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Theorem musum 24717
 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 24719. (Contributed by Mario Carneiro, 2-Jul-2015.)
Assertion
Ref Expression
musum (𝑁 ∈ ℕ → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛𝑁} (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.
StepHypRef Expression
1 fveq2 6103 . . . . . . . 8 (𝑛 = 𝑘 → (μ‘𝑛) = (μ‘𝑘))
21neeq1d 2841 . . . . . . 7 (𝑛 = 𝑘 → ((μ‘𝑛) ≠ 0 ↔ (μ‘𝑘) ≠ 0))
3 breq1 4586 . . . . . . 7 (𝑛 = 𝑘 → (𝑛𝑁𝑘𝑁))
42, 3anbi12d 743 . . . . . 6 (𝑛 = 𝑘 → (((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁) ↔ ((μ‘𝑘) ≠ 0 ∧ 𝑘𝑁)))
54elrab 3331 . . . . 5 (𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} ↔ (𝑘 ∈ ℕ ∧ ((μ‘𝑘) ≠ 0 ∧ 𝑘𝑁)))
6 muval2 24660 . . . . . 6 ((𝑘 ∈ ℕ ∧ (μ‘𝑘) ≠ 0) → (μ‘𝑘) = (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑘})))
76adantrr 749 . . . . 5 ((𝑘 ∈ ℕ ∧ ((μ‘𝑘) ≠ 0 ∧ 𝑘𝑁)) → (μ‘𝑘) = (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑘})))
85, 7sylbi 206 . . . 4 (𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} → (μ‘𝑘) = (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑘})))
98adantl 481 . . 3 ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)}) → (μ‘𝑘) = (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑘})))
109sumeq2dv 14281 . 2 (𝑁 ∈ ℕ → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} (μ‘𝑘) = Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑘})))
11 simpr 476 . . . . 5 (((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁) → 𝑛𝑁)
1211a1i 11 . . . 4 ((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁) → 𝑛𝑁))
1312ss2rabdv 3646 . . 3 (𝑁 ∈ ℕ → {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} ⊆ {𝑛 ∈ ℕ ∣ 𝑛𝑁})
14 ssrab2 3650 . . . . . 6 {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} ⊆ ℕ
15 simpr 476 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)}) → 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)})
1614, 15sseldi 3566 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)}) → 𝑘 ∈ ℕ)
17 mucl 24667 . . . . 5 (𝑘 ∈ ℕ → (μ‘𝑘) ∈ ℤ)
1816, 17syl 17 . . . 4 ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)}) → (μ‘𝑘) ∈ ℤ)
1918zcnd 11359 . . 3 ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)}) → (μ‘𝑘) ∈ ℂ)
20 difrab 3860 . . . . . . 7 ({𝑛 ∈ ℕ ∣ 𝑛𝑁} ∖ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)}) = {𝑛 ∈ ℕ ∣ (𝑛𝑁 ∧ ¬ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁))}
21 pm3.21 463 . . . . . . . . . . 11 (𝑛𝑁 → ((μ‘𝑛) ≠ 0 → ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)))
2221necon1bd 2800 . . . . . . . . . 10 (𝑛𝑁 → (¬ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁) → (μ‘𝑛) = 0))
2322imp 444 . . . . . . . . 9 ((𝑛𝑁 ∧ ¬ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)) → (μ‘𝑛) = 0)
2423a1i 11 . . . . . . . 8 (𝑛 ∈ ℕ → ((𝑛𝑁 ∧ ¬ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)) → (μ‘𝑛) = 0))
2524ss2rabi 3647 . . . . . . 7 {𝑛 ∈ ℕ ∣ (𝑛𝑁 ∧ ¬ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁))} ⊆ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0}
2620, 25eqsstri 3598 . . . . . 6 ({𝑛 ∈ ℕ ∣ 𝑛𝑁} ∖ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)}) ⊆ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0}
2726sseli 3564 . . . . 5 (𝑘 ∈ ({𝑛 ∈ ℕ ∣ 𝑛𝑁} ∖ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)}) → 𝑘 ∈ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0})
281eqeq1d 2612 . . . . . . 7 (𝑛 = 𝑘 → ((μ‘𝑛) = 0 ↔ (μ‘𝑘) = 0))
2928elrab 3331 . . . . . 6 (𝑘 ∈ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0} ↔ (𝑘 ∈ ℕ ∧ (μ‘𝑘) = 0))
3029simprbi 479 . . . . 5 (𝑘 ∈ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0} → (μ‘𝑘) = 0)
3127, 30syl 17 . . . 4 (𝑘 ∈ ({𝑛 ∈ ℕ ∣ 𝑛𝑁} ∖ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)}) → (μ‘𝑘) = 0)
3231adantl 481 . . 3 ((𝑁 ∈ ℕ ∧ 𝑘 ∈ ({𝑛 ∈ ℕ ∣ 𝑛𝑁} ∖ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)})) → (μ‘𝑘) = 0)
