Step |
Hyp |
Ref |
Expression |
1 |
|
fveq2 |
⊢ ( 𝑛 = 𝑘 → ( μ ‘ 𝑛 ) = ( μ ‘ 𝑘 ) ) |
2 |
1
|
neeq1d |
⊢ ( 𝑛 = 𝑘 → ( ( μ ‘ 𝑛 ) ≠ 0 ↔ ( μ ‘ 𝑘 ) ≠ 0 ) ) |
3 |
|
breq1 |
⊢ ( 𝑛 = 𝑘 → ( 𝑛 ∥ 𝑁 ↔ 𝑘 ∥ 𝑁 ) ) |
4 |
2 3
|
anbi12d |
⊢ ( 𝑛 = 𝑘 → ( ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) ↔ ( ( μ ‘ 𝑘 ) ≠ 0 ∧ 𝑘 ∥ 𝑁 ) ) ) |
5 |
4
|
elrab |
⊢ ( 𝑘 ∈ { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } ↔ ( 𝑘 ∈ ℕ ∧ ( ( μ ‘ 𝑘 ) ≠ 0 ∧ 𝑘 ∥ 𝑁 ) ) ) |
6 |
|
muval2 |
⊢ ( ( 𝑘 ∈ ℕ ∧ ( μ ‘ 𝑘 ) ≠ 0 ) → ( μ ‘ 𝑘 ) = ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘 } ) ) ) |
7 |
6
|
adantrr |
⊢ ( ( 𝑘 ∈ ℕ ∧ ( ( μ ‘ 𝑘 ) ≠ 0 ∧ 𝑘 ∥ 𝑁 ) ) → ( μ ‘ 𝑘 ) = ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘 } ) ) ) |
8 |
5 7
|
sylbi |
⊢ ( 𝑘 ∈ { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } → ( μ ‘ 𝑘 ) = ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘 } ) ) ) |
9 |
8
|
adantl |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑘 ∈ { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } ) → ( μ ‘ 𝑘 ) = ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘 } ) ) ) |
10 |
9
|
sumeq2dv |
⊢ ( 𝑁 ∈ ℕ → Σ 𝑘 ∈ { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } ( μ ‘ 𝑘 ) = Σ 𝑘 ∈ { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘 } ) ) ) |
11 |
|
simpr |
⊢ ( ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) → 𝑛 ∥ 𝑁 ) |
12 |
11
|
a1i |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ ) → ( ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) → 𝑛 ∥ 𝑁 ) ) |
13 |
12
|
ss2rabdv |
⊢ ( 𝑁 ∈ ℕ → { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } ⊆ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁 } ) |
14 |
|
ssrab2 |
⊢ { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } ⊆ ℕ |
15 |
|
simpr |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑘 ∈ { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } ) → 𝑘 ∈ { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } ) |
16 |
14 15
|
sselid |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑘 ∈ { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } ) → 𝑘 ∈ ℕ ) |
17 |
|
mucl |
⊢ ( 𝑘 ∈ ℕ → ( μ ‘ 𝑘 ) ∈ ℤ ) |
18 |
16 17
|
syl |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑘 ∈ { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } ) → ( μ ‘ 𝑘 ) ∈ ℤ ) |
19 |
18
|
zcnd |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑘 ∈ { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } ) → ( μ ‘ 𝑘 ) ∈ ℂ ) |
20 |
|
difrab |
⊢ ( { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁 } ∖ { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } ) = { 𝑛 ∈ ℕ ∣ ( 𝑛 ∥ 𝑁 ∧ ¬ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) ) } |
21 |
|
pm3.21 |
⊢ ( 𝑛 ∥ 𝑁 → ( ( μ ‘ 𝑛 ) ≠ 0 → ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) ) ) |
22 |
21
|
necon1bd |
⊢ ( 𝑛 ∥ 𝑁 → ( ¬ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) → ( μ ‘ 𝑛 ) = 0 ) ) |
23 |
22
|
imp |
⊢ ( ( 𝑛 ∥ 𝑁 ∧ ¬ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) ) → ( μ ‘ 𝑛 ) = 0 ) |
24 |
23
|
a1i |
⊢ ( 𝑛 ∈ ℕ → ( ( 𝑛 ∥ 𝑁 ∧ ¬ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) ) → ( μ ‘ 𝑛 ) = 0 ) ) |
25 |
24
|
ss2rabi |
⊢ { 𝑛 ∈ ℕ ∣ ( 𝑛 ∥ 𝑁 ∧ ¬ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) ) } ⊆ { 𝑛 ∈ ℕ ∣ ( μ ‘ 𝑛 ) = 0 } |
26 |
20 25
|
eqsstri |
⊢ ( { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁 } ∖ { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } ) ⊆ { 𝑛 ∈ ℕ ∣ ( μ ‘ 𝑛 ) = 0 } |
27 |
26
|
sseli |
⊢ ( 𝑘 ∈ ( { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁 } ∖ { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } ) → 𝑘 ∈ { 𝑛 ∈ ℕ ∣ ( μ ‘ 𝑛 ) = 0 } ) |
28 |
|
fveqeq2 |
⊢ ( 𝑛 = 𝑘 → ( ( μ ‘ 𝑛 ) = 0 ↔ ( μ ‘ 𝑘 ) = 0 ) ) |
29 |
28
|
elrab |
⊢ ( 𝑘 ∈ { 𝑛 ∈ ℕ ∣ ( μ ‘ 𝑛 ) = 0 } ↔ ( 𝑘 ∈ ℕ ∧ ( μ ‘ 𝑘 ) = 0 ) ) |
30 |
29
|
simprbi |
⊢ ( 𝑘 ∈ { 𝑛 ∈ ℕ ∣ ( μ ‘ 𝑛 ) = 0 } → ( μ ‘ 𝑘 ) = 0 ) |
31 |
27 30
|
syl |
⊢ ( 𝑘 ∈ ( { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁 } ∖ { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } ) → ( μ ‘ 𝑘 ) = 0 ) |
