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Theorem musum 27252

Description: The sum of the Möbius function over the divisors of 𝑁 gives one if 𝑁 = 1, but otherwise always sums to zero. Theorem 2.1 in [ApostolNT] p. 25. This makes the Möbius function useful for inverting divisor sums; see also muinv 27254. (Contributed by Mario Carneiro, 2-Jul-2015.)

**Assertion**
Ref	Expression
musum	⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} (μ‘𝑘) = if(𝑁 = 1, 1, 0))

Distinct variable group: 𝑘,𝑛,𝑁

Proof of Theorem musum Dummy variables 𝑚 𝑝 𝑞 𝑠 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6920	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (μ‘𝑛) = (μ‘𝑘))
2	1	neeq1d 3006	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ((μ‘𝑛) ≠ 0 ↔ (μ‘𝑘) ≠ 0))
3		breq1 5169	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑛 ∥ 𝑁 ↔ 𝑘 ∥ 𝑁))
4	2, 3	anbi12d 631	. . . . . 6 ⊢ (𝑛 = 𝑘 → (((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁) ↔ ((μ‘𝑘) ≠ 0 ∧ 𝑘 ∥ 𝑁)))
5	4	elrab 3708	. . . . 5 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} ↔ (𝑘 ∈ ℕ ∧ ((μ‘𝑘) ≠ 0 ∧ 𝑘 ∥ 𝑁)))
6		muval2 27195	. . . . . 6 ⊢ ((𝑘 ∈ ℕ ∧ (μ‘𝑘) ≠ 0) → (μ‘𝑘) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})))
7	6	adantrr 716	. . . . 5 ⊢ ((𝑘 ∈ ℕ ∧ ((μ‘𝑘) ≠ 0 ∧ 𝑘 ∥ 𝑁)) → (μ‘𝑘) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})))
8	5, 7	sylbi 217	. . . 4 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} → (μ‘𝑘) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})))
9	8	adantl 481	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) → (μ‘𝑘) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})))
10	9	sumeq2dv 15750	. 2 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} (μ‘𝑘) = Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})))
11		simpr 484	. . . . 5 ⊢ (((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁) → 𝑛 ∥ 𝑁)
12	11	a1i 11	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁) → 𝑛 ∥ 𝑁))
13	12	ss2rabdv 4099	. . 3 ⊢ (𝑁 ∈ ℕ → {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} ⊆ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁})
14		ssrab2 4103	. . . . . 6 ⊢ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} ⊆ ℕ
15		simpr 484	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) → 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)})
16	14, 15	sselid 4006	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) → 𝑘 ∈ ℕ)
17		mucl 27202	. . . . 5 ⊢ (𝑘 ∈ ℕ → (μ‘𝑘) ∈ ℤ)
18	16, 17	syl 17	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) → (μ‘𝑘) ∈ ℤ)
19	18	zcnd 12748	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) → (μ‘𝑘) ∈ ℂ)
20		difrab 4337	. . . . . . 7 ⊢ ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} ∖ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) = {𝑛 ∈ ℕ ∣ (𝑛 ∥ 𝑁 ∧ ¬ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁))}
21		pm3.21 471	. . . . . . . . . . 11 ⊢ (𝑛 ∥ 𝑁 → ((μ‘𝑛) ≠ 0 → ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)))
22	21	necon1bd 2964	. . . . . . . . . 10 ⊢ (𝑛 ∥ 𝑁 → (¬ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁) → (μ‘𝑛) = 0))
23	22	imp 406	. . . . . . . . 9 ⊢ ((𝑛 ∥ 𝑁 ∧ ¬ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)) → (μ‘𝑛) = 0)
24	23	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((𝑛 ∥ 𝑁 ∧ ¬ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)) → (μ‘𝑛) = 0))
25	24	ss2rabi 4100	. . . . . . 7 ⊢ {𝑛 ∈ ℕ ∣ (𝑛 ∥ 𝑁 ∧ ¬ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁))} ⊆ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0}
26	20, 25	eqsstri 4043	. . . . . 6 ⊢ ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} ∖ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) ⊆ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0}
27	26	sseli 4004	. . . . 5 ⊢ (𝑘 ∈ ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} ∖ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) → 𝑘 ∈ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0})
28		fveqeq2 6929	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ((μ‘𝑛) = 0 ↔ (μ‘𝑘) = 0))
29	28	elrab 3708	. . . . . 6 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0} ↔ (𝑘 ∈ ℕ ∧ (μ‘𝑘) = 0))
30	29	simprbi 496	. . . . 5 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0} → (μ‘𝑘) = 0)
31	27, 30	syl 17	. . . 4 ⊢ (𝑘 ∈ ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} ∖ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) → (μ‘𝑘) = 0)
32	31	adantl 481	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} ∖ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)})) → (μ‘𝑘) = 0)
33		fzfid 14024	. . . 4 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ∈ Fin)
34		dvdsssfz1 16366	. . . 4 ⊢ (𝑁 ∈ ℕ → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} ⊆ (1...𝑁))
35	33, 34	ssfid 9329	. . 3 ⊢ (𝑁 ∈ ℕ → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} ∈ Fin)
