bitsinv1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > bitsinv1		Structured version Visualization version GIF version

Theorem bitsinv1 16383

Description: There is an explicit inverse to the bits function for nonnegative integers (which can be extended to negative integers using bitscmp 16379), part 1. (Contributed by Mario Carneiro, 7-Sep-2016.)

**Assertion**
Ref	Expression
bitsinv1	⊢ (𝑁 ∈ ℕ₀ → Σ𝑛 ∈ (bits‘𝑁)(2↑𝑛) = 𝑁)

Distinct variable group: 𝑛,𝑁

Proof of Theorem bitsinv1 Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7417	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (0..^𝑥) = (0..^0))
2		fzo0 13656	. . . . . . . . . . 11 ⊢ (0..^0) = ∅
3	1, 2	eqtrdi 2789	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (0..^𝑥) = ∅)
4	3	ineq2d 4213	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((bits‘𝑁) ∩ (0..^𝑥)) = ((bits‘𝑁) ∩ ∅))
5		in0 4392	. . . . . . . . 9 ⊢ ((bits‘𝑁) ∩ ∅) = ∅
6	4, 5	eqtrdi 2789	. . . . . . . 8 ⊢ (𝑥 = 0 → ((bits‘𝑁) ∩ (0..^𝑥)) = ∅)
7	6	sumeq1d 15647	. . . . . . 7 ⊢ (𝑥 = 0 → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑥))(2↑𝑛) = Σ𝑛 ∈ ∅ (2↑𝑛))
8		sum0 15667	. . . . . . 7 ⊢ Σ𝑛 ∈ ∅ (2↑𝑛) = 0
9	7, 8	eqtrdi 2789	. . . . . 6 ⊢ (𝑥 = 0 → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑥))(2↑𝑛) = 0)
10		oveq2 7417	. . . . . . . 8 ⊢ (𝑥 = 0 → (2↑𝑥) = (2↑0))
11		2cn 12287	. . . . . . . . 9 ⊢ 2 ∈ ℂ
12		exp0 14031	. . . . . . . . 9 ⊢ (2 ∈ ℂ → (2↑0) = 1)
13	11, 12	ax-mp 5	. . . . . . . 8 ⊢ (2↑0) = 1
14	10, 13	eqtrdi 2789	. . . . . . 7 ⊢ (𝑥 = 0 → (2↑𝑥) = 1)
15	14	oveq2d 7425	. . . . . 6 ⊢ (𝑥 = 0 → (𝑁 mod (2↑𝑥)) = (𝑁 mod 1))
16	9, 15	eqeq12d 2749	. . . . 5 ⊢ (𝑥 = 0 → (Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑥))(2↑𝑛) = (𝑁 mod (2↑𝑥)) ↔ 0 = (𝑁 mod 1)))
17	16	imbi2d 341	. . . 4 ⊢ (𝑥 = 0 → ((𝑁 ∈ ℕ₀ → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑥))(2↑𝑛) = (𝑁 mod (2↑𝑥))) ↔ (𝑁 ∈ ℕ₀ → 0 = (𝑁 mod 1))))
18		oveq2 7417	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (0..^𝑥) = (0..^𝑘))
19	18	ineq2d 4213	. . . . . . 7 ⊢ (𝑥 = 𝑘 → ((bits‘𝑁) ∩ (0..^𝑥)) = ((bits‘𝑁) ∩ (0..^𝑘)))
20	19	sumeq1d 15647	. . . . . 6 ⊢ (𝑥 = 𝑘 → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑥))(2↑𝑛) = Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑘))(2↑𝑛))
21		oveq2 7417	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (2↑𝑥) = (2↑𝑘))
22	21	oveq2d 7425	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝑁 mod (2↑𝑥)) = (𝑁 mod (2↑𝑘)))
23	20, 22	eqeq12d 2749	. . . . 5 ⊢ (𝑥 = 𝑘 → (Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑥))(2↑𝑛) = (𝑁 mod (2↑𝑥)) ↔ Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑘))(2↑𝑛) = (𝑁 mod (2↑𝑘))))
