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Theorem bitsres 15033

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 472	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → 𝐴 ∈ ℤ)
2		2nn 11062	. . . . . . . 8 ⊢ 2 ∈ ℕ
3	2	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → 2 ∈ ℕ)
4		simpr 476	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℕ₀)
5	3, 4	nnexpcld 12892	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℕ)
6	1, 5	zmodcld 12553	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 mod (2↑𝑁)) ∈ ℕ₀)
7	6	nn0zd 11356	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 mod (2↑𝑁)) ∈ ℤ)
8	7	znegcld 11360	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → -(𝐴 mod (2↑𝑁)) ∈ ℤ)
9		sadadd 15027	. . 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 691	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘-(𝐴 mod (2↑𝑁))) sadd (bits‘𝐴)) = (bits‘(-(𝐴 mod (2↑𝑁)) + 𝐴)))
11		sadadd 15027	. . . . . 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 691	. . . . 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 11359	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → -(𝐴 mod (2↑𝑁)) ∈ ℂ)
14	7	zcnd 11359	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 mod (2↑𝑁)) ∈ ℂ)
15	13, 14	addcomd 10117	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-(𝐴 mod (2↑𝑁)) + (𝐴 mod (2↑𝑁))) = ((𝐴 mod (2↑𝑁)) + -(𝐴 mod (2↑𝑁))))
16	14	negidd 10261	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 mod (2↑𝑁)) + -(𝐴 mod (2↑𝑁))) = 0)
17	15, 16	eqtrd 2644	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-(𝐴 mod (2↑𝑁)) + (𝐴 mod (2↑𝑁))) = 0)
18	17	fveq2d 6107	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘(-(𝐴 mod (2↑𝑁)) + (𝐴 mod (2↑𝑁)))) = (bits‘0))
19		0bits 14999	. . . . . 6 ⊢ (bits‘0) = ∅
20	18, 19	syl6eq 2660	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘(-(𝐴 mod (2↑𝑁)) + (𝐴 mod (2↑𝑁)))) = ∅)
21	12, 20	eqtrd 2644	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘-(𝐴 mod (2↑𝑁))) sadd (bits‘(𝐴 mod (2↑𝑁)))) = ∅)
22	21	oveq1d 6564	. . 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 14986	. . . . 5 ⊢ (bits‘-(𝐴 mod (2↑𝑁))) ⊆ ℕ₀
24		bitsss 14986	. . . . 5 ⊢ (bits‘(𝐴 mod (2↑𝑁))) ⊆ ℕ₀
25		inss1 3795	. . . . . 6 ⊢ ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)) ⊆ (bits‘𝐴)
26		bitsss 14986	. . . . . . 7 ⊢ (bits‘𝐴) ⊆ ℕ₀
27	26	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘𝐴) ⊆ ℕ₀)
28	25, 27	syl5ss 3579	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)) ⊆ ℕ₀)
29		sadass 15031	. . . . 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 1420	. . . 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 14996	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘(𝐴 mod (2↑𝑁))) = ((bits‘𝐴) ∩ (0..^𝑁)))
32	31	oveq1d 6564	. . . . . 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 3795	. . . . . . . . . 10 ⊢ ((bits‘𝐴) ∩ (0..^𝑁)) ⊆ (bits‘𝐴)
34	33, 27	syl5ss 3579	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘𝐴) ∩ (0..^𝑁)) ⊆ ℕ₀)
35		fzouzdisj 12373	. . . . . . . . . . . 12 ⊢ ((0..^𝑁) ∩ (ℤ_≥‘𝑁)) = ∅
36	35	ineq2i 3773	. . . . . . . . . . 11 ⊢ ((bits‘𝐴) ∩ ((0..^𝑁) ∩ (ℤ_≥‘𝑁))) = ((bits‘𝐴) ∩ ∅)
37		inindi 3792	. . . . . . . . . . 11 ⊢ ((bits‘𝐴) ∩ ((0..^𝑁) ∩ (ℤ_≥‘𝑁))) = (((bits‘𝐴) ∩ (0..^𝑁)) ∩ ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)))
38		in0 3920	. . . . . . . . . . 11 ⊢ ((bits‘𝐴) ∩ ∅) = ∅
39	36, 37, 38	3eqtr3i 2640	. . . . . . . . . 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 15025	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((bits‘𝐴) ∩ (0..^𝑁)) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = (((bits‘𝐴) ∩ (0..^𝑁)) ∪ ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))))
42		indi 3832	. . . . . . . 8 ⊢ ((bits‘𝐴) ∩ ((0..^𝑁) ∪ (ℤ_≥‘𝑁))) = (((bits‘𝐴) ∩ (0..^𝑁)) ∪ ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)))
