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Theorem bitsres 16511

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 12340	. . . . . . . 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 14285	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℕ)
6	1, 5	zmodcld 13933	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 mod (2↑𝑁)) ∈ ℕ₀)
7	6	nn0zd 12641	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 mod (2↑𝑁)) ∈ ℤ)
8	7	znegcld 12726	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → -(𝐴 mod (2↑𝑁)) ∈ ℤ)
9		sadadd 16505	. . 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 584	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘-(𝐴 mod (2↑𝑁))) sadd (bits‘𝐴)) = (bits‘(-(𝐴 mod (2↑𝑁)) + 𝐴)))
11		sadadd 16505	. . . . . 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 584	. . . . 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 12725	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → -(𝐴 mod (2↑𝑁)) ∈ ℂ)
14	7	zcnd 12725	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 mod (2↑𝑁)) ∈ ℂ)
15	13, 14	addcomd 11464	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-(𝐴 mod (2↑𝑁)) + (𝐴 mod (2↑𝑁))) = ((𝐴 mod (2↑𝑁)) + -(𝐴 mod (2↑𝑁))))
16	14	negidd 11611	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 mod (2↑𝑁)) + -(𝐴 mod (2↑𝑁))) = 0)
17	15, 16	eqtrd 2776	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-(𝐴 mod (2↑𝑁)) + (𝐴 mod (2↑𝑁))) = 0)
18	17	fveq2d 6909	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘(-(𝐴 mod (2↑𝑁)) + (𝐴 mod (2↑𝑁)))) = (bits‘0))
19		0bits 16477	. . . . . 6 ⊢ (bits‘0) = ∅
20	18, 19	eqtrdi 2792	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘(-(𝐴 mod (2↑𝑁)) + (𝐴 mod (2↑𝑁)))) = ∅)
21	12, 20	eqtrd 2776	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘-(𝐴 mod (2↑𝑁))) sadd (bits‘(𝐴 mod (2↑𝑁)))) = ∅)
22	21	oveq1d 7447	. . 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 16464	. . . . 5 ⊢ (bits‘-(𝐴 mod (2↑𝑁))) ⊆ ℕ₀
24		bitsss 16464	. . . . 5 ⊢ (bits‘(𝐴 mod (2↑𝑁))) ⊆ ℕ₀
25		inss1 4236	. . . . . 6 ⊢ ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)) ⊆ (bits‘𝐴)
26		bitsss 16464	. . . . . . 7 ⊢ (bits‘𝐴) ⊆ ℕ₀
27	26	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘𝐴) ⊆ ℕ₀)
28	25, 27	sstrid 3994	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)) ⊆ ℕ₀)
29		sadass 16509	. . . . 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 1466	. . . 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 16474	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘(𝐴 mod (2↑𝑁))) = ((bits‘𝐴) ∩ (0..^𝑁)))
32	31	oveq1d 7447	. . . . . 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 4236	. . . . . . . . . 10 ⊢ ((bits‘𝐴) ∩ (0..^𝑁)) ⊆ (bits‘𝐴)
34	33, 27	sstrid 3994	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘𝐴) ∩ (0..^𝑁)) ⊆ ℕ₀)
35		fzouzdisj 13736	. . . . . . . . . . . 12 ⊢ ((0..^𝑁) ∩ (ℤ_≥‘𝑁)) = ∅
36	35	ineq2i 4216	. . . . . . . . . . 11 ⊢ ((bits‘𝐴) ∩ ((0..^𝑁) ∩ (ℤ_≥‘𝑁))) = ((bits‘𝐴) ∩ ∅)
37		inindi 4234	. . . . . . . . . . 11 ⊢ ((bits‘𝐴) ∩ ((0..^𝑁) ∩ (ℤ_≥‘𝑁))) = (((bits‘𝐴) ∩ (0..^𝑁)) ∩ ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)))
38		in0 4394	. . . . . . . . . . 11 ⊢ ((bits‘𝐴) ∩ ∅) = ∅
39	36, 37, 38	3eqtr3i 2772	. . . . . . . . . 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 16503	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((bits‘𝐴) ∩ (0..^𝑁)) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = (((bits‘𝐴) ∩ (0..^𝑁)) ∪ ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))))
42		indi 4283	. . . . . . . 8 ⊢ ((bits‘𝐴) ∩ ((0..^𝑁) ∪ (ℤ_≥‘𝑁))) = (((bits‘𝐴) ∩ (0..^𝑁)) ∪ ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)))
43	41, 42	eqtr4di 2794	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((bits‘𝐴) ∩ (0..^𝑁)) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = ((bits‘𝐴) ∩ ((0..^𝑁) ∪ (ℤ_≥‘𝑁))))
44		nn0uz 12921	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
