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Theorem musum 27154

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 27156. (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 6840	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (μ‘𝑛) = (μ‘𝑘))
2	1	neeq1d 2991	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ((μ‘𝑛) ≠ 0 ↔ (μ‘𝑘) ≠ 0))
3		breq1 5088	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑛 ∥ 𝑁 ↔ 𝑘 ∥ 𝑁))
4	2, 3	anbi12d 633	. . . . . 6 ⊢ (𝑛 = 𝑘 → (((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁) ↔ ((μ‘𝑘) ≠ 0 ∧ 𝑘 ∥ 𝑁)))
5	4	elrab 3634	. . . . 5 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} ↔ (𝑘 ∈ ℕ ∧ ((μ‘𝑘) ≠ 0 ∧ 𝑘 ∥ 𝑁)))
6		muval2 27097	. . . . . 6 ⊢ ((𝑘 ∈ ℕ ∧ (μ‘𝑘) ≠ 0) → (μ‘𝑘) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})))
7	6	adantrr 718	. . . . 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 15664	. 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 4015	. . 3 ⊢ (𝑁 ∈ ℕ → {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} ⊆ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁})
14		ssrab2 4020	. . . . . 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 3919	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) → 𝑘 ∈ ℕ)
17		mucl 27104	. . . . 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 12634	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) → (μ‘𝑘) ∈ ℂ)
20		difrab 4258	. . . . . . 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 2950	. . . . . . . . . 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 4016	. . . . . . 7 ⊢ {𝑛 ∈ ℕ ∣ (𝑛 ∥ 𝑁 ∧ ¬ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁))} ⊆ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0}
26	20, 25	eqsstri 3968	. . . . . 6 ⊢ ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} ∖ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) ⊆ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0}
27	26	sseli 3917	. . . . 5 ⊢ (𝑘 ∈ ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} ∖ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) → 𝑘 ∈ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0})
28		fveqeq2 6849	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ((μ‘𝑛) = 0 ↔ (μ‘𝑘) = 0))
29	28	elrab 3634	. . . . . 6 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0} ↔ (𝑘 ∈ ℕ ∧ (μ‘𝑘) = 0))
30	29	simprbi 497	. . . . 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		dvdsfi 16759	. . 3 ⊢ (𝑁 ∈ ℕ → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} ∈ Fin)
34	13, 19, 32, 33	fsumss 15687	. 2 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} (μ‘𝑘) = Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} (μ‘𝑘))
35		fveq2 6840	. . . . 5 ⊢ (𝑥 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘} → (♯‘𝑥) = (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘}))
36	35	oveq2d 7383	. . . 4 ⊢ (𝑥 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘} → (-1↑(♯‘𝑥)) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})))
37	33, 13	ssfid 9179	. . . 4 ⊢ (𝑁 ∈ ℕ → {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} ∈ Fin)
38		eqid 2736	. . . . 5 ⊢ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} = {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}
39		eqid 2736	. . . . 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})
40		oveq1 7374	. . . . . . . 8 ⊢ (𝑞 = 𝑝 → (𝑞 pCnt 𝑥) = (𝑝 pCnt 𝑥))
41	40	cbvmptv 5189	. . . . . . 7 ⊢ (𝑞 ∈ ℙ ↦ (𝑞 pCnt 𝑥)) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑥))
42		oveq2 7375	. . . . . . . 8 ⊢ (𝑥 = 𝑚 → (𝑝 pCnt 𝑥) = (𝑝 pCnt 𝑚))
43	42	mpteq2dv 5179	. . . . . . 7 ⊢ (𝑥 = 𝑚 → (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑥)) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑚)))
44	41, 43	eqtrid 2783	. . . . . 6 ⊢ (𝑥 = 𝑚 → (𝑞 ∈ ℙ ↦ (𝑞 pCnt 𝑥)) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑚)))
45	44	cbvmptv 5189	. . . . 5 ⊢ (𝑥 ∈ ℕ ↦ (𝑞 ∈ ℙ ↦ (𝑞 pCnt 𝑥))) = (𝑚 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑚)))
46	38, 39, 45	sqff1o 27145	. . . 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})
47		breq2 5089	. . . . . . 7 ⊢ (𝑚 = 𝑘 → (𝑝 ∥ 𝑚 ↔ 𝑝 ∥ 𝑘))
48	47	rabbidv 3396	. . . . . 6 ⊢ (𝑚 = 𝑘 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑚} = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})
49		prmex 16646	. . . . . . 7 ⊢ ℙ ∈ V
50	49	rabex 5280	. . . . . 6 ⊢ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘} ∈ V
51	48, 39, 50	fvmpt 6947	. . . . 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})
52	51	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})
53		neg1cn 12144	. . . . 5 ⊢ -1 ∈ ℂ
54		prmdvdsfi 27070	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin)
