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Theorem musum 27408

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 27410. (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 6885	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (μ‘𝑛) = (μ‘𝑘))
2	1	neeq1d 3019	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ((μ‘𝑛) ≠ 0 ↔ (μ‘𝑘) ≠ 0))
3		breq1 5114	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑛 ∥ 𝑁 ↔ 𝑘 ∥ 𝑁))
4	2, 3	anbi12d 644	. . . . . 6 ⊢ (𝑛 = 𝑘 → (((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁) ↔ ((μ‘𝑘) ≠ 0 ∧ 𝑘 ∥ 𝑁)))
5	4	elrab 3652	. . . . 5 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} ↔ (𝑘 ∈ ℕ ∧ ((μ‘𝑘) ≠ 0 ∧ 𝑘 ∥ 𝑁)))
6		muval2 27351	. . . . . 6 ⊢ ((𝑘 ∈ ℕ ∧ (μ‘𝑘) ≠ 0) → (μ‘𝑘) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})))
7	6	adantrr 730	. . . . 5 ⊢ ((𝑘 ∈ ℕ ∧ ((μ‘𝑘) ≠ 0 ∧ 𝑘 ∥ 𝑁)) → (μ‘𝑘) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})))
8	5, 7	sylbi 220	. . . 4 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} → (μ‘𝑘) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})))
9	8	adantl 487	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) → (μ‘𝑘) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})))
10	9	sumeq2dv 15779	. 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 490	. . . . 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 4030	. . 3 ⊢ (𝑁 ∈ ℕ → {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} ⊆ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁})
14		ssrab2 4035	. . . . . 6 ⊢ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} ⊆ ℕ
15		simpr 490	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) → 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)})
16	14, 15	sselid 3936	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) → 𝑘 ∈ ℕ)
17		mucl 27358	. . . . 5 ⊢ (𝑘 ∈ ℕ → (μ‘𝑘) ∈ ℤ)
18	16, 17	syl 18	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) → (μ‘𝑘) ∈ ℤ)
19	18	zcnd 12719	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) → (μ‘𝑘) ∈ ℂ)
20		difrab 4271	. . . . . . 7 ⊢ ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} ∖ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) = {𝑛 ∈ ℕ ∣ (𝑛 ∥ 𝑁 ∧ ¬ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁))}
21		pm3.21 477	. . . . . . . . . . 11 ⊢ (𝑛 ∥ 𝑁 → ((μ‘𝑛) ≠ 0 → ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)))
22	21	necon1bd 2978	. . . . . . . . . 10 ⊢ (𝑛 ∥ 𝑁 → (¬ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁) → (μ‘𝑛) = 0))
23	22	imp 412	. . . . . . . . 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 4031	. . . . . . 7 ⊢ {𝑛 ∈ ℕ ∣ (𝑛 ∥ 𝑁 ∧ ¬ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁))} ⊆ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0}
26	20, 25	eqsstri 3984	. . . . . 6 ⊢ ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} ∖ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) ⊆ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0}
27	26	sseli 3934	. . . . 5 ⊢ (𝑘 ∈ ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} ∖ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) → 𝑘 ∈ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0})
28		fveqeq2 6894	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ((μ‘𝑛) = 0 ↔ (μ‘𝑘) = 0))
29	28	elrab 3652	. . . . . 6 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0} ↔ (𝑘 ∈ ℕ ∧ (μ‘𝑘) = 0))
30	29	simprbi 503	. . . . 5 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0} → (μ‘𝑘) = 0)
31	27, 30	syl 18	. . . 4 ⊢ (𝑘 ∈ ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} ∖ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}) → (μ‘𝑘) = 0)
32	31	adantl 487	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} ∖ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)})) → (μ‘𝑘) = 0)
33		dvdsfi 16872	. . 3 ⊢ (𝑁 ∈ ℕ → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} ∈ Fin)
34	13, 19, 32, 33	fsumss 15801	. 2 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} (μ‘𝑘) = Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} (μ‘𝑘))
35		fveq2 6885	. . . . 5 ⊢ (𝑥 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘} → (♯‘𝑥) = (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘}))
36	35	oveq2d 7435	. . . 4 ⊢ (𝑥 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘} → (-1↑(♯‘𝑥)) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})))
37	33, 13	ssfid 9236	. . . 4 ⊢ (𝑁 ∈ ℕ → {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} ∈ Fin)
38		eqid 2765	. . . . 5 ⊢ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)} = {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛 ∥ 𝑁)}
39		eqid 2765	. . . . 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 7426	. . . . . . . 8 ⊢ (𝑞 = 𝑝 → (𝑞 pCnt 𝑥) = (𝑝 pCnt 𝑥))