45	4, 44	eleqtrdi 2850	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ (ℤ_≥‘0))
46		fzouzsplit 13735	. . . . . . . . . . 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 2788	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ℕ₀ = ((0..^𝑁) ∪ (ℤ_≥‘𝑁)))
49	26, 48	sseqtrid 4025	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘𝐴) ⊆ ((0..^𝑁) ∪ (ℤ_≥‘𝑁)))
50		dfss2 3968	. . . . . . . 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 2776	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((bits‘𝐴) ∩ (0..^𝑁)) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = (bits‘𝐴))
53	32, 52	eqtrd 2776	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘(𝐴 mod (2↑𝑁))) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = (bits‘𝐴))
54	53	oveq2d 7448	. . . 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 2776	. . 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 16507	. . . 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 2784	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘-(𝐴 mod (2↑𝑁))) sadd (bits‘𝐴)) = ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)))
59	1	zcnd 12725	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → 𝐴 ∈ ℂ)
60	13, 59	addcomd 11464	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-(𝐴 mod (2↑𝑁)) + 𝐴) = (𝐴 + -(𝐴 mod (2↑𝑁))))
61	59, 14	negsubd 11627	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 + -(𝐴 mod (2↑𝑁))) = (𝐴 − (𝐴 mod (2↑𝑁))))
62	59, 14	subcld 11621	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 − (𝐴 mod (2↑𝑁))) ∈ ℂ)
63	5	nncnd 12283	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℂ)
64	5	nnne0d 12317	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ≠ 0)
65	62, 63, 64	divcan1d 12045	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)) · (2↑𝑁)) = (𝐴 − (𝐴 mod (2↑𝑁))))
66	1	zred 12724	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → 𝐴 ∈ ℝ)
67	5	nnrpd 13076	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℝ⁺)
68		moddiffl 13923	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (2↑𝑁) ∈ ℝ⁺) → ((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)) = (⌊‘(𝐴 / (2↑𝑁))))
69	66, 67, 68	syl2anc 584	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)) = (⌊‘(𝐴 / (2↑𝑁))))
70	69	oveq1d 7447	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)) · (2↑𝑁)) = ((⌊‘(𝐴 / (2↑𝑁))) · (2↑𝑁)))
71	61, 65, 70	3eqtr2d 2782	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 + -(𝐴 mod (2↑𝑁))) = ((⌊‘(𝐴 / (2↑𝑁))) · (2↑𝑁)))
72	60, 71	eqtrd 2776	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-(𝐴 mod (2↑𝑁)) + 𝐴) = ((⌊‘(𝐴 / (2↑𝑁))) · (2↑𝑁)))
73	72	fveq2d 6909	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘(-(𝐴 mod (2↑𝑁)) + 𝐴)) = (bits‘((⌊‘(𝐴 / (2↑𝑁))) · (2↑𝑁))))
74	10, 58, 73	3eqtr3d 2784	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 2107 ∪ cun 3948 ∩ cin 3949 ⊆ wss 3950 ∅c0 4332 ‘cfv 6560 (class class class)co 7432 ℝcr 11155 0cc0 11156 + caddc 11159 · cmul 11161 − cmin 11493 -cneg 11494 / cdiv 11921 ℕcn 12267 2c2 12322 ℕ₀cn0 12528 ℤcz 12615 ℤ_≥cuz 12879 ℝ⁺crp 13035 ..^cfzo 13695 ⌊cfl 13831 mod cmo 13910 ↑cexp 14103 bitscbits 16457 sadd csad 16458

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1511 df-tru 1542 df-fal 1552 df-had 1593 df-cad 1606 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-disj 5110 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-er 8746 df-map 8869 df-pm 8870 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-inf 9484 df-oi 9551 df-dju 9942 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-n0 12529 df-xnn0 12602 df-z 12616 df-uz 12880 df-rp 13036 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-exp 14104 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-clim 15525 df-sum 15724 df-dvds 16292 df-bits 16460 df-sad 16489

This theorem is referenced by: bitsuz 16512

W3C validator