43	41, 42	syl6eqr 2662	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((bits‘𝐴) ∩ (0..^𝑁)) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = ((bits‘𝐴) ∩ ((0..^𝑁) ∪ (ℤ_≥‘𝑁))))
44		nn0uz 11598	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
45	4, 44	syl6eleq 2698	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ (ℤ_≥‘0))
46		fzouzsplit 12372	. . . . . . . . . . 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	syl5eq 2656	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ℕ₀ = ((0..^𝑁) ∪ (ℤ_≥‘𝑁)))
49	26, 48	syl5sseq 3616	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘𝐴) ⊆ ((0..^𝑁) ∪ (ℤ_≥‘𝑁)))
50		df-ss 3554	. . . . . . . 8 ⊢ ((bits‘𝐴) ⊆ ((0..^𝑁) ∪ (ℤ_≥‘𝑁)) ↔ ((bits‘𝐴) ∩ ((0..^𝑁) ∪ (ℤ_≥‘𝑁))) = (bits‘𝐴))
51	49, 50	sylib 207	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘𝐴) ∩ ((0..^𝑁) ∪ (ℤ_≥‘𝑁))) = (bits‘𝐴))
52	43, 51	eqtrd 2644	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((bits‘𝐴) ∩ (0..^𝑁)) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = (bits‘𝐴))
53	32, 52	eqtrd 2644	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘(𝐴 mod (2↑𝑁))) sadd ((bits‘𝐴) ∩ (ℤ_≥‘𝑁))) = (bits‘𝐴))
54	53	oveq2d 6565	. . . 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 2644	. . 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 15029	. . . 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 2652	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((bits‘-(𝐴 mod (2↑𝑁))) sadd (bits‘𝐴)) = ((bits‘𝐴) ∩ (ℤ_≥‘𝑁)))
59	1	zcnd 11359	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → 𝐴 ∈ ℂ)
60	13, 59	addcomd 10117	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-(𝐴 mod (2↑𝑁)) + 𝐴) = (𝐴 + -(𝐴 mod (2↑𝑁))))
61	59, 14	negsubd 10277	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 + -(𝐴 mod (2↑𝑁))) = (𝐴 − (𝐴 mod (2↑𝑁))))
62	59, 14	subcld 10271	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 − (𝐴 mod (2↑𝑁))) ∈ ℂ)
63	5	nncnd 10913	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℂ)
64	5	nnne0d 10942	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ≠ 0)
65	62, 63, 64	divcan1d 10681	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)) · (2↑𝑁)) = (𝐴 − (𝐴 mod (2↑𝑁))))
66	1	zred 11358	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → 𝐴 ∈ ℝ)
67	5	nnrpd 11746	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℝ⁺)
68		moddiffl 12543	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (2↑𝑁) ∈ ℝ⁺) → ((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)) = (⌊‘(𝐴 / (2↑𝑁))))
69	66, 67, 68	syl2anc 691	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)) = (⌊‘(𝐴 / (2↑𝑁))))
70	69	oveq1d 6564	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (((𝐴 − (𝐴 mod (2↑𝑁))) / (2↑𝑁)) · (2↑𝑁)) = ((⌊‘(𝐴 / (2↑𝑁))) · (2↑𝑁)))
71	61, 65, 70	3eqtr2d 2650	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴 + -(𝐴 mod (2↑𝑁))) = ((⌊‘(𝐴 / (2↑𝑁))) · (2↑𝑁)))
72	60, 71	eqtrd 2644	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-(𝐴 mod (2↑𝑁)) + 𝐴) = ((⌊‘(𝐴 / (2↑𝑁))) · (2↑𝑁)))
73	72	fveq2d 6107	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (bits‘(-(𝐴 mod (2↑𝑁)) + 𝐴)) = (bits‘((⌊‘(𝐴 / (2↑𝑁))) · (2↑𝑁))))
74	10, 58, 73	3eqtr3d 2652	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 383 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 + caddc 9818 · cmul 9820 − cmin 10145 -cneg 10146 / cdiv 10563 ℕcn 10897 2c2 10947 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ℝ⁺crp 11708 ..^cfzo 12334 ⌊cfl 12453 mod cmo 12530 ↑cexp 12722 bitscbits 14979 sadd csad 14980

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-xor 1457 df-tru 1478 df-fal 1481 df-had 1524 df-cad 1537 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-dvds 14822 df-bits 14982 df-sad 15011

This theorem is referenced by: bitsuz 15034

W3C validator