24	23	imbi2d 341	. . . 4 ⊢ (𝑥 = 𝑘 → ((𝑁 ∈ ℕ₀ → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑥))(2↑𝑛) = (𝑁 mod (2↑𝑥))) ↔ (𝑁 ∈ ℕ₀ → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑘))(2↑𝑛) = (𝑁 mod (2↑𝑘)))))
25		oveq2 7417	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (0..^𝑥) = (0..^(𝑘 + 1)))
26	25	ineq2d 4213	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → ((bits‘𝑁) ∩ (0..^𝑥)) = ((bits‘𝑁) ∩ (0..^(𝑘 + 1))))
27	26	sumeq1d 15647	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑥))(2↑𝑛) = Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^(𝑘 + 1)))(2↑𝑛))
28		oveq2 7417	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (2↑𝑥) = (2↑(𝑘 + 1)))
29	28	oveq2d 7425	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝑁 mod (2↑𝑥)) = (𝑁 mod (2↑(𝑘 + 1))))
30	27, 29	eqeq12d 2749	. . . . 5 ⊢ (𝑥 = (𝑘 + 1) → (Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑥))(2↑𝑛) = (𝑁 mod (2↑𝑥)) ↔ Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^(𝑘 + 1)))(2↑𝑛) = (𝑁 mod (2↑(𝑘 + 1)))))
31	30	imbi2d 341	. . . 4 ⊢ (𝑥 = (𝑘 + 1) → ((𝑁 ∈ ℕ₀ → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑥))(2↑𝑛) = (𝑁 mod (2↑𝑥))) ↔ (𝑁 ∈ ℕ₀ → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^(𝑘 + 1)))(2↑𝑛) = (𝑁 mod (2↑(𝑘 + 1))))))
32		oveq2 7417	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (0..^𝑥) = (0..^𝑁))
33	32	ineq2d 4213	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((bits‘𝑁) ∩ (0..^𝑥)) = ((bits‘𝑁) ∩ (0..^𝑁)))
34	33	sumeq1d 15647	. . . . . 6 ⊢ (𝑥 = 𝑁 → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑥))(2↑𝑛) = Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑁))(2↑𝑛))
35		oveq2 7417	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (2↑𝑥) = (2↑𝑁))
36	35	oveq2d 7425	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑁 mod (2↑𝑥)) = (𝑁 mod (2↑𝑁)))
37	34, 36	eqeq12d 2749	. . . . 5 ⊢ (𝑥 = 𝑁 → (Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑥))(2↑𝑛) = (𝑁 mod (2↑𝑥)) ↔ Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑁))(2↑𝑛) = (𝑁 mod (2↑𝑁))))
38	37	imbi2d 341	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝑁 ∈ ℕ₀ → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑥))(2↑𝑛) = (𝑁 mod (2↑𝑥))) ↔ (𝑁 ∈ ℕ₀ → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑁))(2↑𝑛) = (𝑁 mod (2↑𝑁)))))
39		nn0z 12583	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
40		zmod10 13852	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 mod 1) = 0)
41	39, 40	syl 17	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 mod 1) = 0)
42	41	eqcomd 2739	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → 0 = (𝑁 mod 1))