33 fzfid 12634 . . . 4 (𝑁 ∈ ℕ → (1...𝑁) ∈ Fin)
34 dvdsssfz1 14878 . . . 4 (𝑁 ∈ ℕ → {𝑛 ∈ ℕ ∣ 𝑛𝑁} ⊆ (1...𝑁))
35 ssfi 8065 . . . 4 (((1...𝑁) ∈ Fin ∧ {𝑛 ∈ ℕ ∣ 𝑛𝑁} ⊆ (1...𝑁)) → {𝑛 ∈ ℕ ∣ 𝑛𝑁} ∈ Fin)
3633, 34, 35syl2anc 691 . . 3 (𝑁 ∈ ℕ → {𝑛 ∈ ℕ ∣ 𝑛𝑁} ∈ Fin)
3713, 19, 32, 36fsumss 14303 . 2 (𝑁 ∈ ℕ → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} (μ‘𝑘) = Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛𝑁} (μ‘𝑘))
38 fveq2 6103 . . . . 5 (𝑥 = {𝑝 ∈ ℙ ∣ 𝑝𝑘} → (#‘𝑥) = (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑘}))
3938oveq2d 6565 . . . 4 (𝑥 = {𝑝 ∈ ℙ ∣ 𝑝𝑘} → (-1↑(#‘𝑥)) = (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑘})))
40 ssfi 8065 . . . . 5 (({𝑛 ∈ ℕ ∣ 𝑛𝑁} ∈ Fin ∧ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} ⊆ {𝑛 ∈ ℕ ∣ 𝑛𝑁}) → {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} ∈ Fin)
4136, 13, 40syl2anc 691 . . . 4 (𝑁 ∈ ℕ → {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} ∈ Fin)
42 eqid 2610 . . . . 5 {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} = {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)}
43 eqid 2610 . . . . 5 (𝑚 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} ↦ {𝑝 ∈ ℙ ∣ 𝑝𝑚}) = (𝑚 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} ↦ {𝑝 ∈ ℙ ∣ 𝑝𝑚})
44 oveq1 6556 . . . . . . . 8 (𝑞 = 𝑝 → (𝑞 pCnt 𝑥) = (𝑝 pCnt 𝑥))
4544cbvmptv 4678 . . . . . . 7 (𝑞 ∈ ℙ ↦ (𝑞 pCnt 𝑥)) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑥))
46 oveq2 6557 . . . . . . . 8 (𝑥 = 𝑚 → (𝑝 pCnt 𝑥) = (𝑝 pCnt 𝑚))
4746mpteq2dv 4673 . . . . . . 7 (𝑥 = 𝑚 → (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑥)) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑚)))
4845, 47syl5eq 2656 . . . . . 6 (𝑥 = 𝑚 → (𝑞 ∈ ℙ ↦ (𝑞 pCnt 𝑥)) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑚)))
4948cbvmptv 4678 . . . . 5 (𝑥 ∈ ℕ ↦ (𝑞 ∈ ℙ ↦ (𝑞 pCnt 𝑥))) = (𝑚 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑚)))
5042, 43, 49sqff1o 24708 . . . 4 (𝑁 ∈ ℕ → (𝑚 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} ↦ {𝑝 ∈ ℙ ∣ 𝑝𝑚}):{𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)}–1-1-onto→𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁})
51 breq2 4587 . . . . . . 7 (𝑚 = 𝑘 → (𝑝𝑚𝑝𝑘))
5251rabbidv 3164 . . . . . 6 (𝑚 = 𝑘 → {𝑝 ∈ ℙ ∣ 𝑝𝑚} = {𝑝 ∈ ℙ ∣ 𝑝𝑘})
53 zex 11263 . . . . . . . 8 ℤ ∈ V
54 prmz 15227 . . . . . . . . 9 (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
5554ssriv 3572 . . . . . . . 8 ℙ ⊆ ℤ
5653, 55ssexi 4731 . . . . . . 7 ℙ ∈ V
5756rabex 4740 . . . . . 6 {𝑝 ∈ ℙ ∣ 𝑝𝑘} ∈ V
5852, 43, 57fvmpt 6191 . . . . 5 (𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} → ((𝑚 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} ↦ {𝑝 ∈ ℙ ∣ 𝑝𝑚})‘𝑘) = {𝑝 ∈ ℙ ∣ 𝑝𝑘})
5958adantl 481 . . . 4 ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)}) → ((𝑚 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} ↦ {𝑝 ∈ ℙ ∣ 𝑝𝑚})‘𝑘) = {𝑝 ∈ ℙ ∣ 𝑝𝑘})
60 neg1cn 11001 . . . . 5 -1 ∈ ℂ
61 prmdvdsfi 24633 . . . . . . 7 (𝑁 ∈ ℕ → {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∈ Fin)
62 elpwi 4117 . . . . . . 7 (𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} → 𝑥 ⊆ {𝑝 ∈ ℙ ∣ 𝑝𝑁})
63 ssfi 8065 . . . . . . 7 (({𝑝 ∈ ℙ ∣ 𝑝𝑁} ∈ Fin ∧ 𝑥 ⊆ {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → 𝑥 ∈ Fin)
6461, 62, 63syl2an 493 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → 𝑥 ∈ Fin)
65 hashcl 13009 . . . . . 6 (𝑥 ∈ Fin → (#‘𝑥) ∈ ℕ0)
6664, 65syl 17 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → (#‘𝑥) ∈ ℕ0)
67 expcl 12740 . . . . 5 ((-1 ∈ ℂ ∧ (#‘𝑥) ∈ ℕ0) → (-1↑(#‘𝑥)) ∈ ℂ)
6860, 66, 67sylancr 694 . . . 4 ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → (-1↑(#‘𝑥)) ∈ ℂ)
6939, 41, 50, 59, 68fsumf1o 14301 . . 3 (𝑁 ∈ ℕ → Σ𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} (-1↑(#‘𝑥)) = Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑘})))
70 fzfid 12634 . . . . 5 (𝑁 ∈ ℕ → (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) ∈ Fin)
