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Theorem musum 26919
Description: The sum of the MΓΆbius 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 MΓΆbius function useful for inverting divisor sums; see also muinv 26921. (Contributed by Mario Carneiro, 2-Jul-2015.)
Assertion
Ref Expression
musum (𝑁 ∈ β„• β†’ Ξ£π‘˜ ∈ {𝑛 ∈ β„• ∣ 𝑛 βˆ₯ 𝑁} (ΞΌβ€˜π‘˜) = if(𝑁 = 1, 1, 0))
Distinct variable group:   π‘˜,𝑛,𝑁

Proof of Theorem musum
Dummy variables π‘š 𝑝 π‘ž 𝑠 π‘₯ 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6891 . . . . . . . 8 (𝑛 = π‘˜ β†’ (ΞΌβ€˜π‘›) = (ΞΌβ€˜π‘˜))
21neeq1d 3000 . . . . . . 7 (𝑛 = π‘˜ β†’ ((ΞΌβ€˜π‘›) β‰  0 ↔ (ΞΌβ€˜π‘˜) β‰  0))
3 breq1 5151 . . . . . . 7 (𝑛 = π‘˜ β†’ (𝑛 βˆ₯ 𝑁 ↔ π‘˜ βˆ₯ 𝑁))
42, 3anbi12d 631 . . . . . 6 (𝑛 = π‘˜ β†’ (((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁) ↔ ((ΞΌβ€˜π‘˜) β‰  0 ∧ π‘˜ βˆ₯ 𝑁)))
54elrab 3683 . . . . 5 (π‘˜ ∈ {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)} ↔ (π‘˜ ∈ β„• ∧ ((ΞΌβ€˜π‘˜) β‰  0 ∧ π‘˜ βˆ₯ 𝑁)))
6 muval2 26862 . . . . . 6 ((π‘˜ ∈ β„• ∧ (ΞΌβ€˜π‘˜) β‰  0) β†’ (ΞΌβ€˜π‘˜) = (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘˜})))
76adantrr 715 . . . . 5 ((π‘˜ ∈ β„• ∧ ((ΞΌβ€˜π‘˜) β‰  0 ∧ π‘˜ βˆ₯ 𝑁)) β†’ (ΞΌβ€˜π‘˜) = (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘˜})))
85, 7sylbi 216 . . . 4 (π‘˜ ∈ {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)} β†’ (ΞΌβ€˜π‘˜) = (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘˜})))
98adantl 482 . . 3 ((𝑁 ∈ β„• ∧ π‘˜ ∈ {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)}) β†’ (ΞΌβ€˜π‘˜) = (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘˜})))
109sumeq2dv 15653 . 2 (𝑁 ∈ β„• β†’ Ξ£π‘˜ ∈ {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)} (ΞΌβ€˜π‘˜) = Ξ£π‘˜ ∈ {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)} (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘˜})))
11 simpr 485 . . . . 5 (((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁) β†’ 𝑛 βˆ₯ 𝑁)
1211a1i 11 . . . 4 ((𝑁 ∈ β„• ∧ 𝑛 ∈ β„•) β†’ (((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁) β†’ 𝑛 βˆ₯ 𝑁))
1312ss2rabdv 4073 . . 3 (𝑁 ∈ β„• β†’ {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)} βŠ† {𝑛 ∈ β„• ∣ 𝑛 βˆ₯ 𝑁})
14 ssrab2 4077 . . . . . 6 {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)} βŠ† β„•
15 simpr 485 . . . . . 6 ((𝑁 ∈ β„• ∧ π‘˜ ∈ {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)}) β†’ π‘˜ ∈ {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)})
1614, 15sselid 3980 . . . . 5 ((𝑁 ∈ β„• ∧ π‘˜ ∈ {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)}) β†’ π‘˜ ∈ β„•)
17 mucl 26869 . . . . 5 (π‘˜ ∈ β„• β†’ (ΞΌβ€˜π‘˜) ∈ β„€)
1816, 17syl 17 . . . 4 ((𝑁 ∈ β„• ∧ π‘˜ ∈ {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)}) β†’ (ΞΌβ€˜π‘˜) ∈ β„€)
1918zcnd 12671 . . 3 ((𝑁 ∈ β„• ∧ π‘˜ ∈ {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)}) β†’ (ΞΌβ€˜π‘˜) ∈ β„‚)
20 difrab 4308 . . . . . . 7 ({𝑛 ∈ β„• ∣ 𝑛 βˆ₯ 𝑁} βˆ– {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)}) = {𝑛 ∈ β„• ∣ (𝑛 βˆ₯ 𝑁 ∧ Β¬ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁))}
21 pm3.21 472 . . . . . . . . . . 11 (𝑛 βˆ₯ 𝑁 β†’ ((ΞΌβ€˜π‘›) β‰  0 β†’ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)))
2221necon1bd 2958 . . . . . . . . . 10 (𝑛 βˆ₯ 𝑁 β†’ (Β¬ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁) β†’ (ΞΌβ€˜π‘›) = 0))
2322imp 407 . . . . . . . . 9 ((𝑛 βˆ₯ 𝑁 ∧ Β¬ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)) β†’ (ΞΌβ€˜π‘›) = 0)
