴) ∩ (0..^𝑁)) ∪ ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))))
42		indi 4237	. . . . . . . 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 12793	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
45	4, 44	eleqtrdi 2847	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ (ℤ_≥‘0))
46		fzouzsplit 13614	. . . . . . . . . . 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 3977	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘𝐴) ⊆ ((0..^𝑁) ∪ (ℤ_≥‘𝑁)))
50		dfss2 3920	. . . . . . . 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 7376	. . . 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 16400	. . . 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 12601	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → 𝐴 ∈ ℂ)
60	13, 59	addcomd 11339	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-(𝐴 mod (2↑𝑁)) + 𝐴) = (𝐴 + -(𝐴 mod (2↑𝑁))))
61	59, 14	negsubd 11502	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 + -(𝐴 mod (2↑𝑁))) = (𝐴 − (𝐴 mod (2↑𝑁))))
62	59, 14	subcld 11496	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 − (𝐴 mod (2↑𝑁))) ∈ ℂ)
63	5	nncnd 12165	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℂ)
64	5	nnne0d 12199	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ≠ 0)
65	62, 63, 64	divcan1d 11922	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)) · (2↑𝑁)) = (𝐴 − (𝐴 mod (2↑𝑁))))
66	1	zred 12600	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → 𝐴 ∈ ℝ)
67	5	nnrpd 12951	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℝ⁺)
68		moddiffl 13806	. . . . . . 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 7375	. . . . 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 6839	. 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 3900 ∩ cin 3901 ⊆ wss 3902 ∅c0 4286 ‘cfv 6493 (class class class)co 7360 ℝcr 11029 0cc0 11030 + caddc 11033 · cmul 11035 − cmin 11368 -cneg 11369 / cdiv 11798 ℕcn 12149 2c2 12204 ℕ₀cn0 12405 ℤcz 12492 ℤ_≥cuz 12755 ℝ⁺crp 12909 ..^cfzo 13574 ⌊cfl 13714 mod cmo 13793 ↑cexp 13988 bitscbits 16350 sadd csad 16351

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 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-inf2 9554 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108

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 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-disj 5067 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-er 8637 df-map 8769 df-pm 8770 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-inf 9350 df-oi 9419 df-dju 9817 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-n0 12406 df-xnn0 12479 df-z 12493 df-uz 12756 df-rp 12910 df-fz 13428 df-fzo 13575 df-fl 13716 df-mod 13794 df-seq 13929 df-exp 13989 df-hash 14258 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-clim 15415 df-sum 15614 df-dvds 16184 df-bits 16353 df-sad 16382

This theorem is referenced by: bitsuz 16405

W3C validator