7161adantr 480 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))) → {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∈ Fin)
72 pwfi 8144 . . . . . . 7 ({𝑝 ∈ ℙ ∣ 𝑝𝑁} ∈ Fin ↔ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∈ Fin)
7371, 72sylib 207 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))) → 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∈ Fin)
74 ssrab2 3650 . . . . . 6 {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} ⊆ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}
75 ssfi 8065 . . . . . 6 ((𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∈ Fin ∧ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} ⊆ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} ∈ Fin)
7673, 74, 75sylancl 693 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))) → {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} ∈ Fin)
77 simprr 792 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ (𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧})) → 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧})
78 fveq2 6103 . . . . . . . . . . 11 (𝑠 = 𝑥 → (#‘𝑠) = (#‘𝑥))
7978eqeq1d 2612 . . . . . . . . . 10 (𝑠 = 𝑥 → ((#‘𝑠) = 𝑧 ↔ (#‘𝑥) = 𝑧))
8079elrab 3331 . . . . . . . . 9 (𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} ↔ (𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∧ (#‘𝑥) = 𝑧))
8180simprbi 479 . . . . . . . 8 (𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} → (#‘𝑥) = 𝑧)
8277, 81syl 17 . . . . . . 7 ((𝑁 ∈ ℕ ∧ (𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧})) → (#‘𝑥) = 𝑧)
8382ralrimivva 2954 . . . . . 6 (𝑁 ∈ ℕ → ∀𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))∀𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} (#‘𝑥) = 𝑧)
84 invdisj 4571 . . . . . 6 (∀𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))∀𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} (#‘𝑥) = 𝑧Disj 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})){𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧})
8583, 84syl 17 . . . . 5 (𝑁 ∈ ℕ → Disj 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})){𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧})
8661adantr 480 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ (𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧})) → {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∈ Fin)
8774, 77sseldi 3566 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ (𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧})) → 𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁})
8887, 62syl 17 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ (𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧})) → 𝑥 ⊆ {𝑝 ∈ ℙ ∣ 𝑝𝑁})
8986, 88, 63syl2anc 691 . . . . . . 7 ((𝑁 ∈ ℕ ∧ (𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧})) → 𝑥 ∈ Fin)
9089, 65syl 17 . . . . . 6 ((𝑁 ∈ ℕ ∧ (𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧})) → (#‘𝑥) ∈ ℕ0)
9160, 90, 67sylancr 694 . . . . 5 ((𝑁 ∈ ℕ ∧ (𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧})) → (-1↑(#‘𝑥)) ∈ ℂ)
9270, 76, 85, 91fsumiun 14394 . . . 4 (𝑁 ∈ ℕ → Σ𝑥 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})){𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} (-1↑(#‘𝑥)) = Σ𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))Σ𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} (-1↑(#‘𝑥)))
9361adantr 480 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∈ Fin)
94 elpwi 4117 . . . . . . . . . . . . 13 (𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} → 𝑠 ⊆ {𝑝 ∈ ℙ ∣ 𝑝𝑁})
9594adantl 481 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → 𝑠 ⊆ {𝑝 ∈ ℙ ∣ 𝑝𝑁})
96 ssdomg 7887 . . . . . . . . . . . 12 ({𝑝 ∈ ℙ ∣ 𝑝𝑁} ∈ Fin → (𝑠 ⊆ {𝑝 ∈ ℙ ∣ 𝑝𝑁} → 𝑠 ≼ {𝑝 ∈ ℙ ∣ 𝑝𝑁}))
9793, 95, 96sylc 63 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → 𝑠 ≼ {𝑝 ∈ ℙ ∣ 𝑝𝑁})
98 ssfi 8065 . . . . . . . . . . . . 13 (({𝑝 ∈ ℙ ∣ 𝑝𝑁} ∈ Fin ∧ 𝑠 ⊆ {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → 𝑠 ∈ Fin)
