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Theorem musum 24962
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 24964. (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 6229 . . . . . . . 8 (𝑛 = 𝑘 → (μ‘𝑛) = (μ‘𝑘))
21neeq1d 2882 . . . . . . 7 (𝑛 = 𝑘 → ((μ‘𝑛) ≠ 0 ↔ (μ‘𝑘) ≠ 0))
3 breq1 4688 . . . . . . 7 (𝑛 = 𝑘 → (𝑛𝑁𝑘𝑁))
42, 3anbi12d 747 . . . . . 6 (𝑛 = 𝑘 → (((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁) ↔ ((μ‘𝑘) ≠ 0 ∧ 𝑘𝑁)))
54elrab 3396 . . . . 5 (𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} ↔ (𝑘 ∈ ℕ ∧ ((μ‘𝑘) ≠ 0 ∧ 𝑘𝑁)))
6 muval2 24905 . . . . . 6 ((𝑘 ∈ ℕ ∧ (μ‘𝑘) ≠ 0) → (μ‘𝑘) = (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑘})))
76adantrr 753 . . . . 5 ((𝑘 ∈ ℕ ∧ ((μ‘𝑘) ≠ 0 ∧ 𝑘𝑁)) → (μ‘𝑘) = (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑘})))
85, 7sylbi 207 . . . 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 14477 . 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 3716 . . 3 (𝑁 ∈ ℕ → {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} ⊆ {𝑛 ∈ ℕ ∣ 𝑛𝑁})
14 ssrab2 3720 . . . . . 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 3634 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)}) → 𝑘 ∈ ℕ)
17 mucl 24912 . . . . 5 (𝑘 ∈ ℕ → (μ‘𝑘) ∈ ℤ)
1816, 17syl 17 . . . 4 ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)}) → (μ‘𝑘) ∈ ℤ)
1918zcnd 11521 . . 3 ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)}) → (μ‘𝑘) ∈ ℂ)
20 difrab 3934 . . . . . . 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 2841 . . . . . . . . . 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 3717 . . . . . . 7 {𝑛 ∈ ℕ ∣ (𝑛𝑁 ∧ ¬ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁))} ⊆ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0}
2620, 25eqsstri 3668 . . . . . 6 ({𝑛 ∈ ℕ ∣ 𝑛𝑁} ∖ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)}) ⊆ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0}
2726sseli 3632 . . . . 5 (𝑘 ∈ ({𝑛 ∈ ℕ ∣ 𝑛𝑁} ∖ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)}) → 𝑘 ∈ {𝑛 ∈ ℕ ∣ (μ‘𝑛) = 0})
281eqeq1d 2653 . . . . . . 7 (𝑛 = 𝑘 → ((μ‘𝑛) = 0 ↔ (μ‘𝑘) = 0))
2928elrab 3396 . . . . . 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 12812 . . . 4 (𝑁 ∈ ℕ → (1...𝑁) ∈ Fin)
34 dvdsssfz1 15087 . . . 4 (𝑁 ∈ ℕ → {𝑛 ∈ ℕ ∣ 𝑛𝑁} ⊆ (1...𝑁))
35 ssfi 8221 . . . 4 (((1...𝑁) ∈ Fin ∧ {𝑛 ∈ ℕ ∣ 𝑛𝑁} ⊆ (1...𝑁)) → {𝑛 ∈ ℕ ∣ 𝑛𝑁} ∈ Fin)
3633, 34, 35syl2anc 694 . . 3 (𝑁 ∈ ℕ → {𝑛 ∈ ℕ ∣ 𝑛𝑁} ∈ Fin)
3713, 19, 32, 36fsumss 14500 . 2 (𝑁 ∈ ℕ → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} (μ‘𝑘) = Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛𝑁} (μ‘𝑘))
38 fveq2 6229 . . . . 5 (𝑥 = {𝑝 ∈ ℙ ∣ 𝑝𝑘} → (#‘𝑥) = (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑘}))
3938oveq2d 6706 . . . 4 (𝑥 = {𝑝 ∈ ℙ ∣ 𝑝𝑘} → (-1↑(#‘𝑥)) = (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑘})))
40 ssfi 8221 . . . . 5 (({𝑛 ∈ ℕ ∣ 𝑛𝑁} ∈ Fin ∧ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} ⊆ {𝑛 ∈ ℕ ∣ 𝑛𝑁}) → {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} ∈ Fin)
4136, 13, 40syl2anc 694 . . . 4 (𝑁 ∈ ℕ → {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} ∈ Fin)
42 eqid 2651 . . . . 5 {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} = {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)}
43 eqid 2651 . . . . 5 (𝑚 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} ↦ {𝑝 ∈ ℙ ∣ 𝑝𝑚}) = (𝑚 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} ↦ {𝑝 ∈ ℙ ∣ 𝑝𝑚})
44 oveq1 6697 . . . . . . . 8 (𝑞 = 𝑝 → (𝑞 pCnt 𝑥) = (𝑝 pCnt 𝑥))
4544cbvmptv 4783 . . . . . . 7 (𝑞 ∈ ℙ ↦ (𝑞 pCnt 𝑥)) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑥))
46 oveq2 6698 . . . . . . . 8 (𝑥 = 𝑚 → (𝑝 pCnt 𝑥) = (𝑝 pCnt 𝑚))
4746mpteq2dv 4778 . . . . . . 7 (𝑥 = 𝑚 → (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑥)) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑚)))
4845, 47syl5eq 2697 . . . . . 6 (𝑥 = 𝑚 → (𝑞 ∈ ℙ ↦ (𝑞 pCnt 𝑥)) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑚)))
4948cbvmptv 4783 . . . . 5 (𝑥 ∈ ℕ ↦ (𝑞 ∈ ℙ ↦ (𝑞 pCnt 𝑥))) = (𝑚 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑚)))
