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Theorem bitsinv1 16385

Description: There is an explicit inverse to the bits function for nonnegative integers (which can be extended to negative integers using bitscmp 16381), 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 7419	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (0..^𝑥) = (0..^0))
2		fzo0 13658	. . . . . . . . . . 11 ⊢ (0..^0) = ∅
3	1, 2	eqtrdi 2788	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (0..^𝑥) = ∅)
4	3	ineq2d 4212	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((bits‘𝑁) ∩ (0..^𝑥)) = ((bits‘𝑁) ∩ ∅))
5		in0 4391	. . . . . . . . 9 ⊢ ((bits‘𝑁) ∩ ∅) = ∅
6	4, 5	eqtrdi 2788	. . . . . . . 8 ⊢ (𝑥 = 0 → ((bits‘𝑁) ∩ (0..^𝑥)) = ∅)
7	6	sumeq1d 15649	. . . . . . 7 ⊢ (𝑥 = 0 → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑥))(2↑𝑛) = Σ𝑛 ∈ ∅ (2↑𝑛))
8		sum0 15669	. . . . . . 7 ⊢ Σ𝑛 ∈ ∅ (2↑𝑛) = 0
9	7, 8	eqtrdi 2788	. . . . . 6 ⊢ (𝑥 = 0 → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑥))(2↑𝑛) = 0)
10		oveq2 7419	. . . . . . . 8 ⊢ (𝑥 = 0 → (2↑𝑥) = (2↑0))
11		2cn 12289	. . . . . . . . 9 ⊢ 2 ∈ ℂ
12		exp0 14033	. . . . . . . . 9 ⊢ (2 ∈ ℂ → (2↑0) = 1)
13	11, 12	ax-mp 5	. . . . . . . 8 ⊢ (2↑0) = 1
14	10, 13	eqtrdi 2788	. . . . . . 7 ⊢ (𝑥 = 0 → (2↑𝑥) = 1)
15	14	oveq2d 7427	. . . . . 6 ⊢ (𝑥 = 0 → (𝑁 mod (2↑𝑥)) = (𝑁 mod 1))
16	9, 15	eqeq12d 2748	. . . . 5 ⊢ (𝑥 = 0 → (Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑥))(2↑𝑛) = (𝑁 mod (2↑𝑥)) ↔ 0 = (𝑁 mod 1)))
17	16	imbi2d 340	. . . 4 ⊢ (𝑥 = 0 → ((𝑁 ∈ ℕ₀ → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑥))(2↑𝑛) = (𝑁 mod (2↑𝑥))) ↔ (𝑁 ∈ ℕ₀ → 0 = (𝑁 mod 1))))
18		oveq2 7419	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (0..^𝑥) = (0..^𝑘))
19	18	ineq2d 4212	. . . . . . 7 ⊢ (𝑥 = 𝑘 → ((bits‘𝑁) ∩ (0..^𝑥)) = ((bits‘𝑁) ∩ (0..^𝑘)))
20	19	sumeq1d 15649	. . . . . 6 ⊢ (𝑥 = 𝑘 → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑥))(2↑𝑛) = Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑘))(2↑𝑛))
21		oveq2 7419	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (2↑𝑥) = (2↑𝑘))
22	21	oveq2d 7427	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝑁 mod (2↑𝑥)) = (𝑁 mod (2↑𝑘)))
23	20, 22	eqeq12d 2748	. . . . 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 340	. . . 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 7419	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (0..^𝑥) = (0..^(𝑘 + 1)))
26	25	ineq2d 4212	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → ((bits‘𝑁) ∩ (0..^𝑥)) = ((bits‘𝑁) ∩ (0..^(𝑘 + 1))))
27	26	sumeq1d 15649	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑥))(2↑𝑛) = Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^(𝑘 + 1)))(2↑𝑛))
28		oveq2 7419	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (2↑𝑥) = (2↑(𝑘 + 1)))
29	28	oveq2d 7427	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝑁 mod (2↑𝑥)) = (𝑁 mod (2↑(𝑘 + 1))))
30	27, 29	eqeq12d 2748	. . . . 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 340	. . . 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 7419	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (0..^𝑥) = (0..^𝑁))