2423a1i 11 . . . . . . . 8 (𝑛 ∈ β„• β†’ ((𝑛 βˆ₯ 𝑁 ∧ Β¬ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)) β†’ (ΞΌβ€˜π‘›) = 0))
2524ss2rabi 4074 . . . . . . 7 {𝑛 ∈ β„• ∣ (𝑛 βˆ₯ 𝑁 ∧ Β¬ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁))} βŠ† {𝑛 ∈ β„• ∣ (ΞΌβ€˜π‘›) = 0}
2620, 25eqsstri 4016 . . . . . 6 ({𝑛 ∈ β„• ∣ 𝑛 βˆ₯ 𝑁} βˆ– {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)}) βŠ† {𝑛 ∈ β„• ∣ (ΞΌβ€˜π‘›) = 0}
2726sseli 3978 . . . . 5 (π‘˜ ∈ ({𝑛 ∈ β„• ∣ 𝑛 βˆ₯ 𝑁} βˆ– {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)}) β†’ π‘˜ ∈ {𝑛 ∈ β„• ∣ (ΞΌβ€˜π‘›) = 0})
28 fveqeq2 6900 . . . . . . 7 (𝑛 = π‘˜ β†’ ((ΞΌβ€˜π‘›) = 0 ↔ (ΞΌβ€˜π‘˜) = 0))
2928elrab 3683 . . . . . 6 (π‘˜ ∈ {𝑛 ∈ β„• ∣ (ΞΌβ€˜π‘›) = 0} ↔ (π‘˜ ∈ β„• ∧ (ΞΌβ€˜π‘˜) = 0))
3029simprbi 497 . . . . 5 (π‘˜ ∈ {𝑛 ∈ β„• ∣ (ΞΌβ€˜π‘›) = 0} β†’ (ΞΌβ€˜π‘˜) = 0)
3127, 30syl 17 . . . 4 (π‘˜ ∈ ({𝑛 ∈ β„• ∣ 𝑛 βˆ₯ 𝑁} βˆ– {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)}) β†’ (ΞΌβ€˜π‘˜) = 0)
3231adantl 482 . . 3 ((𝑁 ∈ β„• ∧ π‘˜ ∈ ({𝑛 ∈ β„• ∣ 𝑛 βˆ₯ 𝑁} βˆ– {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)})) β†’ (ΞΌβ€˜π‘˜) = 0)
33 fzfid 13942 . . . 4 (𝑁 ∈ β„• β†’ (1...𝑁) ∈ Fin)
34 dvdsssfz1 16265 . . . 4 (𝑁 ∈ β„• β†’ {𝑛 ∈ β„• ∣ 𝑛 βˆ₯ 𝑁} βŠ† (1...𝑁))
3533, 34ssfid 9269 . . 3 (𝑁 ∈ β„• β†’ {𝑛 ∈ β„• ∣ 𝑛 βˆ₯ 𝑁} ∈ Fin)
3613, 19, 32, 35fsumss 15675 . 2 (𝑁 ∈ β„• β†’ Ξ£π‘˜ ∈ {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)} (ΞΌβ€˜π‘˜) = Ξ£π‘˜ ∈ {𝑛 ∈ β„• ∣ 𝑛 βˆ₯ 𝑁} (ΞΌβ€˜π‘˜))
37 fveq2 6891 . . . . 5 (π‘₯ = {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘˜} β†’ (β™―β€˜π‘₯) = (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘˜}))
3837oveq2d 7427 . . . 4 (π‘₯ = {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘˜} β†’ (-1↑(β™―β€˜π‘₯)) = (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘˜})))
3935, 13ssfid 9269 . . . 4 (𝑁 ∈ β„• β†’ {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)} ∈ Fin)
40 eqid 2732 . . . . 5 {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)} = {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)}
41 eqid 2732 . . . . 5 (π‘š ∈ {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)} ↦ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘š}) = (π‘š ∈ {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)} ↦ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘š})
42 oveq1 7418 . . . . . . . 8 (π‘ž = 𝑝 β†’ (π‘ž pCnt π‘₯) = (𝑝 pCnt π‘₯))
4342cbvmptv 5261 . . . . . . 7 (π‘ž ∈ β„™ ↦ (π‘ž pCnt π‘₯)) = (𝑝 ∈ β„™ ↦ (𝑝 pCnt π‘₯))
44 oveq2 7419 . . . . . . . 8 (π‘₯ = π‘š β†’ (𝑝 pCnt π‘₯) = (𝑝 pCnt π‘š))
4544mpteq2dv 5250 . . . . . . 7 (π‘₯ = π‘š β†’ (𝑝 ∈ β„™ ↦ (𝑝 pCnt π‘₯)) = (𝑝 ∈ β„™ ↦ (𝑝 pCnt π‘š)))
4643, 45eqtrid 2784 . . . . . 6 (π‘₯ = π‘š β†’ (π‘ž ∈ β„™ ↦ (π‘ž pCnt π‘₯)) = (𝑝 ∈ β„™ ↦ (𝑝 pCnt π‘š)))
4746cbvmptv 5261 . . . . 5 (π‘₯ ∈ β„• ↦ (π‘ž ∈ β„™ ↦ (π‘ž pCnt π‘₯))) = (π‘š ∈ β„• ↦ (𝑝 ∈ β„™ ↦ (𝑝 pCnt π‘š)))
4840, 41, 47sqff1o 26910 . . . 4 (𝑁 ∈ β„• β†’ (π‘š ∈ {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)} ↦ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘š}):{𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)}–1-1-onto→𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})
49 breq2 5152 . . . . . . 7 (π‘š = π‘˜ β†’ (𝑝 βˆ₯ π‘š ↔ 𝑝 βˆ₯ π‘˜))
5049rabbidv 3440 . . . . . 6 (π‘š = π‘˜ β†’ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘š} = {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘˜})
51 prmex 16618 . . . . . . 7 β„™ ∈ V
5251rabex 5332 . . . . . 6 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘˜} ∈ V