32 |
31
|
adantl |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑘 ∈ ( { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁 } ∖ { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } ) ) → ( μ ‘ 𝑘 ) = 0 ) |
33 |
|
fzfid |
⊢ ( 𝑁 ∈ ℕ → ( 1 ... 𝑁 ) ∈ Fin ) |
34 |
|
dvdsssfz1 |
⊢ ( 𝑁 ∈ ℕ → { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁 } ⊆ ( 1 ... 𝑁 ) ) |
35 |
33 34
|
ssfid |
⊢ ( 𝑁 ∈ ℕ → { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁 } ∈ Fin ) |
36 |
13 19 32 35
|
fsumss |
⊢ ( 𝑁 ∈ ℕ → Σ 𝑘 ∈ { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } ( μ ‘ 𝑘 ) = Σ 𝑘 ∈ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁 } ( μ ‘ 𝑘 ) ) |
37 |
|
fveq2 |
⊢ ( 𝑥 = { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘 } → ( ♯ ‘ 𝑥 ) = ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘 } ) ) |
38 |
37
|
oveq2d |
⊢ ( 𝑥 = { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘 } → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘 } ) ) ) |
39 |
35 13
|
ssfid |
⊢ ( 𝑁 ∈ ℕ → { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } ∈ Fin ) |
40 |
|
eqid |
⊢ { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } = { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } |
41 |
|
eqid |
⊢ ( 𝑚 ∈ { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } ↦ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑚 } ) = ( 𝑚 ∈ { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } ↦ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑚 } ) |
42 |
|
oveq1 |
⊢ ( 𝑞 = 𝑝 → ( 𝑞 pCnt 𝑥 ) = ( 𝑝 pCnt 𝑥 ) ) |
43 |
42
|
cbvmptv |
⊢ ( 𝑞 ∈ ℙ ↦ ( 𝑞 pCnt 𝑥 ) ) = ( 𝑝 ∈ ℙ ↦ ( 𝑝 pCnt 𝑥 ) ) |
44 |
|
oveq2 |
⊢ ( 𝑥 = 𝑚 → ( 𝑝 pCnt 𝑥 ) = ( 𝑝 pCnt 𝑚 ) ) |
45 |
44
|
mpteq2dv |
⊢ ( 𝑥 = 𝑚 → ( 𝑝 ∈ ℙ ↦ ( 𝑝 pCnt 𝑥 ) ) = ( 𝑝 ∈ ℙ ↦ ( 𝑝 pCnt 𝑚 ) ) ) |
46 |
43 45
|
eqtrid |
⊢ ( 𝑥 = 𝑚 → ( 𝑞 ∈ ℙ ↦ ( 𝑞 pCnt 𝑥 ) ) = ( 𝑝 ∈ ℙ ↦ ( 𝑝 pCnt 𝑚 ) ) ) |
47 |
46
|
cbvmptv |
⊢ ( 𝑥 ∈ ℕ ↦ ( 𝑞 ∈ ℙ ↦ ( 𝑞 pCnt 𝑥 ) ) ) = ( 𝑚 ∈ ℕ ↦ ( 𝑝 ∈ ℙ ↦ ( 𝑝 pCnt 𝑚 ) ) ) |
48 |
40 41 47
|
sqff1o |
⊢ ( 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 } ) |
49 |
|
breq2 |
⊢ ( 𝑚 = 𝑘 → ( 𝑝 ∥ 𝑚 ↔ 𝑝 ∥ 𝑘 ) ) |
50 |
49
|
rabbidv |
⊢ ( 𝑚 = 𝑘 → { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑚 } = { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘 } ) |
51 |
|
prmex |
⊢ ℙ ∈ V |
52 |
51
|
rabex |
⊢ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘 } ∈ V |
53 |
50 41 52
|
fvmpt |
⊢ ( 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 } ) |
54 |
53
|
adantl |
⊢ ( ( 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 } ) |
55 |
|
neg1cn |
⊢ - 1 ∈ ℂ |
56 |
|
prmdvdsfi |
⊢ ( 𝑁 ∈ ℕ → { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∈ Fin ) |
57 |
|
elpwi |
⊢ ( 𝑥 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } → 𝑥 ⊆ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) |
58 |
|
ssfi |
⊢ ( ( { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∈ Fin ∧ 𝑥 ⊆ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) → 𝑥 ∈ Fin ) |
59 |
56 57 58
|
syl2an |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) → 𝑥 ∈ Fin ) |
60 |
|
hashcl |
⊢ ( 𝑥 ∈ Fin → ( ♯ ‘ 𝑥 ) ∈ ℕ0 ) |
61 |
59 60
|
syl |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) → ( ♯ ‘ 𝑥 ) ∈ ℕ0 ) |
62 |
|
expcl |
⊢ ( ( - 1 ∈ ℂ ∧ ( ♯ ‘ 𝑥 ) ∈ ℕ0 ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) ∈ ℂ ) |
63 |
55 61 62
|
sylancr |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) ∈ ℂ ) |
64 |
38 39 48 54 63
|
fsumf1o |
⊢ ( 𝑁 ∈ ℕ → Σ 𝑥 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = Σ 𝑘 ∈ { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘 } ) ) ) |
65 |
|
fzfid |
⊢ ( 𝑁 ∈ ℕ → ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ∈ Fin ) |
66 |
56
|
adantr |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ) → { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∈ Fin ) |
67 |
|
pwfi |
⊢ ( { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∈ Fin ↔ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∈ Fin ) |