41	40	cbvmptv 5217	. . . . . . 7 ⊢ (𝑞 ∈ ℙ ↦ (𝑞 pCnt 𝑥)) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑥))
42		oveq2 7427	. . . . . . . 8 ⊢ (𝑥 = 𝑚 → (𝑝 pCnt 𝑥) = (𝑝 pCnt 𝑚))
43	42	mpteq2dv 5207	. . . . . . 7 ⊢ (𝑥 = 𝑚 → (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑥)) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑚)))
44	41, 43	eqtrid 2812	. . . . . 6 ⊢ (𝑥 = 𝑚 → (𝑞 ∈ ℙ ↦ (𝑞 pCnt 𝑥)) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑚)))
45	44	cbvmptv 5217	. . . . 5 ⊢ (𝑥 ∈ ℕ ↦ (𝑞 ∈ ℙ ↦ (𝑞 pCnt 𝑥))) = (𝑚 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑚)))
46	38, 39, 45	sqff1o 27399	. . . 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 5115	. . . . . . 7 ⊢ (𝑚 = 𝑘 → (𝑝 ∥ 𝑚 ↔ 𝑝 ∥ 𝑘))
48	47	rabbidv 3425	. . . . . 6 ⊢ (𝑚 = 𝑘 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑚} = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘})
49		prmex 16759	. . . . . . 7 ⊢ ℙ ∈ V
50	49	rabex 5311	. . . . . 6 ⊢ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘} ∈ V
51	48, 39, 50	fvmpt 6993	. . . . 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 487	. . . 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 12220	. . . . 5 ⊢ -1 ∈ ℂ
54		prmdvdsfi 27324	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin)
55		elpwi 4571	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} → 𝑥 ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
56		ssfi 9164	. . . . . . 7 ⊢ (({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin ∧ 𝑥 ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑥 ∈ Fin)
57	54, 55, 56	syl2an 608	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑥 ∈ Fin)
58		hashcl 14412	. . . . . 6 ⊢ (𝑥 ∈ Fin → (♯‘𝑥) ∈ ℕ₀)
59	57, 58	syl 18	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘𝑥) ∈ ℕ₀)
60		expcl 14135	. . . . 5 ⊢ ((-1 ∈ ℂ ∧ (♯‘𝑥) ∈ ℕ₀) → (-1↑(♯‘𝑥)) ∈ ℂ)
61	53, 59, 60	sylancr 599	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (-1↑(♯‘𝑥)) ∈ ℂ)
62	36, 37, 46, 52, 61	fsumf1o 15799	. . 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 14029	. . . . 5 ⊢ (𝑁 ∈ ℕ → (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ∈ Fin)
64	54	adantr 486	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin)
65		pwfi 9285	. . . . . . 7 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin ↔ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin)
66	64, 65	sylib 221	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) → 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin)
67		ssrab2 4035	. . . . . 6 ⊢ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} ⊆ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}
68		ssfi 9164	. . . . . 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 598	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) → {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} ∈ Fin)
70		simprr 785	. . . . . . . 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 6894	. . . . . . . . . 10 ⊢ (𝑠 = 𝑥 → ((♯‘𝑠) = 𝑧 ↔ (♯‘𝑥) = 𝑧))
72	71	elrab 3652	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} ↔ (𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∧ (♯‘𝑥) = 𝑧))
73	72	simprbi 503	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} → (♯‘𝑥) = 𝑧)
74	70, 73	syl 18	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧})) → (♯‘𝑥) = 𝑧)
75	74	ralrimivva 3210	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ∀𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))∀𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧} (♯‘𝑥) = 𝑧)
76		invdisj 5097	. . . . . 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 18	. . . . 5 ⊢ (𝑁 ∈ ℕ → Disj 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})){𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧})
78	54	adantr 486	. . . . . . . 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 3936	. . . . . . . . 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 18	. . . . . . . 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 9236	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧})) → 𝑥 ∈ Fin)
82	81, 58	syl 18	. . . . . 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 599	. . . . 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 15898	. . . 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 5019	. . . . . 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 486	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin)
87		elpwi 4571	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} → 𝑠 ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
88	87	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑠 ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