43		oveq1 7416	. . . . . . 7 ⊢ (Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑘))(2↑𝑛) = (𝑁 mod (2↑𝑘)) → (Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑘))(2↑𝑛) + Σ𝑛 ∈ ((bits‘𝑁) ∩ {𝑘})(2↑𝑛)) = ((𝑁 mod (2↑𝑘)) + Σ𝑛 ∈ ((bits‘𝑁) ∩ {𝑘})(2↑𝑛)))
44		fzonel 13646	. . . . . . . . . . . . 13 ⊢ ¬ 𝑘 ∈ (0..^𝑘)
45	44	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → ¬ 𝑘 ∈ (0..^𝑘))
46		disjsn 4716	. . . . . . . . . . . 12 ⊢ (((0..^𝑘) ∩ {𝑘}) = ∅ ↔ ¬ 𝑘 ∈ (0..^𝑘))
47	45, 46	sylibr 233	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → ((0..^𝑘) ∩ {𝑘}) = ∅)
48	47	ineq2d 4213	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → ((bits‘𝑁) ∩ ((0..^𝑘) ∩ {𝑘})) = ((bits‘𝑁) ∩ ∅))
49		inindi 4227	. . . . . . . . . 10 ⊢ ((bits‘𝑁) ∩ ((0..^𝑘) ∩ {𝑘})) = (((bits‘𝑁) ∩ (0..^𝑘)) ∩ ((bits‘𝑁) ∩ {𝑘}))
50	48, 49, 5	3eqtr3g 2796	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → (((bits‘𝑁) ∩ (0..^𝑘)) ∩ ((bits‘𝑁) ∩ {𝑘})) = ∅)
51		simpr 486	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
52		nn0uz 12864	. . . . . . . . . . . . 13 ⊢ ℕ₀ = (ℤ_≥‘0)
53	51, 52	eleqtrdi 2844	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ (ℤ_≥‘0))
54		fzosplitsn 13740	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (ℤ_≥‘0) → (0..^(𝑘 + 1)) = ((0..^𝑘) ∪ {𝑘}))
55	53, 54	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → (0..^(𝑘 + 1)) = ((0..^𝑘) ∪ {𝑘}))
56	55	ineq2d 4213	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → ((bits‘𝑁) ∩ (0..^(𝑘 + 1))) = ((bits‘𝑁) ∩ ((0..^𝑘) ∪ {𝑘})))
57		indi 4274	. . . . . . . . . 10 ⊢ ((bits‘𝑁) ∩ ((0..^𝑘) ∪ {𝑘})) = (((bits‘𝑁) ∩ (0..^𝑘)) ∪ ((bits‘𝑁) ∩ {𝑘}))
58	56, 57	eqtrdi 2789	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → ((bits‘𝑁) ∩ (0..^(𝑘 + 1))) = (((bits‘𝑁) ∩ (0..^𝑘)) ∪ ((bits‘𝑁) ∩ {𝑘})))
59		fzofi 13939	. . . . . . . . . . 11 ⊢ (0..^(𝑘 + 1)) ∈ Fin
60		inss2 4230	. . . . . . . . . . 11 ⊢ ((bits‘𝑁) ∩ (0..^(𝑘 + 1))) ⊆ (0..^(𝑘 + 1))
61		ssfi 9173	. . . . . . . . . . 11 ⊢ (((0..^(𝑘 + 1)) ∈ Fin ∧ ((bits‘𝑁) ∩ (0..^(𝑘 + 1))) ⊆ (0..^(𝑘 + 1))) → ((bits‘𝑁) ∩ (0..^(𝑘 + 1))) ∈ Fin)
62	59, 60, 61	mp2an 691	. . . . . . . . . 10 ⊢ ((bits‘𝑁) ∩ (0..^(𝑘 + 1))) ∈ Fin
63	62	a1i 11	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → ((bits‘𝑁) ∩ (0..^(𝑘 + 1))) ∈ Fin)
64		2nn 12285	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ
65	64	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑛 ∈ ((bits‘𝑁) ∩ (0..^(𝑘 + 1)))) → 2 ∈ ℕ)
66		simpr 486	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑛 ∈ ((bits‘𝑁) ∩ (0..^(𝑘 + 1)))) → 𝑛 ∈ ((bits‘𝑁) ∩ (0..^(𝑘 + 1))))
67	66	elin2d 4200	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑛 ∈ ((bits‘𝑁) ∩ (0..^(𝑘 + 1)))) → 𝑛 ∈ (0..^(𝑘 + 1)))