36	13, 19, 32, 35	fsumss 15773	. 2 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} (μ‘𝑘) = Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} (μ‘𝑘))
37		fveq2 6920	. . . . 5 ⊢ (𝑥 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘} → (♯‘𝑥) = (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘}))
38	37	oveq2d 7464	. . . 4 ⊢ (𝑥 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘} → (-1↑(♯‘𝑥)) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})))
39	35, 13	ssfid 9329	. . . 4 ⊢ (𝑁 ∈ ℕ → {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} ∈ Fin)
40		eqid 2740	. . . . 5 ⊢ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} = {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}
41		eqid 2740	. . . . 5 ⊢ (𝑚 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} ↦ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑚}) = (𝑚 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} ↦ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑚})
42		oveq1 7455	. . . . . . . 8 ⊢ (𝑞 = 𝑝 → (𝑞 pCnt 𝑥) = (𝑝 pCnt 𝑥))
43	42	cbvmptv 5279	. . . . . . 7 ⊢ (𝑞 ∈ ℙ ↦ (𝑞 pCnt 𝑥)) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑥))
44		oveq2 7456	. . . . . . . 8 ⊢ (𝑥 = 𝑚 → (𝑝 pCnt 𝑥) = (𝑝 pCnt 𝑚))
45	44	mpteq2dv 5268	. . . . . . 7 ⊢ (𝑥 = 𝑚 → (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑥)) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑚)))
46	43, 45	eqtrid 2792	. . . . . 6 ⊢ (𝑥 = 𝑚 → (𝑞 ∈ ℙ ↦ (𝑞 pCnt 𝑥)) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑚)))
47	46	cbvmptv 5279	. . . . 5 ⊢ (𝑥 ∈ ℕ ↦ (𝑞 ∈ ℙ ↦ (𝑞 pCnt 𝑥))) = (𝑚 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑚)))
48	40, 41, 47	sqff1o 27243	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑚 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} ↦ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑚}):{𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}–1-1-onto→𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
49		breq2 5170	. . . . . . 7 ⊢ (𝑚 = 𝑘 → (𝑝 ∥ 𝑚 ↔ 𝑝 ∥ 𝑘))
50	49	rabbidv 3451	. . . . . 6 ⊢ (𝑚 = 𝑘 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑚} = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})
51		prmex 16724	. . . . . . 7 ⊢ ℙ ∈ V
52	51	rabex 5357	. . . . . 6 ⊢ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘} ∈ V
53	50, 41, 52	fvmpt 7029	. . . . 5 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} → ((𝑚 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} ↦ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑚})‘𝑘) = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})
54	53	adantl 481	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) → ((𝑚 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} ↦ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑚})‘𝑘) = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})
55		neg1cn 12407	. . . . 5 ⊢ -1 ∈ ℂ
56		prmdvdsfi 27168	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin)
57		elpwi 4629	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} → 𝑥 ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
58		ssfi 9240	. . . . . . 7 ⊢ (({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin ∧ 𝑥 ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑥 ∈ Fin)
59	56, 57, 58	syl2an 595	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑥 ∈ Fin)
60		hashcl 14405	. . . . . 6 ⊢ (𝑥 ∈ Fin → (♯‘𝑥) ∈ ℕ₀)
61	59, 60	syl 17	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘𝑥) ∈ ℕ₀)
62		expcl 14130	. . . . 5 ⊢ ((-1 ∈ ℂ ∧ (♯‘𝑥) ∈ ℕ₀) → (-1↑(♯‘𝑥)) ∈ ℂ)
63	55, 61, 62	sylancr 586	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (-1↑(♯‘𝑥)) ∈ ℂ)
64	38, 39, 48, 54, 63	fsumf1o 15771	. . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} (-1↑(♯‘𝑥)) = Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})))
65		fzfid 14024	. . . . 5 ⊢ (𝑁 ∈ ℕ → (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ∈ Fin)
66	56	adantr 480	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin)
67		pwfi 9385	. . . . . . 7 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin ↔ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin)
68	66, 67	sylib 218	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) → 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin)
69		ssrab2 4103	. . . . . 6 ⊢ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} ⊆ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}
70		ssfi 9240	. . . . . 6 ⊢ ((𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin ∧ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} ⊆ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} ∈ Fin)
71	68, 69, 70	sylancl 585	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) → {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} ∈ Fin)
72		simprr 772	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧})) → 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧})