5042, 43, 49sqff1o 24953 . . . 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 4689 . . . . . . 7 (𝑚 = 𝑘 → (𝑝𝑚𝑝𝑘))
5251rabbidv 3220 . . . . . 6 (𝑚 = 𝑘 → {𝑝 ∈ ℙ ∣ 𝑝𝑚} = {𝑝 ∈ ℙ ∣ 𝑝𝑘})
53 zex 11424 . . . . . . . 8 ℤ ∈ V
54 prmz 15436 . . . . . . . . 9 (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
5554ssriv 3640 . . . . . . . 8 ℙ ⊆ ℤ
5653, 55ssexi 4836 . . . . . . 7 ℙ ∈ V
5756rabex 4845 . . . . . 6 {𝑝 ∈ ℙ ∣ 𝑝𝑘} ∈ V
5852, 43, 57fvmpt 6321 . . . . 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 11162 . . . . 5 -1 ∈ ℂ
61 prmdvdsfi 24878 . . . . . . 7 (𝑁 ∈ ℕ → {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∈ Fin)
62 elpwi 4201 . . . . . . 7 (𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} → 𝑥 ⊆ {𝑝 ∈ ℙ ∣ 𝑝𝑁})
63 ssfi 8221 . . . . . . 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 13185 . . . . . 6 (𝑥 ∈ Fin → (#‘𝑥) ∈ ℕ0)
6664, 65syl 17 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → (#‘𝑥) ∈ ℕ0)
67 expcl 12918 . . . . 5 ((-1 ∈ ℂ ∧ (#‘𝑥) ∈ ℕ0) → (-1↑(#‘𝑥)) ∈ ℂ)
6860, 66, 67sylancr 696 . . . 4 ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → (-1↑(#‘𝑥)) ∈ ℂ)
6939, 41, 50, 59, 68fsumf1o 14498 . . 3 (𝑁 ∈ ℕ → Σ𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} (-1↑(#‘𝑥)) = Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑘})))
70 fzfid 12812 . . . . 5 (𝑁 ∈ ℕ → (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) ∈ Fin)
7161adantr 480 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))) → {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∈ Fin)
72 pwfi 8302 . . . . . . 7 ({𝑝 ∈ ℙ ∣ 𝑝𝑁} ∈ Fin ↔ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∈ Fin)
7371, 72sylib 208 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))) → 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∈ Fin)
74 ssrab2 3720 . . . . . 6 {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} ⊆ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}
75 ssfi 8221 . . . . . 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 695 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))) → {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} ∈ Fin)
77 simprr 811 . . . . . . . 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 6229 . . . . . . . . . . 11 (𝑠 = 𝑥 → (#‘𝑠) = (#‘𝑥))
7978eqeq1d 2653 . . . . . . . . . 10 (𝑠 = 𝑥 → ((#‘𝑠) = 𝑧 ↔ (#‘𝑥) = 𝑧))
8079elrab 3396 . . . . . . . . 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 3000 . . . . . 6 (𝑁 ∈ ℕ → ∀𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))∀𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} (#‘𝑥) = 𝑧)
84 invdisj 4670 . . . . . 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 3634 . . . . . . . . 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 694 . . . . . . 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 696 . . . . 5 ((𝑁 ∈ ℕ ∧ (𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧})) → (-1↑(#‘𝑥)) ∈ ℂ)
9270, 76, 85, 91fsumiun 14597 . . . 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 4201 . . . . . . . . . . . . 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 8043 . . . . . . . . . . . 12 ({𝑝 ∈ ℙ ∣ 𝑝𝑁} ∈ Fin → (𝑠 ⊆ {𝑝 ∈ ℙ ∣ 𝑝𝑁} → 𝑠 ≼ {𝑝 ∈ ℙ ∣ 𝑝𝑁}))
9793, 95, 96sylc 65 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → 𝑠 ≼ {𝑝 ∈ ℙ ∣ 𝑝𝑁})
98 ssfi 8221 . . . . . . . . . . . . 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 13206 . . . . . . . . . . . 12 ((𝑠 ∈ Fin ∧ {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∈ Fin) → ((#‘𝑠) ≤ (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}) ↔ 𝑠 ≼ {𝑝 ∈ ℙ ∣ 𝑝𝑁}))
10199, 93, 100syl2anc 694 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → ((#‘𝑠) ≤ (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}) ↔ 𝑠 ≼ {𝑝 ∈ ℙ ∣ 𝑝𝑁}))
10297, 101mpbird 247 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → (#‘𝑠) ≤ (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))
103 hashcl 13185 . . . . . . . . . . . . 13 (𝑠 ∈ Fin → (#‘𝑠) ∈ ℕ0)
10499, 103syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → (#‘𝑠) ∈ ℕ0)
105 nn0uz 11760 . . . . . . . . . . . 12 0 = (ℤ‘0)
