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Theorem musum 27128

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 27130. (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 6822	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (μ‘𝑛) = (μ‘𝑘))
2	1	neeq1d 2987	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ((μ‘𝑛) ≠ 0 ↔ (μ‘𝑘) ≠ 0))
3		breq1 5092	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑛 ∥ 𝑁 ↔ 𝑘 ∥ 𝑁))
4	2, 3	anbi12d 632	. . . . . 6 ⊢ (𝑛 = 𝑘 → (((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁) ↔ ((μ‘𝑘) ≠ 0 ∧ 𝑘 ∥ 𝑁)))
5	4	elrab 3642	. . . . 5 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} ↔ (𝑘 ∈ ℕ ∧ ((μ‘𝑘) ≠ 0 ∧ 𝑘 ∥ 𝑁)))
6		muval2 27071	. . . . . 6 ⊢ ((𝑘 ∈ ℕ ∧ (μ‘𝑘) ≠ 0) → (μ‘𝑘) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})))
7	6	adantrr 717	. . . . 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 15609	. 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 4021	. . 3 ⊢ (𝑁 ∈ ℕ → {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} ⊆ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁})
14		ssrab2 4027	. . . . . 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 3927	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) → 𝑘 ∈ ℕ)
17		mucl 27078	. . . . 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 12578	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) → (μ‘𝑘) ∈ ℂ)
20		difrab 4265	. . . . . . 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 2946	. . . . . . . . . 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 4023	. . . . . . 7 ⊢ {𝑛 ∈ ℕ ∣ (𝑛 ∥ 𝑁 ∧ ¬ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁))} ⊆ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0}
26	20, 25	eqsstri 3976	. . . . . 6 ⊢ ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} ∖ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) ⊆ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0}
27	26	sseli 3925	. . . . 5 ⊢ (𝑘 ∈ ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} ∖ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) → 𝑘 ∈ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0})
28		fveqeq2 6831	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ((μ‘𝑛) = 0 ↔ (μ‘𝑘) = 0))
29	28	elrab 3642	. . . . . 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		dvdsfi 16700	. . 3 ⊢ (𝑁 ∈ ℕ → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} ∈ Fin)
34	13, 19, 32, 33	fsumss 15632	. 2 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} (μ‘𝑘) = Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} (μ‘𝑘))
35		fveq2 6822	. . . . 5 ⊢ (𝑥 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘} → (♯‘𝑥) = (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘}))
36	35	oveq2d 7362	. . . 4 ⊢ (𝑥 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘} → (-1↑(♯‘𝑥)) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})))
37	33, 13	ssfid 9153	. . . 4 ⊢ (𝑁 ∈ ℕ → {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} ∈ Fin)
38		eqid 2731	. . . . 5 ⊢ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} = {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}
39		eqid 2731	. . . . 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 7353	. . . . . . . 8 ⊢ (𝑞 = 𝑝 → (𝑞 pCnt 𝑥) = (𝑝 pCnt 𝑥))
41	40	cbvmptv 5193	. . . . . . 7 ⊢ (𝑞 ∈ ℙ ↦ (𝑞 pCnt 𝑥)) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑥))
42		oveq2 7354	. . . . . . . 8 ⊢ (𝑥 = 𝑚 → (𝑝 pCnt 𝑥) = (𝑝 pCnt 𝑚))
43	42	mpteq2dv 5183	. . . . . . 7 ⊢ (𝑥 = 𝑚 → (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑥)) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑚)))
44	41, 43	eqtrid 2778	. . . . . 6 ⊢ (𝑥 = 𝑚 → (𝑞 ∈ ℙ ↦ (𝑞 pCnt 𝑥)) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑚)))
45	44	cbvmptv 5193	. . . . 5 ⊢ (𝑥 ∈ ℕ ↦ (𝑞 ∈ ℙ ↦ (𝑞 pCnt 𝑥))) = (𝑚 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑚)))
46	38, 39, 45	sqff1o 27119	. . . 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 5093	. . . . . . 7 ⊢ (𝑚 = 𝑘 → (𝑝 ∥ 𝑚 ↔ 𝑝 ∥ 𝑘))
48	47	rabbidv 3402	. . . . . 6 ⊢ (𝑚 = 𝑘 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑚} = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})
49		prmex 16588	. . . . . . 7 ⊢ ℙ ∈ V
50	49	rabex 5275	. . . . . 6 ⊢ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘} ∈ V
51	48, 39, 50	fvmpt 6929	. . . . 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 12110	. . . . 5 ⊢ -1 ∈ ℂ
54		prmdvdsfi 27044	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin)
55		elpwi 4554	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} → 𝑥 ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