68		elfzouz 13636	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (0..^(𝑘 + 1)) → 𝑛 ∈ (ℤ_≥‘0))
69	67, 68	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑛 ∈ ((bits‘𝑁) ∩ (0..^(𝑘 + 1)))) → 𝑛 ∈ (ℤ_≥‘0))
70	69, 52	eleqtrrdi 2845	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑛 ∈ ((bits‘𝑁) ∩ (0..^(𝑘 + 1)))) → 𝑛 ∈ ℕ₀)
71	65, 70	nnexpcld 14208	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑛 ∈ ((bits‘𝑁) ∩ (0..^(𝑘 + 1)))) → (2↑𝑛) ∈ ℕ)
72	71	nncnd 12228	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑛 ∈ ((bits‘𝑁) ∩ (0..^(𝑘 + 1)))) → (2↑𝑛) ∈ ℂ)
73	50, 58, 63, 72	fsumsplit 15687	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^(𝑘 + 1)))(2↑𝑛) = (Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑘))(2↑𝑛) + Σ𝑛 ∈ ((bits‘𝑁) ∩ {𝑘})(2↑𝑛)))
74		bitsinv1lem 16382	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ ℕ₀) → (𝑁 mod (2↑(𝑘 + 1))) = ((𝑁 mod (2↑𝑘)) + if(𝑘 ∈ (bits‘𝑁), (2↑𝑘), 0)))
75	39, 74	sylan 581	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → (𝑁 mod (2↑(𝑘 + 1))) = ((𝑁 mod (2↑𝑘)) + if(𝑘 ∈ (bits‘𝑁), (2↑𝑘), 0)))
76		eqeq2 2745	. . . . . . . . . . 11 ⊢ ((2↑𝑘) = if(𝑘 ∈ (bits‘𝑁), (2↑𝑘), 0) → (Σ𝑛 ∈ ((bits‘𝑁) ∩ {𝑘})(2↑𝑛) = (2↑𝑘) ↔ Σ𝑛 ∈ ((bits‘𝑁) ∩ {𝑘})(2↑𝑛) = if(𝑘 ∈ (bits‘𝑁), (2↑𝑘), 0)))
77		eqeq2 2745	. . . . . . . . . . 11 ⊢ (0 = if(𝑘 ∈ (bits‘𝑁), (2↑𝑘), 0) → (Σ𝑛 ∈ ((bits‘𝑁) ∩ {𝑘})(2↑𝑛) = 0 ↔ Σ𝑛 ∈ ((bits‘𝑁) ∩ {𝑘})(2↑𝑛) = if(𝑘 ∈ (bits‘𝑁), (2↑𝑘), 0)))
78		simpr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑘 ∈ (bits‘𝑁)) → 𝑘 ∈ (bits‘𝑁))
79	78	snssd 4813	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑘 ∈ (bits‘𝑁)) → {𝑘} ⊆ (bits‘𝑁))
80		sseqin2 4216	. . . . . . . . . . . . . 14 ⊢ ({𝑘} ⊆ (bits‘𝑁) ↔ ((bits‘𝑁) ∩ {𝑘}) = {𝑘})
81	79, 80	sylib 217	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑘 ∈ (bits‘𝑁)) → ((bits‘𝑁) ∩ {𝑘}) = {𝑘})
82	81	sumeq1d 15647	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑘 ∈ (bits‘𝑁)) → Σ𝑛 ∈ ((bits‘𝑁) ∩ {𝑘})(2↑𝑛) = Σ𝑛 ∈ {𝑘} (2↑𝑛))
83		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑘 ∈ (bits‘𝑁)) → 𝑘 ∈ ℕ₀)
84	64	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑘 ∈ (bits‘𝑁)) → 2 ∈ ℕ)
85	84, 83	nnexpcld 14208	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑘 ∈ (bits‘𝑁)) → (2↑𝑘) ∈ ℕ)
86	85	nncnd 12228	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑘 ∈ (bits‘𝑁)) → (2↑𝑘) ∈ ℂ)
87		oveq2 7417	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (2↑𝑛) = (2↑𝑘))
88	87	sumsn 15692	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ₀ ∧ (2↑𝑘) ∈ ℂ) → Σ𝑛 ∈ {𝑘} (2↑𝑛) = (2↑𝑘))
89	83, 86, 88	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑘 ∈ (bits‘𝑁)) → Σ𝑛 ∈ {𝑘} (2↑𝑛) = (2↑𝑘))