9961, 94, 98syl2an 493 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → 𝑠 ∈ Fin)
100 hashdom 13029 . . . . . . . . . . . 12 ((𝑠 ∈ Fin ∧ {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∈ Fin) → ((#‘𝑠) ≤ (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}) ↔ 𝑠 ≼ {𝑝 ∈ ℙ ∣ 𝑝𝑁}))
10199, 93, 100syl2anc 691 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → ((#‘𝑠) ≤ (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}) ↔ 𝑠 ≼ {𝑝 ∈ ℙ ∣ 𝑝𝑁}))
10297, 101mpbird 246 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → (#‘𝑠) ≤ (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))
103 hashcl 13009 . . . . . . . . . . . . 13 (𝑠 ∈ Fin → (#‘𝑠) ∈ ℕ0)
10499, 103syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → (#‘𝑠) ∈ ℕ0)
105 nn0uz 11598 . . . . . . . . . . . 12 0 = (ℤ‘0)
106104, 105syl6eleq 2698 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → (#‘𝑠) ∈ (ℤ‘0))
107 hashcl 13009 . . . . . . . . . . . . . 14 ({𝑝 ∈ ℙ ∣ 𝑝𝑁} ∈ Fin → (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}) ∈ ℕ0)
10861, 107syl 17 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ → (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}) ∈ ℕ0)
109108adantr 480 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}) ∈ ℕ0)
110109nn0zd 11356 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}) ∈ ℤ)
111 elfz5 12205 . . . . . . . . . . 11 (((#‘𝑠) ∈ (ℤ‘0) ∧ (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}) ∈ ℤ) → ((#‘𝑠) ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) ↔ (#‘𝑠) ≤ (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})))
112106, 110, 111syl2anc 691 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → ((#‘𝑠) ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) ↔ (#‘𝑠) ≤ (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})))
113102, 112mpbird 246 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → (#‘𝑠) ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})))
114 eqidd 2611 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → (#‘𝑠) = (#‘𝑠))
115 eqeq2 2621 . . . . . . . . . 10 (𝑧 = (#‘𝑠) → ((#‘𝑠) = 𝑧 ↔ (#‘𝑠) = (#‘𝑠)))
116115rspcev 3282 . . . . . . . . 9 (((#‘𝑠) ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) ∧ (#‘𝑠) = (#‘𝑠)) → ∃𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))(#‘𝑠) = 𝑧)
117113, 114, 116syl2anc 691 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → ∃𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))(#‘𝑠) = 𝑧)
118117ralrimiva 2949 . . . . . . 7 (𝑁 ∈ ℕ → ∀𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}∃𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))(#‘𝑠) = 𝑧)
119 rabid2 3096 . . . . . . 7 (𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} = {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ ∃𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))(#‘𝑠) = 𝑧} ↔ ∀𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}∃𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))(#‘𝑠) = 𝑧)
120118, 119sylibr 223 . . . . . 6 (𝑁 ∈ ℕ → 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} = {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ ∃𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))(#‘𝑠) = 𝑧})
121 iunrab 4503 . . . . . 6 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})){𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} = {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ ∃𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))(#‘𝑠) = 𝑧}
122120, 121syl6reqr 2663 . . . . 5 (𝑁 ∈ ℕ → 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})){𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} = 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁})
123122sumeq1d 14279 . . . 4 (𝑁 ∈ ℕ → Σ𝑥 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})){𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} (-1↑(#‘𝑥)) = Σ𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} (-1↑(#‘𝑥)))