56		ssfi 9082	. . . . . . 7 ⊢ (({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin ∧ 𝑥 ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑥 ∈ Fin)
57	54, 55, 56	syl2an 596	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑥 ∈ Fin)
58		hashcl 14263	. . . . . 6 ⊢ (𝑥 ∈ Fin → (♯‘𝑥) ∈ ℕ₀)
59	57, 58	syl 17	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘𝑥) ∈ ℕ₀)
60		expcl 13986	. . . . 5 ⊢ ((-1 ∈ ℂ ∧ (♯‘𝑥) ∈ ℕ₀) → (-1↑(♯‘𝑥)) ∈ ℂ)
61	53, 59, 60	sylancr 587	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (-1↑(♯‘𝑥)) ∈ ℂ)
62	36, 37, 46, 52, 61	fsumf1o 15630	. . 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 13880	. . . . 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 9203	. . . . . . 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 4027	. . . . . 6 ⊢ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} ⊆ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}
68		ssfi 9082	. . . . . 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 586	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) → {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} ∈ Fin)
70		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})
71		fveqeq2 6831	. . . . . . . . . 10 ⊢ (𝑠 = 𝑥 → ((♯‘𝑠) = 𝑧 ↔ (♯‘𝑥) = 𝑧))
72	71	elrab 3642	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} ↔ (𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∧ (♯‘𝑥) = 𝑧))
73	72	simprbi 496	. . . . . . . 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 3175	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ∀𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))∀𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} (♯‘𝑥) = 𝑧)
76		invdisj 5075	. . . . . 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 3927	. . . . . . . . 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 9153	. . . . . . 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 587	. . . . 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 15728	. . . 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 4999	. . . . . 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 4554	. . . . . . . . . . . . 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 8922	. . . . . . . . . . . 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 9082	. . . . . . . . . . . . 13 ⊢ (({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin ∧ 𝑠 ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑠 ∈ Fin)
92	54, 87, 91	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑠 ∈ Fin)
93		hashdom 14286	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ Fin ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin) → ((♯‘𝑠) ≤ (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ↔ 𝑠 ≼ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))
94	92, 86, 93	syl2anc 584	. . . . . . . . . . 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 14263	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ Fin → (♯‘𝑠) ∈ ℕ₀)
97	92, 96	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘𝑠) ∈ ℕ₀)
98		nn0uz 12774	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
99	97, 98	eleqtrdi 2841	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘𝑠) ∈ (ℤ_≥‘0))
100		hashcl 14263	. . . . . . . . . . . . . 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 12494	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℤ)
104		elfz5 13416	. . . . . . . . . . 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 584	. . . . . . . . . 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 2732	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘𝑠) = (♯‘𝑠))
108		eqeq2 2743	. . . . . . . . . 10 ⊢ (𝑧 = (♯‘𝑠) → ((♯‘𝑠) = 𝑧 ↔ (♯‘𝑠) = (♯‘𝑠)))
109	108	rspcev 3572	. . . . . . . . 9 ⊢ (((♯‘𝑠) ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ∧ (♯‘𝑠) = (♯‘𝑠)) → ∃𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(♯‘𝑠) = 𝑧)
110	106, 107, 109	syl2anc 584	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → ∃𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(♯‘𝑠) = 𝑧)
111	110	ralrimiva 3124	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ∀𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}∃𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(♯‘𝑠) = 𝑧)
112		rabid2 3428	. . . . . . 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 2785	. . . . 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 15607	. . . 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 13520	. . . . . . . . . 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 13986	. . . . . . . . 9 ⊢ ((-1 ∈ ℂ ∧ 𝑧 ∈ ℕ₀) → (-1↑𝑧) ∈ ℂ)