73		fveqeq2 6929	. . . . . . . . . 10 ⊢ (𝑠 = 𝑥 → ((♯‘𝑠) = 𝑧 ↔ (♯‘𝑥) = 𝑧))
74	73	elrab 3708	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} ↔ (𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∧ (♯‘𝑥) = 𝑧))
75	74	simprbi 496	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} → (♯‘𝑥) = 𝑧)
76	72, 75	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧})) → (♯‘𝑥) = 𝑧)
77	76	ralrimivva 3208	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ∀𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))∀𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} (♯‘𝑥) = 𝑧)
78		invdisj 5152	. . . . . 6 ⊢ (∀𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))∀𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} (♯‘𝑥) = 𝑧 → Disj 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})){𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧})
79	77, 78	syl 17	. . . . 5 ⊢ (𝑁 ∈ ℕ → Disj 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})){𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧})
80	56	adantr 480	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧})) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin)
81	69, 72	sselid 4006	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧})) → 𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
82	81, 57	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧})) → 𝑥 ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
83	80, 82	ssfid 9329	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧})) → 𝑥 ∈ Fin)
84	83, 60	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧})) → (♯‘𝑥) ∈ ℕ₀)
85	55, 84, 62	sylancr 586	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧})) → (-1↑(♯‘𝑥)) ∈ ℂ)
86	65, 71, 79, 85	fsumiun 15869	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑥 ∈ ∪ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})){𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} (-1↑(♯‘𝑥)) = Σ𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))Σ𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} (-1↑(♯‘𝑥)))
87		iunrab 5075	. . . . . 6 ⊢ ∪ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})){𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} = {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ ∃𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(♯‘𝑠) = 𝑧}
88	56	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin)
89		elpwi 4629	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} → 𝑠 ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
90	89	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑠 ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
91		ssdomg 9060	. . . . . . . . . . . 12 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin → (𝑠 ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} → 𝑠 ≼ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))
92	88, 90, 91	sylc 65	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑠 ≼ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
93		ssfi 9240	. . . . . . . . . . . . 13 ⊢ (({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin ∧ 𝑠 ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑠 ∈ Fin)
94	56, 89, 93	syl2an 595	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑠 ∈ Fin)
95		hashdom 14428	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ Fin ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin) → ((♯‘𝑠) ≤ (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ↔ 𝑠 ≼ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))
96	94, 88, 95	syl2anc 583	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → ((♯‘𝑠) ≤ (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ↔ 𝑠 ≼ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))
97	92, 96	mpbird 257	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘𝑠) ≤ (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))
98		hashcl 14405	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ Fin → (♯‘𝑠) ∈ ℕ₀)
99	94, 98	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘𝑠) ∈ ℕ₀)
100		nn0uz 12945	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
101	99, 100	eleqtrdi 2854	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘𝑠) ∈ (ℤ_≥‘0))
102		hashcl 14405	. . . . . . . . . . . . . 14 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℕ₀)
103	56, 102	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℕ₀)
104	103	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℕ₀)
105	104	nn0zd 12665	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℤ)
106		elfz5 13576	. . . . . . . . . . 11 ⊢ (((♯‘𝑠) ∈ (ℤ_≥‘0) ∧ (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℤ) → ((♯‘𝑠) ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ↔ (♯‘𝑠) ≤ (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})))