89		ssdomg 9003	. . . . . . . . . . . 12 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin → (𝑠 ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} → 𝑠 ≼ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))
90	86, 88, 89	sylc 66	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑠 ≼ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})
91		ssfi 9164	. . . . . . . . . . . . 13 ⊢ (({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin ∧ 𝑠 ⊆ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑠 ∈ Fin)
92	54, 87, 91	syl2an 608	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → 𝑠 ∈ Fin)
93		hashdom 14435	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ Fin ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin) → ((♯‘𝑠) ≤ (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ↔ 𝑠 ≼ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))
94	92, 86, 93	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → ((♯‘𝑠) ≤ (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ↔ 𝑠 ≼ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))
95	90, 94	mpbird 260	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘𝑠) ≤ (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))
96		hashcl 14412	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ Fin → (♯‘𝑠) ∈ ℕ₀)
97	92, 96	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘𝑠) ∈ ℕ₀)
98		nn0uz 12918	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
99	97, 98	eleqtrdi 2875	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘𝑠) ∈ (ℤ_≥‘0))
100		hashcl 14412	. . . . . . . . . . . . . 14 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℕ₀)
101	54, 100	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℕ₀)
102	101	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℕ₀)
103	102	nn0zd 12633	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℤ)
104		elfz5 13562	. . . . . . . . . . 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 596	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → ((♯‘𝑠) ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ↔ (♯‘𝑠) ≤ (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})))
106	95, 105	mpbird 260	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘𝑠) ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})))
107		eqidd 2766	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → (♯‘𝑠) = (♯‘𝑠))
108		eqeq2 2777	. . . . . . . . . 10 ⊢ (𝑧 = (♯‘𝑠) → ((♯‘𝑠) = 𝑧 ↔ (♯‘𝑠) = (♯‘𝑠)))
109	108	rspcev 3583	. . . . . . . . 9 ⊢ (((♯‘𝑠) ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) ∧ (♯‘𝑠) = (♯‘𝑠)) → ∃𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(♯‘𝑠) = 𝑧)
110	106, 107, 109	syl2anc 596	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) → ∃𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(♯‘𝑠) = 𝑧)
111	110	ralrimiva 3159	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ∀𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}∃𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(♯‘𝑠) = 𝑧)
112		rabid2 3451	. . . . . . 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 237	. . . . . 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 2819	. . . . 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 15777	. . . 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 13667	. . . . . . . . . 10 ⊢ (𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → 𝑧 ∈ ℕ₀)
117	116	adantl 487	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) → 𝑧 ∈ ℕ₀)
118		expcl 14135	. . . . . . . . 9 ⊢ ((-1 ∈ ℂ ∧ 𝑧 ∈ ℕ₀) → (-1↑𝑧) ∈ ℂ)
119	53, 117, 118	sylancr 599	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) → (-1↑𝑧) ∈ ℂ)
120		fsumconst 15866	. . . . . . . 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 596	. . . . . . 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 487	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∣ (♯‘𝑠) = 𝑧}) → (♯‘𝑥) = 𝑧)
123	122	oveq2d 7435	. . . . . . . 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 15779	. . . . . . 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 13570	. . . . . . . . 9 ⊢ (𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) → 𝑧 ∈ ℤ)
126		hashbc 14510	. . . . . . . . 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 608	. . . . . . . 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 7434	. . . . . . 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 2810	. . . . . 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 15779	. . . . 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 12375	. . . . . . 7 ⊢ (1 + -1) = 0
132	131	oveq1i 7429	. . . . . 6 ⊢ ((1 + -1)↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))