124 elfznn0 12302 . . . . . . . . . 10 (𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) → 𝑧 ∈ ℕ0)
125124adantl 481 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))) → 𝑧 ∈ ℕ0)
126 expcl 12740 . . . . . . . . 9 ((-1 ∈ ℂ ∧ 𝑧 ∈ ℕ0) → (-1↑𝑧) ∈ ℂ)
12760, 125, 126sylancr 694 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))) → (-1↑𝑧) ∈ ℂ)
128 fsumconst 14364 . . . . . . . 8 (({𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} ∈ Fin ∧ (-1↑𝑧) ∈ ℂ) → Σ𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} (-1↑𝑧) = ((#‘{𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧}) · (-1↑𝑧)))
12976, 127, 128syl2anc 691 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))) → Σ𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} (-1↑𝑧) = ((#‘{𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧}) · (-1↑𝑧)))
13081adantl 481 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧}) → (#‘𝑥) = 𝑧)
131130oveq2d 6565 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧}) → (-1↑(#‘𝑥)) = (-1↑𝑧))
132131sumeq2dv 14281 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))) → Σ𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} (-1↑(#‘𝑥)) = Σ𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} (-1↑𝑧))
133 elfzelz 12213 . . . . . . . . 9 (𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) → 𝑧 ∈ ℤ)
134 hashbc 13094 . . . . . . . . 9 (({𝑝 ∈ ℙ ∣ 𝑝𝑁} ∈ Fin ∧ 𝑧 ∈ ℤ) → ((#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})C𝑧) = (#‘{𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧}))
13561, 133, 134syl2an 493 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))) → ((#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})C𝑧) = (#‘{𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧}))
136135oveq1d 6564 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))) → (((#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})C𝑧) · (-1↑𝑧)) = ((#‘{𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧}) · (-1↑𝑧)))
137129, 132, 1363eqtr4d 2654 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))) → Σ𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} (-1↑(#‘𝑥)) = (((#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})C𝑧) · (-1↑𝑧)))
138137sumeq2dv 14281 . . . . 5 (𝑁 ∈ ℕ → Σ𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))Σ𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} (-1↑(#‘𝑥)) = Σ𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))(((#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})C𝑧) · (-1↑𝑧)))
139 1pneg1e0 11006 . . . . . . 7 (1 + -1) = 0
140139oveq1i 6559 . . . . . 6 ((1 + -1)↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) = (0↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))
141 binom1p 14402 . . . . . . 7 ((-1 ∈ ℂ ∧ (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}) ∈ ℕ0) → ((1 + -1)↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) = Σ𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))(((#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})C𝑧) · (-1↑𝑧)))
14260, 108, 141sylancr 694 . . . . . 6 (𝑁 ∈ ℕ → ((1 + -1)↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) = Σ𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))(((#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})C𝑧) · (-1↑𝑧)))
143140, 142syl5eqr 2658 . . . . 5 (𝑁 ∈ ℕ → (0↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) = Σ𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))(((#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})C𝑧) · (-1↑𝑧)))
144 eqeq2 2621 . . . . . 6 (1 = if(𝑁 = 1, 1, 0) → ((0↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) = 1 ↔ (0↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) = if(𝑁 = 1, 1, 0)))
145 eqeq2 2621 . . . . . 6 (0 = if(𝑁 = 1, 1, 0) → ((0↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) = 0 ↔ (0↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) = if(𝑁 = 1, 1, 0)))