107	101, 105, 106	syl2anc 583	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → ((♯‘𝑠) ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ↔ (♯‘𝑠) ≤ (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})))
108	97, 107	mpbird 257	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘𝑠) ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})))
109		eqidd 2741	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘𝑠) = (♯‘𝑠))
110		eqeq2 2752	. . . . . . . . . 10 ⊢ (𝑧 = (♯‘𝑠) → ((♯‘𝑠) = 𝑧 ↔ (♯‘𝑠) = (♯‘𝑠)))
111	110	rspcev 3635	. . . . . . . . 9 ⊢ (((♯‘𝑠) ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ∧ (♯‘𝑠) = (♯‘𝑠)) → ∃𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(♯‘𝑠) = 𝑧)
112	108, 109, 111	syl2anc 583	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → ∃𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(♯‘𝑠) = 𝑧)
113	112	ralrimiva 3152	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ∀𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}∃𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(♯‘𝑠) = 𝑧)
114		rabid2 3478	. . . . . . 7 ⊢ (𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} = {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ ∃𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(♯‘𝑠) = 𝑧} ↔ ∀𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}∃𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(♯‘𝑠) = 𝑧)
115	113, 114	sylibr 234	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} = {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ ∃𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(♯‘𝑠) = 𝑧})
116	87, 115	eqtr4id 2799	. . . . 5 ⊢ (𝑁 ∈ ℕ → ∪ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})){𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} = 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
117	116	sumeq1d 15748	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑥 ∈ ∪ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})){𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} (-1↑(♯‘𝑥)) = Σ𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} (-1↑(♯‘𝑥)))
118		elfznn0 13677	. . . . . . . . . 10 ⊢ (𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → 𝑧 ∈ ℕ₀)
119	118	adantl 481	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) → 𝑧 ∈ ℕ₀)
120		expcl 14130	. . . . . . . . 9 ⊢ ((-1 ∈ ℂ ∧ 𝑧 ∈ ℕ₀) → (-1↑𝑧) ∈ ℂ)
121	55, 119, 120	sylancr 586	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) → (-1↑𝑧) ∈ ℂ)
122		fsumconst 15838	. . . . . . . 8 ⊢ (({𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} ∈ Fin ∧ (-1↑𝑧) ∈ ℂ) → Σ𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} (-1↑𝑧) = ((♯‘{𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧}) · (-1↑𝑧)))
123	71, 121, 122	syl2anc 583	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) → Σ𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} (-1↑𝑧) = ((♯‘{𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧}) · (-1↑𝑧)))
124	75	adantl 481	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧}) → (♯‘𝑥) = 𝑧)
125	124	oveq2d 7464	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧}) → (-1↑(♯‘𝑥)) = (-1↑𝑧))
126	125	sumeq2dv 15750	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) → Σ𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} (-1↑(♯‘𝑥)) = Σ𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} (-1↑𝑧))
127		elfzelz 13584	. . . . . . . . 9 ⊢ (𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → 𝑧 ∈ ℤ)
128		hashbc 14502	. . . . . . . . 9 ⊢ (({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin ∧ 𝑧 ∈ ℤ) → ((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})C𝑧) = (♯‘{𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧}))
129	56, 127, 128	syl2an 595	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) → ((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})C𝑧) = (♯‘{𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧}))
130	129	oveq1d 7463	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) → (((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})C𝑧) · (-1↑𝑧)) = ((♯‘{𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧}) · (-1↑𝑧)))
131	123, 126, 130	3eqtr4d 2790	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) → Σ𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} (-1↑(♯‘𝑥)) = (((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})C𝑧) · (-1↑𝑧)))
132	131	sumeq2dv 15750	. . . . 5 ⊢ (𝑁 ∈ ℕ → Σ𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))Σ𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} (-1↑(♯‘𝑥)) = Σ𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})C𝑧) · (-1↑𝑧)))
133		1pneg1e0 12412	. . . . . . 7 ⊢ (1 + -1) = 0
134	133	oveq1i 7458	. . . . . 6 ⊢ ((1 + -1)↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))
135		binom1p 15879	. . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℕ₀) → ((1 + -1)↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = Σ𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})C𝑧) · (-1↑𝑧)))