90	82, 89	eqtrd 2773	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑘 ∈ (bits‘𝑁)) → Σ𝑛 ∈ ((bits‘𝑁) ∩ {𝑘})(2↑𝑛) = (2↑𝑘))
91		simpr 486	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 ∈ (bits‘𝑁)) → ¬ 𝑘 ∈ (bits‘𝑁))
92		disjsn 4716	. . . . . . . . . . . . . 14 ⊢ (((bits‘𝑁) ∩ {𝑘}) = ∅ ↔ ¬ 𝑘 ∈ (bits‘𝑁))
93	91, 92	sylibr 233	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 ∈ (bits‘𝑁)) → ((bits‘𝑁) ∩ {𝑘}) = ∅)
94	93	sumeq1d 15647	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 ∈ (bits‘𝑁)) → Σ𝑛 ∈ ((bits‘𝑁) ∩ {𝑘})(2↑𝑛) = Σ𝑛 ∈ ∅ (2↑𝑛))
95	94, 8	eqtrdi 2789	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 ∈ (bits‘𝑁)) → Σ𝑛 ∈ ((bits‘𝑁) ∩ {𝑘})(2↑𝑛) = 0)
96	76, 77, 90, 95	ifbothda 4567	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → Σ𝑛 ∈ ((bits‘𝑁) ∩ {𝑘})(2↑𝑛) = if(𝑘 ∈ (bits‘𝑁), (2↑𝑘), 0))
97	96	oveq2d 7425	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → ((𝑁 mod (2↑𝑘)) + Σ𝑛 ∈ ((bits‘𝑁) ∩ {𝑘})(2↑𝑛)) = ((𝑁 mod (2↑𝑘)) + if(𝑘 ∈ (bits‘𝑁), (2↑𝑘), 0)))
98	75, 97	eqtr4d 2776	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → (𝑁 mod (2↑(𝑘 + 1))) = ((𝑁 mod (2↑𝑘)) + Σ𝑛 ∈ ((bits‘𝑁) ∩ {𝑘})(2↑𝑛)))
99	73, 98	eqeq12d 2749	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → (Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^(𝑘 + 1)))(2↑𝑛) = (𝑁 mod (2↑(𝑘 + 1))) ↔ (Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑘))(2↑𝑛) + Σ𝑛 ∈ ((bits‘𝑁) ∩ {𝑘})(2↑𝑛)) = ((𝑁 mod (2↑𝑘)) + Σ𝑛 ∈ ((bits‘𝑁) ∩ {𝑘})(2↑𝑛))))
100	43, 99	imbitrrid 245	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → (Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑘))(2↑𝑛) = (𝑁 mod (2↑𝑘)) → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^(𝑘 + 1)))(2↑𝑛) = (𝑁 mod (2↑(𝑘 + 1)))))
101	100	expcom 415	. . . . 5 ⊢ (𝑘 ∈ ℕ₀ → (𝑁 ∈ ℕ₀ → (Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑘))(2↑𝑛) = (𝑁 mod (2↑𝑘)) → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^(𝑘 + 1)))(2↑𝑛) = (𝑁 mod (2↑(𝑘 + 1))))))
102	101	a2d 29	. . . 4 ⊢ (𝑘 ∈ ℕ₀ → ((𝑁 ∈ ℕ₀ → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑘))(2↑𝑛) = (𝑁 mod (2↑𝑘))) → (𝑁 ∈ ℕ₀ → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^(𝑘 + 1)))(2↑𝑛) = (𝑁 mod (2↑(𝑘 + 1))))))
103	17, 24, 31, 38, 42, 102	nn0ind 12657	. . 3 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 ∈ ℕ₀ → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑁))(2↑𝑛) = (𝑁 mod (2↑𝑁))))
104	103	pm2.43i 52	. 2 ⊢ (𝑁 ∈ ℕ₀ → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑁))(2↑𝑛) = (𝑁 mod (2↑𝑁)))
105		id 22	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℕ₀)
106	105, 52	eleqtrdi 2844	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ (ℤ_≥‘0))