5350, 41, 52fvmpt 6998 . . . . 5 (π‘˜ ∈ {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)} β†’ ((π‘š ∈ {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)} ↦ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘š})β€˜π‘˜) = {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘˜})
5453adantl 482 . . . 4 ((𝑁 ∈ β„• ∧ π‘˜ ∈ {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)}) β†’ ((π‘š ∈ {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)} ↦ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘š})β€˜π‘˜) = {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘˜})
55 neg1cn 12330 . . . . 5 -1 ∈ β„‚
56 prmdvdsfi 26835 . . . . . . 7 (𝑁 ∈ β„• β†’ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∈ Fin)
57 elpwi 4609 . . . . . . 7 (π‘₯ ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} β†’ π‘₯ βŠ† {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})
58 ssfi 9175 . . . . . . 7 (({𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∈ Fin ∧ π‘₯ βŠ† {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) β†’ π‘₯ ∈ Fin)
5956, 57, 58syl2an 596 . . . . . 6 ((𝑁 ∈ β„• ∧ π‘₯ ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) β†’ π‘₯ ∈ Fin)
60 hashcl 14320 . . . . . 6 (π‘₯ ∈ Fin β†’ (β™―β€˜π‘₯) ∈ β„•0)
6159, 60syl 17 . . . . 5 ((𝑁 ∈ β„• ∧ π‘₯ ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) β†’ (β™―β€˜π‘₯) ∈ β„•0)
62 expcl 14049 . . . . 5 ((-1 ∈ β„‚ ∧ (β™―β€˜π‘₯) ∈ β„•0) β†’ (-1↑(β™―β€˜π‘₯)) ∈ β„‚)
6355, 61, 62sylancr 587 . . . 4 ((𝑁 ∈ β„• ∧ π‘₯ ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) β†’ (-1↑(β™―β€˜π‘₯)) ∈ β„‚)
6438, 39, 48, 54, 63fsumf1o 15673 . . 3 (𝑁 ∈ β„• β†’ Ξ£π‘₯ ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} (-1↑(β™―β€˜π‘₯)) = Ξ£π‘˜ ∈ {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)} (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘˜})))
65 fzfid 13942 . . . . 5 (𝑁 ∈ β„• β†’ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) ∈ Fin)
6656adantr 481 . . . . . . 7 ((𝑁 ∈ β„• ∧ 𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))) β†’ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∈ Fin)
67 pwfi 9180 . . . . . . 7 ({𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∈ Fin ↔ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∈ Fin)
6866, 67sylib 217 . . . . . 6 ((𝑁 ∈ β„• ∧ 𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))) β†’ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∈ Fin)
69 ssrab2 4077 . . . . . 6 {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧} βŠ† 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}
70 ssfi 9175 . . . . . 6 ((𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∈ Fin ∧ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧} βŠ† 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) β†’ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧} ∈ Fin)
7168, 69, 70sylancl 586 . . . . 5 ((𝑁 ∈ β„• ∧ 𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))) β†’ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧} ∈ Fin)
72 simprr 771 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) ∧ π‘₯ ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧})) β†’ π‘₯ ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧})
73 fveqeq2 6900 . . . . . . . . . 10 (𝑠 = π‘₯ β†’ ((β™―β€˜π‘ ) = 𝑧 ↔ (β™―β€˜π‘₯) = 𝑧))
7473elrab 3683 . . . . . . . . 9 (π‘₯ ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧} ↔ (π‘₯ ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∧ (β™―β€˜π‘₯) = 𝑧))
7574simprbi 497 . . . . . . . 8 (π‘₯ ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧} β†’ (β™―β€˜π‘₯) = 𝑧)