33	32	ineq2d 4212	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((bits‘𝑁) ∩ (0..^𝑥)) = ((bits‘𝑁) ∩ (0..^𝑁)))
34	33	sumeq1d 15649	. . . . . 6 ⊢ (𝑥 = 𝑁 → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑥))(2↑𝑛) = Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑁))(2↑𝑛))
35		oveq2 7419	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (2↑𝑥) = (2↑𝑁))
36	35	oveq2d 7427	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑁 mod (2↑𝑥)) = (𝑁 mod (2↑𝑁)))
37	34, 36	eqeq12d 2748	. . . . 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 340	. . . 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 12585	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
40		zmod10 13854	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 mod 1) = 0)
41	39, 40	syl 17	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 mod 1) = 0)
42	41	eqcomd 2738	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → 0 = (𝑁 mod 1))
43		oveq1 7418	. . . . . . 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 13648	. . . . . . . . . . . . 13 ⊢ ¬ 𝑘 ∈ (0..^𝑘)
45	44	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → ¬ 𝑘 ∈ (0..^𝑘))
46		disjsn 4715	. . . . . . . . . . . 12 ⊢ (((0..^𝑘) ∩ {𝑘}) = ∅ ↔ ¬ 𝑘 ∈ (0..^𝑘))
47	45, 46	sylibr 233	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → ((0..^𝑘) ∩ {𝑘}) = ∅)
48	47	ineq2d 4212	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → ((bits‘𝑁) ∩ ((0..^𝑘) ∩ {𝑘})) = ((bits‘𝑁) ∩ ∅))
49		inindi 4226	. . . . . . . . . 10 ⊢ ((bits‘𝑁) ∩ ((0..^𝑘) ∩ {𝑘})) = (((bits‘𝑁) ∩ (0..^𝑘)) ∩ ((bits‘𝑁) ∩ {𝑘}))
50	48, 49, 5	3eqtr3g 2795	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → (((bits‘𝑁) ∩ (0..^𝑘)) ∩ ((bits‘𝑁) ∩ {𝑘})) = ∅)
51		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
52		nn0uz 12866	. . . . . . . . . . . . 13 ⊢ ℕ₀ = (ℤ_≥‘0)
53	51, 52	eleqtrdi 2843	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ (ℤ_≥‘0))
54		fzosplitsn 13742	. . . . . . . . . . . 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 4212	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → ((bits‘𝑁) ∩ (0..^(𝑘 + 1))) = ((bits‘𝑁) ∩ ((0..^𝑘) ∪ {𝑘})))
57		indi 4273	. . . . . . . . . 10 ⊢ ((bits‘𝑁) ∩ ((0..^𝑘) ∪ {𝑘})) = (((bits‘𝑁) ∩ (0..^𝑘)) ∪ ((bits‘𝑁) ∩ {𝑘}))
58	56, 57	eqtrdi 2788	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → ((bits‘𝑁) ∩ (0..^(𝑘 + 1))) = (((bits‘𝑁) ∩ (0..^𝑘)) ∪ ((bits‘𝑁) ∩ {𝑘})))
59		fzofi 13941	. . . . . . . . . . 11 ⊢ (0..^(𝑘 + 1)) ∈ Fin
60		inss2 4229	. . . . . . . . . . 11 ⊢ ((bits‘𝑁) ∩ (0..^(𝑘 + 1))) ⊆ (0..^(𝑘 + 1))
61		ssfi 9175	. . . . . . . . . . 11 ⊢ (((0..^(𝑘 + 1)) ∈ Fin ∧ ((bits‘𝑁) ∩ (0..^(𝑘 + 1))) ⊆ (0..^(𝑘 + 1))) → ((bits‘𝑁) ∩ (0..^(𝑘 + 1))) ∈ Fin)
62	59, 60, 61	mp2an 690	. . . . . . . . . 10 ⊢ ((bits‘𝑁) ∩ (0..^(𝑘 + 1))) ∈ Fin
63	62	a1i 11	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → ((bits‘𝑁) ∩ (0..^(𝑘 + 1))) ∈ Fin)
64		2nn 12287	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ
65	64	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑛 ∈ ((bits‘𝑁) ∩ (0..^(𝑘 + 1)))) → 2 ∈ ℕ)
66		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑛 ∈ ((bits‘𝑁) ∩ (0..^(𝑘 + 1)))) → 𝑛 ∈ ((bits‘𝑁) ∩ (0..^(𝑘 + 1))))
67	66	elin2d 4199	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑛 ∈ ((bits‘𝑁) ∩ (0..^(𝑘 + 1)))) → 𝑛 ∈ (0..^(𝑘 + 1)))
68		elfzouz 13638	. . . . . . . . . . . . 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 2844	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑛 ∈ ((bits‘𝑁) ∩ (0..^(𝑘 + 1)))) → 𝑛 ∈ ℕ₀)
71	65, 70	nnexpcld 14210	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑛 ∈ ((bits‘𝑁) ∩ (0..^(𝑘 + 1)))) → (2↑𝑛) ∈ ℕ)
72	71	nncnd 12230	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑛 ∈ ((bits‘𝑁) ∩ (0..^(𝑘 + 1)))) → (2↑𝑛) ∈ ℂ)
73	50, 58, 63, 72	fsumsplit 15689	. . . . . . . 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 16384	. . . . . . . . . 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 580	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → (𝑁 mod (2↑(𝑘 + 1))) = ((𝑁 mod (2↑𝑘)) + if(𝑘 ∈ (bits‘𝑁), (2↑𝑘), 0)))
76		eqeq2 2744	. . . . . . . . . . 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 2744	. . . . . . . . . . 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 485	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑘 ∈ (bits‘𝑁)) → 𝑘 ∈ (bits‘𝑁))
79	78	snssd 4812	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑘 ∈ (bits‘𝑁)) → {𝑘} ⊆ (bits‘𝑁))
80		sseqin2 4215	. . . . . . . . . . . . . 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 15649	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑘 ∈ (bits‘𝑁)) → Σ𝑛 ∈ ((bits‘𝑁) ∩ {𝑘})(2↑𝑛) = Σ𝑛 ∈ {𝑘} (2↑𝑛))
83		simplr 767	. . . . . . . . . . . . 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 14210	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑘 ∈ (bits‘𝑁)) → (2↑𝑘) ∈ ℕ)
86	85	nncnd 12230	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑘 ∈ (bits‘𝑁)) → (2↑𝑘) ∈ ℂ)
87		oveq2 7419	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (2↑𝑛) = (2↑𝑘))
88	87	sumsn 15694	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ₀ ∧ (2↑𝑘) ∈ ℂ) → Σ𝑛 ∈ {𝑘} (2↑𝑛) = (2↑𝑘))
89	83, 86, 88	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑘 ∈ (bits‘𝑁)) → Σ𝑛 ∈ {𝑘} (2↑𝑛) = (2↑𝑘))
90	82, 89	eqtrd 2772	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ 𝑘 ∈ (bits‘𝑁)) → Σ𝑛 ∈ ((bits‘𝑁) ∩ {𝑘})(2↑𝑛) = (2↑𝑘))
91		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 ∈ (bits‘𝑁)) → ¬ 𝑘 ∈ (bits‘𝑁))
92		disjsn 4715	. . . . . . . . . . . . . 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 15649	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 ∈ (bits‘𝑁)) → Σ𝑛 ∈ ((bits‘𝑁) ∩ {𝑘})(2↑𝑛) = Σ𝑛 ∈ ∅ (2↑𝑛))
95	94, 8	eqtrdi 2788	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 ∈ (bits‘𝑁)) → Σ𝑛 ∈ ((bits‘𝑁) ∩ {𝑘})(2↑𝑛) = 0)
96	76, 77, 90, 95	ifbothda 4566	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → Σ𝑛 ∈ ((bits‘𝑁) ∩ {𝑘})(2↑𝑛) = if(𝑘 ∈ (bits‘𝑁), (2↑𝑘), 0))