107	64	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → 2 ∈ ℕ)
108	107, 105	nnexpcld 14208	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → (2↑𝑁) ∈ ℕ)
109	108	nnzd 12585	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → (2↑𝑁) ∈ ℤ)
110		2z 12594	. . . . . . . 8 ⊢ 2 ∈ ℤ
111		uzid 12837	. . . . . . . 8 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
112	110, 111	ax-mp 5	. . . . . . 7 ⊢ 2 ∈ (ℤ_≥‘2)
113		bernneq3 14194	. . . . . . 7 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℕ₀) → 𝑁 < (2↑𝑁))
114	112, 113	mpan 689	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 < (2↑𝑁))
115		elfzo2 13635	. . . . . 6 ⊢ (𝑁 ∈ (0..^(2↑𝑁)) ↔ (𝑁 ∈ (ℤ_≥‘0) ∧ (2↑𝑁) ∈ ℤ ∧ 𝑁 < (2↑𝑁)))
116	106, 109, 114, 115	syl3anbrc 1344	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ (0..^(2↑𝑁)))
117		bitsfzo 16376	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝑁 ∈ (0..^(2↑𝑁)) ↔ (bits‘𝑁) ⊆ (0..^𝑁)))
118	39, 105, 117	syl2anc 585	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 ∈ (0..^(2↑𝑁)) ↔ (bits‘𝑁) ⊆ (0..^𝑁)))
119	116, 118	mpbid 231	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (bits‘𝑁) ⊆ (0..^𝑁))
120		df-ss 3966	. . . 4 ⊢ ((bits‘𝑁) ⊆ (0..^𝑁) ↔ ((bits‘𝑁) ∩ (0..^𝑁)) = (bits‘𝑁))
121	119, 120	sylib 217	. . 3 ⊢ (𝑁 ∈ ℕ₀ → ((bits‘𝑁) ∩ (0..^𝑁)) = (bits‘𝑁))
122	121	sumeq1d 15647	. 2 ⊢ (𝑁 ∈ ℕ₀ → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑁))(2↑𝑛) = Σ𝑛 ∈ (bits‘𝑁)(2↑𝑛))
123		nn0re 12481	. . 3 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
124		2rp 12979	. . . . 5 ⊢ 2 ∈ ℝ⁺
125	124	a1i 11	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → 2 ∈ ℝ⁺)
126	125, 39	rpexpcld 14210	. . 3 ⊢ (𝑁 ∈ ℕ₀ → (2↑𝑁) ∈ ℝ⁺)
127		nn0ge0 12497	. . 3 ⊢ (𝑁 ∈ ℕ₀ → 0 ≤ 𝑁)
128		modid 13861	. . 3 ⊢ (((𝑁 ∈ ℝ ∧ (2↑𝑁) ∈ ℝ⁺) ∧ (0 ≤ 𝑁 ∧ 𝑁 < (2↑𝑁))) → (𝑁 mod (2↑𝑁)) = 𝑁)
129	123, 126, 127, 114, 128	syl22anc 838	. 2 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 mod (2↑𝑁)) = 𝑁)
130	104, 122, 129	3eqtr3d 2781	1 ⊢ (𝑁 ∈ ℕ₀ → Σ𝑛 ∈ (bits‘𝑁)(2↑𝑛) = 𝑁)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∪ cun 3947 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 ifcif 4529 {csn 4629 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 Fincfn 8939 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 < clt 11248 ≤ cle 11249 ℕcn 12212 2c2 12267 ℕ₀cn0 12472 ℤcz 12558 ℤ_≥cuz 12822 ℝ⁺crp 12974 ..^cfzo 13627 mod cmo 13834 ↑cexp 14027 Σcsu 15632 bitscbits 16360

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-sum 15633 df-dvds 16198 df-bits 16363

This theorem is referenced by: bitsinv2 16384 bitsf1ocnv 16385 eulerpartlemgc 33361 eulerpartlemgs2 33379

W3C validator