7672, 75syl 17 . . . . . . 7 ((𝑁 ∈ β„• ∧ (𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) ∧ π‘₯ ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧})) β†’ (β™―β€˜π‘₯) = 𝑧)
7776ralrimivva 3200 . . . . . 6 (𝑁 ∈ β„• β†’ βˆ€π‘§ ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))βˆ€π‘₯ ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧} (β™―β€˜π‘₯) = 𝑧)
78 invdisj 5132 . . . . . 6 (βˆ€π‘§ ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))βˆ€π‘₯ ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧} (β™―β€˜π‘₯) = 𝑧 β†’ Disj 𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})){𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧})
7977, 78syl 17 . . . . 5 (𝑁 ∈ β„• β†’ Disj 𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})){𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧})
8056adantr 481 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) ∧ π‘₯ ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧})) β†’ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∈ Fin)
8169, 72sselid 3980 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) ∧ π‘₯ ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧})) β†’ π‘₯ ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})
8281, 57syl 17 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) ∧ π‘₯ ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧})) β†’ π‘₯ βŠ† {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})
8380, 82ssfid 9269 . . . . . . 7 ((𝑁 ∈ β„• ∧ (𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) ∧ π‘₯ ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧})) β†’ π‘₯ ∈ Fin)
8483, 60syl 17 . . . . . 6 ((𝑁 ∈ β„• ∧ (𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) ∧ π‘₯ ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧})) β†’ (β™―β€˜π‘₯) ∈ β„•0)
8555, 84, 62sylancr 587 . . . . 5 ((𝑁 ∈ β„• ∧ (𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) ∧ π‘₯ ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧})) β†’ (-1↑(β™―β€˜π‘₯)) ∈ β„‚)
8665, 71, 79, 85fsumiun 15771 . . . 4 (𝑁 ∈ β„• β†’ Ξ£π‘₯ ∈ βˆͺ 𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})){𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧} (-1↑(β™―β€˜π‘₯)) = Σ𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))Ξ£π‘₯ ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧} (-1↑(β™―β€˜π‘₯)))
87 iunrab 5055 . . . . . 6 βˆͺ 𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})){𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧} = {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ βˆƒπ‘§ ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))(β™―β€˜π‘ ) = 𝑧}
8856adantr 481 . . . . . . . . . . . 12 ((𝑁 ∈ β„• ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) β†’ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∈ Fin)
89 elpwi 4609 . . . . . . . . . . . . 13 (𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} β†’ 𝑠 βŠ† {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})
9089adantl 482 . . . . . . . . . . . 12 ((𝑁 ∈ β„• ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) β†’ 𝑠 βŠ† {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})
91 ssdomg 8998 . . . . . . . . . . . 12 ({𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∈ Fin β†’ (𝑠 βŠ† {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} β†’ 𝑠 β‰Ό {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))
9288, 90, 91sylc 65 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) β†’ 𝑠 β‰Ό {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})