97	96	oveq2d 7427	. . . . . . . . 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 2775	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → (𝑁 mod (2↑(𝑘 + 1))) = ((𝑁 mod (2↑𝑘)) + Σ𝑛 ∈ ((bits‘𝑁) ∩ {𝑘})(2↑𝑛)))
99	73, 98	eqeq12d 2748	. . . . . . 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 414	. . . . 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 12659	. . 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 2843	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ (ℤ_≥‘0))
107	64	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → 2 ∈ ℕ)
108	107, 105	nnexpcld 14210	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → (2↑𝑁) ∈ ℕ)
109	108	nnzd 12587	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → (2↑𝑁) ∈ ℤ)
110		2z 12596	. . . . . . . 8 ⊢ 2 ∈ ℤ
111		uzid 12839	. . . . . . . 8 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
112	110, 111	ax-mp 5	. . . . . . 7 ⊢ 2 ∈ (ℤ_≥‘2)
113		bernneq3 14196	. . . . . . 7 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℕ₀) → 𝑁 < (2↑𝑁))
114	112, 113	mpan 688	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 < (2↑𝑁))
115		elfzo2 13637	. . . . . 6 ⊢ (𝑁 ∈ (0..^(2↑𝑁)) ↔ (𝑁 ∈ (ℤ_≥‘0) ∧ (2↑𝑁) ∈ ℤ ∧ 𝑁 < (2↑𝑁)))
116	106, 109, 114, 115	syl3anbrc 1343	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ (0..^(2↑𝑁)))
117		bitsfzo 16378	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝑁 ∈ (0..^(2↑𝑁)) ↔ (bits‘𝑁) ⊆ (0..^𝑁)))
118	39, 105, 117	syl2anc 584	. . . . 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 3965	. . . 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 15649	. 2 ⊢ (𝑁 ∈ ℕ₀ → Σ𝑛 ∈ ((bits‘𝑁) ∩ (0..^𝑁))(2↑𝑛) = Σ𝑛 ∈ (bits‘𝑁)(2↑𝑛))
123		nn0re 12483	. . 3 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
124		2rp 12981	. . . . 5 ⊢ 2 ∈ ℝ⁺
125	124	a1i 11	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → 2 ∈ ℝ⁺)
126	125, 39	rpexpcld 14212	. . 3 ⊢ (𝑁 ∈ ℕ₀ → (2↑𝑁) ∈ ℝ⁺)
127		nn0ge0 12499	. . 3 ⊢ (𝑁 ∈ ℕ₀ → 0 ≤ 𝑁)
128		modid 13863	. . 3 ⊢ (((𝑁 ∈ ℝ ∧ (2↑𝑁) ∈ ℝ⁺) ∧ (0 ≤ 𝑁 ∧ 𝑁 < (2↑𝑁))) → (𝑁 mod (2↑𝑁)) = 𝑁)
129	123, 126, 127, 114, 128	syl22anc 837	. 2 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 mod (2↑𝑁)) = 𝑁)
130	104, 122, 129	3eqtr3d 2780	1 ⊢ (𝑁 ∈ ℕ₀ → Σ𝑛 ∈ (bits‘𝑁)(2↑𝑛) = 𝑁)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 ifcif 4528 {csn 4628 class class class wbr 5148 ‘cfv 6543 (class class class)co 7411 Fincfn 8941 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 < clt 11250 ≤ cle 11251 ℕcn 12214 2c2 12269 ℕ₀cn0 12474 ℤcz 12560 ℤ_≥cuz 12824 ℝ⁺crp 12976 ..^cfzo 13629 mod cmo 13836 ↑cexp 14029 Σcsu 15634 bitscbits 16362

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-n0 12475 df-z 12561 df-uz 12825 df-rp 12977 df-fz 13487 df-fzo 13630 df-fl 13759 df-mod 13837 df-seq 13969 df-exp 14030 df-hash 14293 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-sum 15635 df-dvds 16200 df-bits 16365

This theorem is referenced by: bitsinv2 16386 bitsf1ocnv 16387 eulerpartlemgc 33430 eulerpartlemgs2 33448

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