106104, 105syl6eleq 2740 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → (#‘𝑠) ∈ (ℤ‘0))
107 hashcl 13185 . . . . . . . . . . . . . 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 11518 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}) ∈ ℤ)
111 elfz5 12372 . . . . . . . . . . 11 (((#‘𝑠) ∈ (ℤ‘0) ∧ (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}) ∈ ℤ) → ((#‘𝑠) ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) ↔ (#‘𝑠) ≤ (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})))
112106, 110, 111syl2anc 694 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → ((#‘𝑠) ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) ↔ (#‘𝑠) ≤ (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})))
113102, 112mpbird 247 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → (#‘𝑠) ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})))
114 eqidd 2652 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → (#‘𝑠) = (#‘𝑠))
115 eqeq2 2662 . . . . . . . . . 10 (𝑧 = (#‘𝑠) → ((#‘𝑠) = 𝑧 ↔ (#‘𝑠) = (#‘𝑠)))
116115rspcev 3340 . . . . . . . . 9 (((#‘𝑠) ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) ∧ (#‘𝑠) = (#‘𝑠)) → ∃𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))(#‘𝑠) = 𝑧)
117113, 114, 116syl2anc 694 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}) → ∃𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))(#‘𝑠) = 𝑧)
118117ralrimiva 2995 . . . . . . 7 (𝑁 ∈ ℕ → ∀𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁}∃𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))(#‘𝑠) = 𝑧)
119 rabid2 3148 . . . . . . 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 224 . . . . . 6 (𝑁 ∈ ℕ → 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} = {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ ∃𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))(#‘𝑠) = 𝑧})
121 iunrab 4599 . . . . . 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 2704 . . . . 5 (𝑁 ∈ ℕ → 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})){𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧} = 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁})
123122sumeq1d 14475 . . . 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 12471 . . . . . . . . . 10 (𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) → 𝑧 ∈ ℕ0)
125124adantl 481 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))) → 𝑧 ∈ ℕ0)
126 expcl 12918 . . . . . . . . 9 ((-1 ∈ ℂ ∧ 𝑧 ∈ ℕ0) → (-1↑𝑧) ∈ ℂ)
12760, 125, 126sylancr 696 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))) → (-1↑𝑧) ∈ ℂ)
128 fsumconst 14566 . . . . . . . 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 694 . . . . . . 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 6706 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))) ∧ 𝑥 ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∣ (#‘𝑠) = 𝑧}) → (-1↑(#‘𝑥)) = (-1↑𝑧))
132131sumeq2dv 14477 . . . . . . 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 12380 . . . . . . . . 9 (𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) → 𝑧 ∈ ℤ)
134 hashbc 13275 . . . . . . . . 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 6705 . . . . . . 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 2695 . . . . . 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 14477 . . . . 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 11167 . . . . . . 7 (1 + -1) = 0
140139oveq1i 6700 . . . . . 6 ((1 + -1)↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) = (0↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))
141 binom1p 14607 . . . . . . 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 696 . . . . . 6 (𝑁 ∈ ℕ → ((1 + -1)↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) = Σ𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))(((#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})C𝑧) · (-1↑𝑧)))
143140, 142syl5eqr 2699 . . . . 5 (𝑁 ∈ ℕ → (0↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) = Σ𝑧 ∈ (0...(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}))(((#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})C𝑧) · (-1↑𝑧)))