93 ssfi 9175 . . . . . . . . . . . . 13 (({𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∈ Fin ∧ 𝑠 βŠ† {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) β†’ 𝑠 ∈ Fin)
9456, 89, 93syl2an 596 . . . . . . . . . . . 12 ((𝑁 ∈ β„• ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) β†’ 𝑠 ∈ Fin)
95 hashdom 14343 . . . . . . . . . . . 12 ((𝑠 ∈ Fin ∧ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∈ Fin) β†’ ((β™―β€˜π‘ ) ≀ (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) ↔ 𝑠 β‰Ό {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))
9694, 88, 95syl2anc 584 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) β†’ ((β™―β€˜π‘ ) ≀ (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) ↔ 𝑠 β‰Ό {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))
9792, 96mpbird 256 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) β†’ (β™―β€˜π‘ ) ≀ (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))
98 hashcl 14320 . . . . . . . . . . . . 13 (𝑠 ∈ Fin β†’ (β™―β€˜π‘ ) ∈ β„•0)
9994, 98syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ β„• ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) β†’ (β™―β€˜π‘ ) ∈ β„•0)
100 nn0uz 12868 . . . . . . . . . . . 12 β„•0 = (β„€β‰₯β€˜0)
10199, 100eleqtrdi 2843 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) β†’ (β™―β€˜π‘ ) ∈ (β„€β‰₯β€˜0))
102 hashcl 14320 . . . . . . . . . . . . . 14 ({𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∈ Fin β†’ (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) ∈ β„•0)
10356, 102syl 17 . . . . . . . . . . . . 13 (𝑁 ∈ β„• β†’ (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) ∈ β„•0)
104103adantr 481 . . . . . . . . . . . 12 ((𝑁 ∈ β„• ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) β†’ (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) ∈ β„•0)
105104nn0zd 12588 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) β†’ (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) ∈ β„€)
106 elfz5 13497 . . . . . . . . . . 11 (((β™―β€˜π‘ ) ∈ (β„€β‰₯β€˜0) ∧ (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) ∈ β„€) β†’ ((β™―β€˜π‘ ) ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) ↔ (β™―β€˜π‘ ) ≀ (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})))
107101, 105, 106syl2anc 584 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) β†’ ((β™―β€˜π‘ ) ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) ↔ (β™―β€˜π‘ ) ≀ (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})))
10897, 107mpbird 256 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) β†’ (β™―β€˜π‘ ) ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})))
109 eqidd 2733 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) β†’ (β™―β€˜π‘ ) = (β™―β€˜π‘ ))
110 eqeq2 2744 . . . . . . . . . 10 (𝑧 = (β™―β€˜π‘ ) β†’ ((β™―β€˜π‘ ) = 𝑧 ↔ (β™―β€˜π‘ ) = (β™―β€˜π‘ )))
111110rspcev 3612 . . . . . . . . 9 (((β™―β€˜π‘ ) ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) ∧ (β™―β€˜π‘ ) = (β™―β€˜π‘ )) β†’ βˆƒπ‘§ ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))(β™―β€˜π‘ ) = 𝑧)
112108, 109, 111syl2anc 584 . . . . . . . 8 ((𝑁 ∈ β„• ∧ 𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) β†’ βˆƒπ‘§ ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))(β™―β€˜π‘ ) = 𝑧)
113112ralrimiva 3146 . . . . . . 7 (𝑁 ∈ β„• β†’ βˆ€π‘  ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}βˆƒπ‘§ ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))(β™―β€˜π‘ ) = 𝑧)