119	53, 117, 118	sylancr 587	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) → (-1↑𝑧) ∈ ℂ)
120		fsumconst 15697	. . . . . . . 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 584	. . . . . . 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 7362	. . . . . . . 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 15609	. . . . . . 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 13424	. . . . . . . . 9 ⊢ (𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → 𝑧 ∈ ℤ)
126		hashbc 14360	. . . . . . . . 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 596	. . . . . . . 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 7361	. . . . . . 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 2776	. . . . . 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 15609	. . . . 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 12239	. . . . . . 7 ⊢ (1 + -1) = 0
132	131	oveq1i 7356	. . . . . 6 ⊢ ((1 + -1)↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))
133		binom1p 15738	. . . . . . 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 587	. . . . . 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 2780	. . . . 5 ⊢ (𝑁 ∈ ℕ → (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = Σ𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})C𝑧) · (-1↑𝑧)))
136		eqeq2 2743	. . . . . 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 2743	. . . . . 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 16617	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℙ → ¬ 𝑝 ∥ 1)
139		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → 𝑁 = 1)
140	139	breq2d 5101	. . . . . . . . . . . . . 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 3123	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → ∀𝑝 ∈ ℙ ¬ 𝑝 ∥ 𝑁)
144		rabeq0 4335	. . . . . . . . . . 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 6826	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) = (♯‘∅))
147		hash0 14274	. . . . . . . . 9 ⊢ (♯‘∅) = 0
148	146, 147	eqtrdi 2782	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) = 0)
149	148	oveq2d 7362	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = (0↑0))
150		0exp0e1 13973	. . . . . . 7 ⊢ (0↑0) = 1
151	149, 150	eqtrdi 2782	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = 1)
152		df-ne 2929	. . . . . . . . . . 11 ⊢ (𝑁 ≠ 1 ↔ ¬ 𝑁 = 1)
153		eluz2b3 12820	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1))
154	153	biimpri 228	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ≠ 1) → 𝑁 ∈ (ℤ_≥‘2))
155	152, 154	sylan2br 595	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → 𝑁 ∈ (ℤ_≥‘2))
156		exprmfct 16615	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ∃𝑝 ∈ ℙ 𝑝 ∥ 𝑁)
157	155, 156	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → ∃𝑝 ∈ ℙ 𝑝 ∥ 𝑁)
158		rabn0 4336	. . . . . . . . 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 14273	. . . . . . . . 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 14046	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = 0)
165	136, 137, 151, 164	ifbothda 4511	. . . . 5 ⊢ (𝑁 ∈ ℕ → (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = if(𝑁 = 1, 1, 0))
166	130, 135, 165	3eqtr2d 2772	. . . 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 2774	. . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} (-1↑(♯‘𝑥)) = if(𝑁 = 1, 1, 0))
168	62, 167	eqtr3d 2768	. 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 2774	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 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∃wrex 3056 {crab 3395 ∖ cdif 3894 ⊆ wss 3897 ∅c0 4280 ifcif 4472 𝒫 cpw 4547 ∪ ciun 4939 Disj wdisj 5056 class class class wbr 5089 ↦ cmpt 5170 ‘cfv 6481 (class class class)co 7346 ≼ cdom 8867 Fincfn 8869 ℂcc 11004 0cc0 11006 1c1 11007 + caddc 11009 · cmul 11011 ≤ cle 11147 -cneg 11345 ℕcn 12125 2c2 12180 ℕ₀cn0 12381 ℤcz 12468 ℤ_≥cuz 12732 ...cfz 13407 ↑cexp 13968 Ccbc 14209 ♯chash 14237 Σcsu 15593 ∥ cdvds 16163 ℙcprime 16582 pCnt cpc 16748 μcmu 27032

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-disj 5057 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-oadd 8389 df-er 8622 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-sup 9326 df-inf 9327 df-oi 9396 df-dju 9794 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-n0 12382 df-xnn0 12455 df-z 12469 df-uz 12733 df-q 12847 df-rp 12891 df-fz 13408 df-fzo 13555 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-sum 15594 df-dvds 16164 df-gcd 16406 df-prm 16583 df-pc 16749 df-mu 27038

This theorem is referenced by: musumsum 27129 muinv 27130

W3C validator