133		binom1p 15910	. . . . . . 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 599	. . . . . 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 2814	. . . . 5 ⊢ (𝑁 ∈ ℕ → (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = Σ𝑧 ∈ (0...(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}))(((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})C𝑧) · (-1↑𝑧)))
136		eqeq2 2777	. . . . . 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 2777	. . . . . 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 16789	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℙ → ¬ 𝑝 ∥ 1)
139		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → 𝑁 = 1)
140	139	breq2d 5123	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (𝑝 ∥ 𝑁 ↔ 𝑝 ∥ 1))
141	140	notbid 321	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (¬ 𝑝 ∥ 𝑁 ↔ ¬ 𝑝 ∥ 1))
142	138, 141	imbitrrid 249	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (𝑝 ∈ ℙ → ¬ 𝑝 ∥ 𝑁))
143	142	ralrimiv 3158	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → ∀𝑝 ∈ ℙ ¬ 𝑝 ∥ 𝑁)
144		rabeq0 4345	. . . . . . . . . . 11 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} = ∅ ↔ ∀𝑝 ∈ ℙ ¬ 𝑝 ∥ 𝑁)
145	143, 144	sylibr 237	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} = ∅)
146	145	fveq2d 6889	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) = (♯‘∅))
147		hash0 14423	. . . . . . . . 9 ⊢ (♯‘∅) = 0
148	146, 147	eqtrdi 2816	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) = 0)
149	148	oveq2d 7435	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = (0↑0))
150		0exp0e1 14122	. . . . . . 7 ⊢ (0↑0) = 1
151	149, 150	eqtrdi 2816	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = 1)
152		df-ne 2961	. . . . . . . . . . 11 ⊢ (𝑁 ≠ 1 ↔ ¬ 𝑁 = 1)
153		eluz2b3 12964	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1))
154	153	biimpri 231	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ≠ 1) → 𝑁 ∈ (ℤ_≥‘2))
155	152, 154	sylan2br 607	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → 𝑁 ∈ (ℤ_≥‘2))
156		exprmfct 16787	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ∃𝑝 ∈ ℙ 𝑝 ∥ 𝑁)
157	155, 156	syl 18	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → ∃𝑝 ∈ ℙ 𝑝 ∥ 𝑁)
158		rabn0 4346	. . . . . . . . 9 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ≠ ∅ ↔ ∃𝑝 ∈ ℙ 𝑝 ∥ 𝑁)
159	157, 158	sylibr 237	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ≠ ∅)
160	54	adantr 486	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin)
161		hashnncl 14422	. . . . . . . . 9 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ∈ Fin → ((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℕ ↔ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ≠ ∅))
162	160, 161	syl 18	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → ((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℕ ↔ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} ≠ ∅))
163	159, 162	mpbird 260	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁}) ∈ ℕ)
164	163	0expd 14195	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = 0)
165	136, 137, 151, 164	ifbothda 4528	. . . . 5 ⊢ (𝑁 ∈ ℕ → (0↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁})) = if(𝑁 = 1, 1, 0))
166	130, 135, 165	3eqtr2d 2806	. . . 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 2808	. . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁} (-1↑(♯‘𝑥)) = if(𝑁 = 1, 1, 0))
168	62, 167	eqtr3d 2802	. 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 2808	1 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} (μ‘𝑘) = if(𝑁 = 1, 1, 0))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∀wral 3081 ∃wrex 3091 {crab 3418 ∖ cdif 3903 ⊆ wss 3906 ∅c0 4286 ifcif 4489 𝒫 cpw 4564 ∪ ciun 4958 Disj wdisj 5078 class class class wbr 5111 ↦ cmpt 5194 ‘cfv 6540 (class class class)co 7419 ≼ cdom 8947 Fincfn 8949 ℂcc 11115 0cc0 11117 1c1 11118 + caddc 11120 · cmul 11122 ≤ cle 11261 -cneg 11459 ℕcn 12250 2c2 12312 ℕ₀cn0 12521 ℤcz 12608 ℤ_≥cuz 12880 ...cfz 13553 ↑cexp 14117 Ccbc 14358 ♯chash 14386 Σcsu 15763 ∥ cdvds 16334 ℙcprime 16753 pCnt cpc 16920 μcmu 27312

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194 ax-pre-sup 11195

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-disj 5079 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-oadd 8463 df-er 8700 df-map 8832 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-sup 9409 df-inf 9410 df-oi 9479 df-dju 9903 df-card 9941 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-div 11889 df-nn 12251 df-2 12320 df-3 12321 df-n0 12522 df-xnn0 12595 df-z 12609 df-uz 12881 df-q 12991 df-rp 13035 df-fz 13554 df-fzo 13702 df-fl 13845 df-mod 13923 df-seq 14058 df-exp 14118 df-fac 14330 df-bc 14359 df-hash 14387 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-clim 15565 df-sum 15764 df-dvds 16335 df-gcd 16577 df-prm 16754 df-pc 16921 df-mu 27318

This theorem is used by: musumsum 27409 muinv 27410

W3C validator