55		elpwi 4548	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} → 𝑥 ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
56		ssfi 9107	. . . . . . 7 ⊢ (({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin ∧ 𝑥 ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑥 ∈ Fin)
57	54, 55, 56	syl2an 597	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑥 ∈ Fin)
58		hashcl 14318	. . . . . 6 ⊢ (𝑥 ∈ Fin → (♯‘𝑥) ∈ ℕ₀)
59	57, 58	syl 17	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘𝑥) ∈ ℕ₀)
60		expcl 14041	. . . . 5 ⊢ ((-1 ∈ ℂ ∧ (♯‘𝑥) ∈ ℕ₀) → (-1↑(♯‘𝑥)) ∈ ℂ)
61	53, 59, 60	sylancr 588	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (-1↑(♯‘𝑥)) ∈ ℂ)
62	36, 37, 46, 52, 61	fsumf1o 15685	. . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} (-1↑(♯‘𝑥)) = Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})))
63		fzfid 13935	. . . . 5 ⊢ (𝑁 ∈ ℕ → (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ∈ Fin)
64	54	adantr 480	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin)
65		pwfi 9229	. . . . . . 7 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin ↔ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin)
66	64, 65	sylib 218	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) → 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin)
67		ssrab2 4020	. . . . . 6 ⊢ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} ⊆ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}
68		ssfi 9107	. . . . . 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)
69	66, 67, 68	sylancl 587	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) → {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} ∈ Fin)
70		simprr 773	. . . . . . . 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})
71		fveqeq2 6849	. . . . . . . . . 10 ⊢ (𝑠 = 𝑥 → ((♯‘𝑠) = 𝑧 ↔ (♯‘𝑥) = 𝑧))
72	71	elrab 3634	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} ↔ (𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∧ (♯‘𝑥) = 𝑧))
73	72	simprbi 497	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} → (♯‘𝑥) = 𝑧)
74	70, 73	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧})) → (♯‘𝑥) = 𝑧)
75	74	ralrimivva 3180	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ∀𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))∀𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} (♯‘𝑥) = 𝑧)
76		invdisj 5071	. . . . . 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})
77	75, 76	syl 17	. . . . 5 ⊢ (𝑁 ∈ ℕ → Disj 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})){𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧})
78	54	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)
79	67, 70	sselid 3919	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧})) → 𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
80	79, 55	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})
81	78, 80	ssfid 9179	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧})) → 𝑥 ∈ Fin)
82	81, 58	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧})) → (♯‘𝑥) ∈ ℕ₀)
83	53, 82, 60	sylancr 588	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧})) → (-1↑(♯‘𝑥)) ∈ ℂ)
84	63, 69, 77, 83	fsumiun 15784	. . . 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)))
85		iunrab 4995	. . . . . 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}
86	54	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin)
87		elpwi 4548	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} → 𝑠 ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
88	87	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑠 ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
89		ssdomg 8947	. . . . . . . . . . . 12 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin → (𝑠 ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} → 𝑠 ≼ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))
90	86, 88, 89	sylc 65	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑠 ≼ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
91		ssfi 9107	. . . . . . . . . . . . 13 ⊢ (({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin ∧ 𝑠 ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑠 ∈ Fin)
92	54, 87, 91	syl2an 597	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑠 ∈ Fin)
93		hashdom 14341	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ Fin ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin) → ((♯‘𝑠) ≤ (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ↔ 𝑠 ≼ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))
94	92, 86, 93	syl2anc 585	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → ((♯‘𝑠) ≤ (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ↔ 𝑠 ≼ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))