114 rabid2 3464 . . . . . . 7 (𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} = {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ βˆƒπ‘§ ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))(β™―β€˜π‘ ) = 𝑧} ↔ βˆ€π‘  ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}βˆƒπ‘§ ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))(β™―β€˜π‘ ) = 𝑧)
115113, 114sylibr 233 . . . . . 6 (𝑁 ∈ β„• β†’ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} = {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ βˆƒπ‘§ ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))(β™―β€˜π‘ ) = 𝑧})
11687, 115eqtr4id 2791 . . . . 5 (𝑁 ∈ β„• β†’ βˆͺ 𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})){𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧} = 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})
117116sumeq1d 15651 . . . 4 (𝑁 ∈ β„• β†’ Ξ£π‘₯ ∈ βˆͺ 𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})){𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧} (-1↑(β™―β€˜π‘₯)) = Ξ£π‘₯ ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} (-1↑(β™―β€˜π‘₯)))
118 elfznn0 13598 . . . . . . . . . 10 (𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) β†’ 𝑧 ∈ β„•0)
119118adantl 482 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ 𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))) β†’ 𝑧 ∈ β„•0)
120 expcl 14049 . . . . . . . . 9 ((-1 ∈ β„‚ ∧ 𝑧 ∈ β„•0) β†’ (-1↑𝑧) ∈ β„‚)
12155, 119, 120sylancr 587 . . . . . . . 8 ((𝑁 ∈ β„• ∧ 𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))) β†’ (-1↑𝑧) ∈ β„‚)
122 fsumconst 15740 . . . . . . . 8 (({𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧} ∈ Fin ∧ (-1↑𝑧) ∈ β„‚) β†’ Ξ£π‘₯ ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧} (-1↑𝑧) = ((β™―β€˜{𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧}) Β· (-1↑𝑧)))
12371, 121, 122syl2anc 584 . . . . . . 7 ((𝑁 ∈ β„• ∧ 𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))) β†’ Ξ£π‘₯ ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧} (-1↑𝑧) = ((β™―β€˜{𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧}) Β· (-1↑𝑧)))
12475adantl 482 . . . . . . . . 9 (((𝑁 ∈ β„• ∧ 𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))) ∧ π‘₯ ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧}) β†’ (β™―β€˜π‘₯) = 𝑧)
125124oveq2d 7427 . . . . . . . 8 (((𝑁 ∈ β„• ∧ 𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))) ∧ π‘₯ ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧}) β†’ (-1↑(β™―β€˜π‘₯)) = (-1↑𝑧))
126125sumeq2dv 15653 . . . . . . 7 ((𝑁 ∈ β„• ∧ 𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))) β†’ Ξ£π‘₯ ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧} (-1↑(β™―β€˜π‘₯)) = Ξ£π‘₯ ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧} (-1↑𝑧))
127 elfzelz 13505 . . . . . . . . 9 (𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) β†’ 𝑧 ∈ β„€)
128 hashbc 14416 . . . . . . . . 9 (({𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∈ Fin ∧ 𝑧 ∈ β„€) β†’ ((β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})C𝑧) = (β™―β€˜{𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧}))
12956, 127, 128syl2an 596 . . . . . . . 8 ((𝑁 ∈ β„• ∧ 𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))) β†’ ((β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})C𝑧) = (β™―β€˜{𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧}))
130129oveq1d 7426 . . . . . . 7 ((𝑁 ∈ β„• ∧ 𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))) β†’ (((β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})C𝑧) Β· (-1↑𝑧)) = ((β™―β€˜{𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧}) Β· (-1↑𝑧)))