136	55, 103, 135	sylancr 586	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ((1 + -1)↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = Σ𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})C𝑧) · (-1↑𝑧)))
137	134, 136	eqtr3id 2794	. . . . 5 ⊢ (𝑁 ∈ ℕ → (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = Σ𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})C𝑧) · (-1↑𝑧)))
138		eqeq2 2752	. . . . . 6 ⊢ (1 = if(𝑁 = 1, 1, 0) → ((0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = 1 ↔ (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = if(𝑁 = 1, 1, 0)))
139		eqeq2 2752	. . . . . 6 ⊢ (0 = if(𝑁 = 1, 1, 0) → ((0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = 0 ↔ (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = if(𝑁 = 1, 1, 0)))
140		nprmdvds1 16753	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℙ → ¬ 𝑝 ∥ 1)
141		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → 𝑁 = 1)
142	141	breq2d 5178	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (𝑝 ∥ 𝑁 ↔ 𝑝 ∥ 1))
143	142	notbid 318	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (¬ 𝑝 ∥ 𝑁 ↔ ¬ 𝑝 ∥ 1))
144	140, 143	imbitrrid 246	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (𝑝 ∈ ℙ → ¬ 𝑝 ∥ 𝑁))
145	144	ralrimiv 3151	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → ∀𝑝 ∈ ℙ ¬ 𝑝 ∥ 𝑁)
146		rabeq0 4411	. . . . . . . . . . 11 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} = ∅ ↔ ∀𝑝 ∈ ℙ ¬ 𝑝 ∥ 𝑁)
147	145, 146	sylibr 234	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} = ∅)
148	147	fveq2d 6924	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) = (♯‘∅))
149		hash0 14416	. . . . . . . . 9 ⊢ (♯‘∅) = 0
150	148, 149	eqtrdi 2796	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) = 0)
151	150	oveq2d 7464	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = (0↑0))
152		0exp0e1 14117	. . . . . . 7 ⊢ (0↑0) = 1
153	151, 152	eqtrdi 2796	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = 1)
154		df-ne 2947	. . . . . . . . . . 11 ⊢ (𝑁 ≠ 1 ↔ ¬ 𝑁 = 1)
155		eluz2b3 12987	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1))
156	155	biimpri 228	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ≠ 1) → 𝑁 ∈ (ℤ_≥‘2))
157	154, 156	sylan2br 594	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → 𝑁 ∈ (ℤ_≥‘2))
158		exprmfct 16751	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ∃𝑝 ∈ ℙ 𝑝 ∥ 𝑁)
159	157, 158	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → ∃𝑝 ∈ ℙ 𝑝 ∥ 𝑁)
160		rabn0 4412	. . . . . . . . 9 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ≠ ∅ ↔ ∃𝑝 ∈ ℙ 𝑝 ∥ 𝑁)
161	159, 160	sylibr 234	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ≠ ∅)
162	56	adantr 480	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin)
163		hashnncl 14415	. . . . . . . . 9 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin → ((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℕ ↔ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ≠ ∅))
164	162, 163	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → ((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℕ ↔ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ≠ ∅))
165	161, 164	mpbird 257	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℕ)
166	165	0expd 14189	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = 0)
167	138, 139, 153, 166	ifbothda 4586	. . . . 5 ⊢ (𝑁 ∈ ℕ → (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = if(𝑁 = 1, 1, 0))
168	132, 137, 167	3eqtr2d 2786	. . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))Σ𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} (-1↑(♯‘𝑥)) = if(𝑁 = 1, 1, 0))
169	86, 117, 168	3eqtr3d 2788	. . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} (-1↑(♯‘𝑥)) = if(𝑁 = 1, 1, 0))
170	64, 169	eqtr3d 2782	. 2 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})) = if(𝑁 = 1, 1, 0))
171	10, 36, 170	3eqtr3d 2788	1 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} (μ‘𝑘) = if(𝑁 = 1, 1, 0))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 {crab 3443 ∖ cdif 3973 ⊆ wss 3976 ∅c0 4352 ifcif 4548 𝒫 cpw 4622 ∪ ciun 5015 Disj wdisj 5133 class class class wbr 5166 ↦ cmpt 5249 ‘cfv 6573 (class class class)co 7448 ≼ cdom 9001 Fincfn 9003 ℂcc 11182 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 ≤ cle 11325 -cneg 11521 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 ℤcz 12639 ℤ_≥cuz 12903 ...cfz 13567 ↑cexp 14112 Ccbc 14351 ♯chash 14379 Σcsu 15734 ∥ cdvds 16302 ℙcprime 16718 pCnt cpc 16883 μcmu 27156

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-disj 5134 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-xnn0 12626 df-z 12640 df-uz 12904 df-q 13014 df-rp 13058 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-sum 15735 df-dvds 16303 df-gcd 16541 df-prm 16719 df-pc 16884 df-mu 27162

This theorem is referenced by: musumsum 27253 muinv 27254

W3C validator