95	90, 94	mpbird 257	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘𝑠) ≤ (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))
96		hashcl 14318	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ Fin → (♯‘𝑠) ∈ ℕ₀)
97	92, 96	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘𝑠) ∈ ℕ₀)
98		nn0uz 12826	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
99	97, 98	eleqtrdi 2846	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘𝑠) ∈ (ℤ_≥‘0))
100		hashcl 14318	. . . . . . . . . . . . . 14 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℕ₀)
101	54, 100	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℕ₀)
102	101	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℕ₀)
103	102	nn0zd 12549	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℤ)
104		elfz5 13470	. . . . . . . . . . 11 ⊢ (((♯‘𝑠) ∈ (ℤ_≥‘0) ∧ (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℤ) → ((♯‘𝑠) ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ↔ (♯‘𝑠) ≤ (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})))
105	99, 103, 104	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → ((♯‘𝑠) ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ↔ (♯‘𝑠) ≤ (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})))
106	95, 105	mpbird 257	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘𝑠) ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})))
107		eqidd 2737	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘𝑠) = (♯‘𝑠))
108		eqeq2 2748	. . . . . . . . . 10 ⊢ (𝑧 = (♯‘𝑠) → ((♯‘𝑠) = 𝑧 ↔ (♯‘𝑠) = (♯‘𝑠)))
109	108	rspcev 3564	. . . . . . . . 9 ⊢ (((♯‘𝑠) ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ∧ (♯‘𝑠) = (♯‘𝑠)) → ∃𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(♯‘𝑠) = 𝑧)
110	106, 107, 109	syl2anc 585	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → ∃𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(♯‘𝑠) = 𝑧)
111	110	ralrimiva 3129	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ∀𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}∃𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(♯‘𝑠) = 𝑧)
112		rabid2 3422	. . . . . . 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)
113	111, 112	sylibr 234	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} = {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ ∃𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(♯‘𝑠) = 𝑧})
114	85, 113	eqtr4id 2790	. . . . 5 ⊢ (𝑁 ∈ ℕ → ∪ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})){𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} = 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
115	114	sumeq1d 15662	. . . 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)))
116		elfznn0 13574	. . . . . . . . . 10 ⊢ (𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → 𝑧 ∈ ℕ₀)
117	116	adantl 481	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) → 𝑧 ∈ ℕ₀)
118		expcl 14041	. . . . . . . . 9 ⊢ ((-1 ∈ ℂ ∧ 𝑧 ∈ ℕ₀) → (-1↑𝑧) ∈ ℂ)
119	53, 117, 118	sylancr 588	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) → (-1↑𝑧) ∈ ℂ)
120		fsumconst 15752	. . . . . . . 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)))
121	69, 119, 120	syl2anc 585	. . . . . . 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)))
122	73	adantl 481	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧}) → (♯‘𝑥) = 𝑧)
123	122	oveq2d 7383	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧}) → (-1↑(♯‘𝑥)) = (-1↑𝑧))
124	123	sumeq2dv 15664	. . . . . . 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))
125		elfzelz 13478	. . . . . . . . 9 ⊢ (𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → 𝑧 ∈ ℤ)
126		hashbc 14415	. . . . . . . . 9 ⊢ (({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin ∧ 𝑧 ∈ ℤ) → ((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})C𝑧) = (♯‘{𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧}))
127	54, 125, 126	syl2an 597	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) → ((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})C𝑧) = (♯‘{𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧}))
128	127	oveq1d 7382	. . . . . . 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)))
129	121, 124, 128	3eqtr4d 2781	. . . . . 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)))
130	129	sumeq2dv 15664	. . . . 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)))
131		1pneg1e0 12295	. . . . . . 7 ⊢ (1 + -1) = 0
132	131	oveq1i 7377	. . . . . 6 ⊢ ((1 + -1)↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))
133		binom1p 15796	. . . . . . 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)))