131123, 126, 1303eqtr4d 2782 . . . . . 6 ((𝑁 ∈ β„• ∧ 𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))) β†’ Ξ£π‘₯ ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧} (-1↑(β™―β€˜π‘₯)) = (((β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})C𝑧) Β· (-1↑𝑧)))
132131sumeq2dv 15653 . . . . 5 (𝑁 ∈ β„• β†’ Σ𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))Ξ£π‘₯ ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧} (-1↑(β™―β€˜π‘₯)) = Σ𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))(((β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})C𝑧) Β· (-1↑𝑧)))
133 1pneg1e0 12335 . . . . . . 7 (1 + -1) = 0
134133oveq1i 7421 . . . . . 6 ((1 + -1)↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) = (0↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))
135 binom1p 15781 . . . . . . 7 ((-1 ∈ β„‚ ∧ (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) ∈ β„•0) β†’ ((1 + -1)↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) = Σ𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))(((β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})C𝑧) Β· (-1↑𝑧)))
13655, 103, 135sylancr 587 . . . . . 6 (𝑁 ∈ β„• β†’ ((1 + -1)↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) = Σ𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))(((β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})C𝑧) Β· (-1↑𝑧)))
137134, 136eqtr3id 2786 . . . . 5 (𝑁 ∈ β„• β†’ (0↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) = Σ𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))(((β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})C𝑧) Β· (-1↑𝑧)))
138 eqeq2 2744 . . . . . 6 (1 = if(𝑁 = 1, 1, 0) β†’ ((0↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) = 1 ↔ (0↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) = if(𝑁 = 1, 1, 0)))
139 eqeq2 2744 . . . . . 6 (0 = if(𝑁 = 1, 1, 0) β†’ ((0↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) = 0 ↔ (0↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) = if(𝑁 = 1, 1, 0)))
140 nprmdvds1 16647 . . . . . . . . . . . . 13 (𝑝 ∈ β„™ β†’ Β¬ 𝑝 βˆ₯ 1)
141 simpr 485 . . . . . . . . . . . . . . 15 ((𝑁 ∈ β„• ∧ 𝑁 = 1) β†’ 𝑁 = 1)
142141breq2d 5160 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ 𝑁 = 1) β†’ (𝑝 βˆ₯ 𝑁 ↔ 𝑝 βˆ₯ 1))
143142notbid 317 . . . . . . . . . . . . 13 ((𝑁 ∈ β„• ∧ 𝑁 = 1) β†’ (Β¬ 𝑝 βˆ₯ 𝑁 ↔ Β¬ 𝑝 βˆ₯ 1))
144140, 143imbitrrid 245 . . . . . . . . . . . 12 ((𝑁 ∈ β„• ∧ 𝑁 = 1) β†’ (𝑝 ∈ β„™ β†’ Β¬ 𝑝 βˆ₯ 𝑁))
145144ralrimiv 3145 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ 𝑁 = 1) β†’ βˆ€π‘ ∈ β„™ Β¬ 𝑝 βˆ₯ 𝑁)
146 rabeq0 4384 . . . . . . . . . . 11 ({𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} = βˆ… ↔ βˆ€π‘ ∈ β„™ Β¬ 𝑝 βˆ₯ 𝑁)
147145, 146sylibr 233 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ 𝑁 = 1) β†’ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} = βˆ…)
148147fveq2d 6895 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ 𝑁 = 1) β†’ (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) = (β™―β€˜βˆ…))
149 hash0 14331 . . . . . . . . 9 (β™―β€˜βˆ…) = 0
150148, 149eqtrdi 2788 . . . . . . . 8 ((𝑁 ∈ β„• ∧ 𝑁 = 1) β†’ (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) = 0)
151150oveq2d 7427 . . . . . . 7 ((𝑁 ∈ β„• ∧ 𝑁 = 1) β†’ (0↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) = (0↑0))
152 0exp0e1 14036 . . . . . . 7 (0↑0) = 1
153151, 152eqtrdi 2788 . . . . . 6 ((𝑁 ∈ β„• ∧ 𝑁 = 1) β†’ (0↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) = 1)
154 df-ne 2941 . . . . . . . . . . 11 (𝑁 β‰  1 ↔ Β¬ 𝑁 = 1)
155 eluz2b3 12910 . . . . . . . . . . . 12 (𝑁 ∈ (β„€β‰₯β€˜2) ↔ (𝑁 ∈ β„• ∧ 𝑁 β‰  1))