146 nprmdvds1 15256 . . . . . . . . . . . . 13 (𝑝 ∈ ℙ → ¬ 𝑝 ∥ 1)
147 simpr 476 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → 𝑁 = 1)
148147breq2d 4595 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (𝑝𝑁𝑝 ∥ 1))
149148notbid 307 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (¬ 𝑝𝑁 ↔ ¬ 𝑝 ∥ 1))
150146, 149syl5ibr 235 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (𝑝 ∈ ℙ → ¬ 𝑝𝑁))
151150ralrimiv 2948 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → ∀𝑝 ∈ ℙ ¬ 𝑝𝑁)
152 rabeq0 3911 . . . . . . . . . . 11 ({𝑝 ∈ ℙ ∣ 𝑝𝑁} = ∅ ↔ ∀𝑝 ∈ ℙ ¬ 𝑝𝑁)
153151, 152sylibr 223 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → {𝑝 ∈ ℙ ∣ 𝑝𝑁} = ∅)
154153fveq2d 6107 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}) = (#‘∅))
155 hash0 13019 . . . . . . . . 9 (#‘∅) = 0
156154, 155syl6eq 2660 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}) = 0)
157156oveq2d 6565 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (0↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) = (0↑0))
158 0exp0e1 12727 . . . . . . 7 (0↑0) = 1
159157, 158syl6eq 2660 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (0↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) = 1)
160 df-ne 2782 . . . . . . . . . . 11 (𝑁 ≠ 1 ↔ ¬ 𝑁 = 1)
161 eluz2b3 11638 . . . . . . . . . . . 12 (𝑁 ∈ (ℤ‘2) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1))
162161biimpri 217 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝑁 ≠ 1) → 𝑁 ∈ (ℤ‘2))
163160, 162sylan2br 492 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → 𝑁 ∈ (ℤ‘2))
164 exprmfct 15254 . . . . . . . . . 10 (𝑁 ∈ (ℤ‘2) → ∃𝑝 ∈ ℙ 𝑝𝑁)
165163, 164syl 17 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → ∃𝑝 ∈ ℙ 𝑝𝑁)
166 rabn0 3912 . . . . . . . . 9 ({𝑝 ∈ ℙ ∣ 𝑝𝑁} ≠ ∅ ↔ ∃𝑝 ∈ ℙ 𝑝𝑁)
167165, 166sylibr 223 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → {𝑝 ∈ ℙ ∣ 𝑝𝑁} ≠ ∅)
16861adantr 480 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∈ Fin)
169 hashnncl 13018 . . . . . . . . 9 ({𝑝 ∈ ℙ ∣ 𝑝𝑁} ∈ Fin → ((#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}) ∈ ℕ ↔ {𝑝 ∈ ℙ ∣ 𝑝𝑁} ≠ ∅))
170168, 169syl 17 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → ((#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}) ∈ ℕ ↔ {𝑝 ∈ ℙ ∣ 𝑝𝑁} ≠ ∅))
171167, 170mpbird 246 . . . . . . 7 ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}) ∈ ℕ)
1721710expd 12886 . . . . . 6 ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → (0↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) = 0)
173144, 145, 159, 172ifbothda 4073 . . . . 5 (𝑁 ∈ ℕ → (0↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) = if(𝑁 = 1, 1, 0))
174138, 143, 1733eqtr2d 2650 . . . 4 (𝑁 ∈ ℕ → Σ𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))Σ𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} (-1↑(#‘𝑥)) = if(𝑁 = 1, 1, 0))
17592, 123, 1743eqtr3d 2652 . . 3 (𝑁 ∈ ℕ → Σ𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} (-1↑(#‘𝑥)) = if(𝑁 = 1, 1, 0))
17669, 175eqtr3d 2646 . 2 (𝑁 ∈ ℕ → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑘})) = if(𝑁 = 1, 1, 0))
17710, 37, 1763eqtr3d 2652 1 (𝑁 ∈ ℕ → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛𝑁} (μ‘𝑘) = if(𝑁 = 1, 1, 0))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  ifcif 4036  𝒫 cpw 4108  ∪ ciun 4455  Disj wdisj 4553   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   ≼ cdom 7839  Fincfn 7841  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   ≤ cle 9954  -cneg 10146  ℕcn 10897  2c2 10947  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197  ↑cexp 12722  Ccbc 12951  #chash 12979  Σcsu 14264   ∥ cdvds 14821  ℙcprime 15223   pCnt cpc 15379  μcmu 24621 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-dvds 14822  df-gcd 15055  df-prm 15224  df-pc 15380  df-mu 24627 This theorem is referenced by:  musumsum  24718  muinv  24719
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