144 eqeq2 2662 . . . . . 6 (1 = if(𝑁 = 1, 1, 0) → ((0↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) = 1 ↔ (0↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) = if(𝑁 = 1, 1, 0)))
145 eqeq2 2662 . . . . . 6 (0 = if(𝑁 = 1, 1, 0) → ((0↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) = 0 ↔ (0↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) = if(𝑁 = 1, 1, 0)))
146 nprmdvds1 15465 . . . . . . . . . . . . 13 (𝑝 ∈ ℙ → ¬ 𝑝 ∥ 1)
147 simpr 476 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → 𝑁 = 1)
148147breq2d 4697 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (𝑝𝑁𝑝 ∥ 1))
149148notbid 307 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (¬ 𝑝𝑁 ↔ ¬ 𝑝 ∥ 1))
150146, 149syl5ibr 236 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (𝑝 ∈ ℙ → ¬ 𝑝𝑁))
151150ralrimiv 2994 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → ∀𝑝 ∈ ℙ ¬ 𝑝𝑁)
152 rabeq0 3990 . . . . . . . . . . 11 ({𝑝 ∈ ℙ ∣ 𝑝𝑁} = ∅ ↔ ∀𝑝 ∈ ℙ ¬ 𝑝𝑁)
153151, 152sylibr 224 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → {𝑝 ∈ ℙ ∣ 𝑝𝑁} = ∅)
154153fveq2d 6233 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}) = (#‘∅))
155 hash0 13196 . . . . . . . . 9 (#‘∅) = 0
156154, 155syl6eq 2701 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}) = 0)
157156oveq2d 6706 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (0↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) = (0↑0))
158 0exp0e1 12905 . . . . . . 7 (0↑0) = 1
159157, 158syl6eq 2701 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑁 = 1) → (0↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) = 1)
160 df-ne 2824 . . . . . . . . . . 11 (𝑁 ≠ 1 ↔ ¬ 𝑁 = 1)
161 eluz2b3 11800 . . . . . . . . . . . 12 (𝑁 ∈ (ℤ‘2) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1))
162161biimpri 218 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ 𝑁 ≠ 1) → 𝑁 ∈ (ℤ‘2))
163160, 162sylan2br 492 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → 𝑁 ∈ (ℤ‘2))
164 exprmfct 15463 . . . . . . . . . 10 (𝑁 ∈ (ℤ‘2) → ∃𝑝 ∈ ℙ 𝑝𝑁)
165163, 164syl 17 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → ∃𝑝 ∈ ℙ 𝑝𝑁)
166 rabn0 3991 . . . . . . . . 9 ({𝑝 ∈ ℙ ∣ 𝑝𝑁} ≠ ∅ ↔ ∃𝑝 ∈ ℙ 𝑝𝑁)
167165, 166sylibr 224 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → {𝑝 ∈ ℙ ∣ 𝑝𝑁} ≠ ∅)
16861adantr 480 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → {𝑝 ∈ ℙ ∣ 𝑝𝑁} ∈ Fin)
169 hashnncl 13195 . . . . . . . . 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 247 . . . . . . 7 ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁}) ∈ ℕ)
1721710expd 13064 . . . . . 6 ((𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1) → (0↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) = 0)
173144, 145, 159, 172ifbothda 4156 . . . . 5 (𝑁 ∈ ℕ → (0↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑁})) = if(𝑁 = 1, 1, 0))
174138, 143, 1733eqtr2d 2691 . . . 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 2693 . . 3 (𝑁 ∈ ℕ → Σ𝑥 ∈ 𝒫 {𝑝 ∈ ℙ ∣ 𝑝𝑁} (-1↑(#‘𝑥)) = if(𝑁 = 1, 1, 0))
17669, 175eqtr3d 2687 . 2 (𝑁 ∈ ℕ → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ ((μ‘𝑛) ≠ 0 ∧ 𝑛𝑁)} (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑘})) = if(𝑁 = 1, 1, 0))
17710, 37, 1763eqtr3d 2693 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 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  cdif 3604  wss 3607  c0 3948  ifcif 4119  𝒫 cpw 4191   ciun 4552  Disj wdisj 4652   class class class wbr 4685  cmpt 4762  cfv 5926  (class class class)co 6690  cdom 7995  Fincfn 7997  cc 9972  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  cle 10113  -cneg 10305  cn 11058  2c2 11108  0cn0 11330  cz 11415  cuz 11725  ...cfz 12364  cexp 12900  Ccbc 13129  #chash 13157  Σcsu 14460  cdvds 15027  cprime 15432   pCnt cpc 15588  μcmu 24866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-fac 13101  df-bc 13130  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-dvds 15028  df-gcd 15264  df-prm 15433  df-pc 15589  df-mu 24872
This theorem is referenced by:  musumsum  24963  muinv  24964
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