156155biimpri 227 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ 𝑁 β‰  1) β†’ 𝑁 ∈ (β„€β‰₯β€˜2))
157154, 156sylan2br 595 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ Β¬ 𝑁 = 1) β†’ 𝑁 ∈ (β„€β‰₯β€˜2))
158 exprmfct 16645 . . . . . . . . . 10 (𝑁 ∈ (β„€β‰₯β€˜2) β†’ βˆƒπ‘ ∈ β„™ 𝑝 βˆ₯ 𝑁)
159157, 158syl 17 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ Β¬ 𝑁 = 1) β†’ βˆƒπ‘ ∈ β„™ 𝑝 βˆ₯ 𝑁)
160 rabn0 4385 . . . . . . . . 9 ({𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} β‰  βˆ… ↔ βˆƒπ‘ ∈ β„™ 𝑝 βˆ₯ 𝑁)
161159, 160sylibr 233 . . . . . . . 8 ((𝑁 ∈ β„• ∧ Β¬ 𝑁 = 1) β†’ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} β‰  βˆ…)
16256adantr 481 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ Β¬ 𝑁 = 1) β†’ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∈ Fin)
163 hashnncl 14330 . . . . . . . . 9 ({𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∈ Fin β†’ ((β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) ∈ β„• ↔ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} β‰  βˆ…))
164162, 163syl 17 . . . . . . . 8 ((𝑁 ∈ β„• ∧ Β¬ 𝑁 = 1) β†’ ((β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) ∈ β„• ↔ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} β‰  βˆ…))
165161, 164mpbird 256 . . . . . . 7 ((𝑁 ∈ β„• ∧ Β¬ 𝑁 = 1) β†’ (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}) ∈ β„•)
1661650expd 14108 . . . . . 6 ((𝑁 ∈ β„• ∧ Β¬ 𝑁 = 1) β†’ (0↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) = 0)
167138, 139, 153, 166ifbothda 4566 . . . . 5 (𝑁 ∈ β„• β†’ (0↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁})) = if(𝑁 = 1, 1, 0))
168132, 137, 1673eqtr2d 2778 . . . 4 (𝑁 ∈ β„• β†’ Σ𝑧 ∈ (0...(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁}))Ξ£π‘₯ ∈ {𝑠 ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} ∣ (β™―β€˜π‘ ) = 𝑧} (-1↑(β™―β€˜π‘₯)) = if(𝑁 = 1, 1, 0))
16986, 117, 1683eqtr3d 2780 . . 3 (𝑁 ∈ β„• β†’ Ξ£π‘₯ ∈ 𝒫 {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝑁} (-1↑(β™―β€˜π‘₯)) = if(𝑁 = 1, 1, 0))
17064, 169eqtr3d 2774 . 2 (𝑁 ∈ β„• β†’ Ξ£π‘˜ ∈ {𝑛 ∈ β„• ∣ ((ΞΌβ€˜π‘›) β‰  0 ∧ 𝑛 βˆ₯ 𝑁)} (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘˜})) = if(𝑁 = 1, 1, 0))
17110, 36, 1703eqtr3d 2780 1 (𝑁 ∈ β„• β†’ Ξ£π‘˜ ∈ {𝑛 ∈ β„• ∣ 𝑛 βˆ₯ 𝑁} (ΞΌβ€˜π‘˜) = if(𝑁 = 1, 1, 0))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432   βˆ– cdif 3945   βŠ† wss 3948  βˆ…c0 4322  ifcif 4528  π’« cpw 4602  βˆͺ ciun 4997  Disj wdisj 5113   class class class wbr 5148   ↦ cmpt 5231  β€˜cfv 6543  (class class class)co 7411   β‰Ό cdom 8939  Fincfn 8941  β„‚cc 11110  0cc0 11112  1c1 11113   + caddc 11115   Β· cmul 11117   ≀ cle 11253  -cneg 11449  β„•cn 12216  2c2 12271  β„•0cn0 12476  β„€cz 12562  β„€β‰₯cuz 12826  ...cfz 13488  β†‘cexp 14031  Ccbc 14266  β™―chash 14294  Ξ£csu 15636   βˆ₯ cdvds 16201  β„™cprime 16612   pCnt cpc 16773  ΞΌcmu 26823
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-disj 5114  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-oadd 8472  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-oi 9507  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-q 12937  df-rp 12979  df-fz 13489  df-fzo 13632  df-fl 13761  df-mod 13839  df-seq 13971  df-exp 14032  df-fac 14238  df-bc 14267  df-hash 14295  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15436  df-sum 15637  df-dvds 16202  df-gcd 16440  df-prm 16613  df-pc 16774  df-mu 26829
This theorem is referenced by:  musumsum  26920  muinv  26921
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