134	53, 101, 133	sylancr 588	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ((1 + -1)↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = Σ𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})C𝑧) · (-1↑𝑧)))
135	132, 134	eqtr3id 2785	. . . . 5 ⊢ (𝑁 ∈ ℕ → (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = Σ𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})C𝑧) · (-1↑𝑧)))
136		eqeq2 2748	. . . . . 6 ⊢ (1 = if(𝑁 = 1, 1, 0) → ((0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = 1 ↔ (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = if(𝑁 = 1, 1, 0)))
137		eqeq2 2748	. . . . . 6 ⊢ (0 = if(𝑁 = 1, 1, 0) → ((0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = 0 ↔ (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = if(𝑁 = 1, 1, 0)))
138		nprmdvds1 16676	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℙ → ¬ 𝑝 ∥ 1)
139		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → 𝑁 = 1)
140	139	breq2d 5097	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (𝑝 ∥ 𝑁 ↔ 𝑝 ∥ 1))
141	140	notbid 318	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (¬ 𝑝 ∥ 𝑁 ↔ ¬ 𝑝 ∥ 1))
142	138, 141	imbitrrid 246	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (𝑝 ∈ ℙ → ¬ 𝑝 ∥ 𝑁))
143	142	ralrimiv 3128	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → ∀𝑝 ∈ ℙ ¬ 𝑝 ∥ 𝑁)
144		rabeq0 4328	. . . . . . . . . . 11 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} = ∅ ↔ ∀𝑝 ∈ ℙ ¬ 𝑝 ∥ 𝑁)
145	143, 144	sylibr 234	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} = ∅)
146	145	fveq2d 6844	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) = (♯‘∅))
147		hash0 14329	. . . . . . . . 9 ⊢ (♯‘∅) = 0
148	146, 147	eqtrdi 2787	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) = 0)
149	148	oveq2d 7383	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = (0↑0))
150		0exp0e1 14028	. . . . . . 7 ⊢ (0↑0) = 1
151	149, 150	eqtrdi 2787	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = 1)
152		df-ne 2933	. . . . . . . . . . 11 ⊢ (𝑁 ≠ 1 ↔ ¬ 𝑁 = 1)
153		eluz2b3 12872	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1))
154	153	biimpri 228	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ≠ 1) → 𝑁 ∈ (ℤ_≥‘2))
155	152, 154	sylan2br 596	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → 𝑁 ∈ (ℤ_≥‘2))
156		exprmfct 16674	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ∃𝑝 ∈ ℙ 𝑝 ∥ 𝑁)
157	155, 156	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → ∃𝑝 ∈ ℙ 𝑝 ∥ 𝑁)
158		rabn0 4329	. . . . . . . . 9 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ≠ ∅ ↔ ∃𝑝 ∈ ℙ 𝑝 ∥ 𝑁)
159	157, 158	sylibr 234	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ≠ ∅)
160	54	adantr 480	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin)
161		hashnncl 14328	. . . . . . . . 9 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin → ((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℕ ↔ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ≠ ∅))
162	160, 161	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → ((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℕ ↔ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ≠ ∅))
163	159, 162	mpbird 257	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℕ)
164	163	0expd 14101	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = 0)
165	136, 137, 151, 164	ifbothda 4505	. . . . 5 ⊢ (𝑁 ∈ ℕ → (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = if(𝑁 = 1, 1, 0))
166	130, 135, 165	3eqtr2d 2777	. . . 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))
167	84, 115, 166	3eqtr3d 2779	. . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} (-1↑(♯‘𝑥)) = if(𝑁 = 1, 1, 0))
168	62, 167	eqtr3d 2773	. 2 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})) = if(𝑁 = 1, 1, 0))
169	10, 34, 168	3eqtr3d 2779	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 1542 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 ∃wrex 3061 {crab 3389 ∖ cdif 3886 ⊆ wss 3889 ∅c0 4273 ifcif 4466 𝒫 cpw 4541 ∪ ciun 4933 Disj wdisj 5052 class class class wbr 5085 ↦ cmpt 5166 ‘cfv 6498 (class class class)co 7367 ≼ cdom 8891 Fincfn 8893 ℂcc 11036 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 ≤ cle 11180 -cneg 11378 ℕcn 12174 2c2 12236 ℕ₀cn0 12437 ℤcz 12524 ℤ_≥cuz 12788 ...cfz 13461 ↑cexp 14023 Ccbc 14264 ♯chash 14292 Σcsu 15648 ∥ cdvds 16221 ℙcprime 16640 pCnt cpc 16807 μcmu 27058

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-disj 5053 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-xnn0 12511 df-z 12525 df-uz 12789 df-q 12899 df-rp 12943 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-sum 15649 df-dvds 16222 df-gcd 16464 df-prm 16641 df-pc 16808 df-mu 27064

This theorem is referenced by: musumsum 27155 muinv 27156

W3C validator