68 |
66 67
|
sylib |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ) → 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∈ Fin ) |
69 |
|
ssrab2 |
⊢ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ⊆ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } |
70 |
|
ssfi |
⊢ ( ( 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 ) |
71 |
68 69 70
|
sylancl |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ) → { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ∈ Fin ) |
72 |
|
simprr |
⊢ ( ( 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 } ) |
73 |
|
fveqeq2 |
⊢ ( 𝑠 = 𝑥 → ( ( ♯ ‘ 𝑠 ) = 𝑧 ↔ ( ♯ ‘ 𝑥 ) = 𝑧 ) ) |
74 |
73
|
elrab |
⊢ ( 𝑥 ∈ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ↔ ( 𝑥 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∧ ( ♯ ‘ 𝑥 ) = 𝑧 ) ) |
75 |
74
|
simprbi |
⊢ ( 𝑥 ∈ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } → ( ♯ ‘ 𝑥 ) = 𝑧 ) |
76 |
72 75
|
syl |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ∧ 𝑥 ∈ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ) ) → ( ♯ ‘ 𝑥 ) = 𝑧 ) |
77 |
76
|
ralrimivva |
⊢ ( 𝑁 ∈ ℕ → ∀ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ∀ 𝑥 ∈ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ( ♯ ‘ 𝑥 ) = 𝑧 ) |
78 |
|
invdisj |
⊢ ( ∀ 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 } ) |
79 |
77 78
|
syl |
⊢ ( 𝑁 ∈ ℕ → Disj 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ) |
80 |
56
|
adantr |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ∧ 𝑥 ∈ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ) ) → { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∈ Fin ) |
81 |
69 72
|
sselid |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ∧ 𝑥 ∈ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ) ) → 𝑥 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) |
82 |
81 57
|
syl |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ∧ 𝑥 ∈ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ) ) → 𝑥 ⊆ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) |
83 |
80 82
|
ssfid |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ∧ 𝑥 ∈ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ) ) → 𝑥 ∈ Fin ) |
84 |
83 60
|
syl |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ∧ 𝑥 ∈ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ) ) → ( ♯ ‘ 𝑥 ) ∈ ℕ0 ) |
85 |
55 84 62
|
sylancr |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ∧ 𝑥 ∈ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ) ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) ∈ ℂ ) |
86 |
65 71 79 85
|
fsumiun |
⊢ ( 𝑁 ∈ ℕ → Σ 𝑥 ∈ ∪ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = Σ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) Σ 𝑥 ∈ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) ) |
87 |
|
iunrab |
⊢ ∪ 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 } |
88 |
56
|
adantr |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) → { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∈ Fin ) |
89 |
|
elpwi |
⊢ ( 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } → 𝑠 ⊆ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) |
90 |
89
|
adantl |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) → 𝑠 ⊆ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) |
91 |
|
ssdomg |
⊢ ( { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∈ Fin → ( 𝑠 ⊆ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } → 𝑠 ≼ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) |
92 |
88 90 91
|
sylc |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) → 𝑠 ≼ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) |
93 |
|
ssfi |
⊢ ( ( { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∈ Fin ∧ 𝑠 ⊆ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) → 𝑠 ∈ Fin ) |
94 |
56 89 93
|
syl2an |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) → 𝑠 ∈ Fin ) |
95 |
|
hashdom |
⊢ ( ( 𝑠 ∈ Fin ∧ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∈ Fin ) → ( ( ♯ ‘ 𝑠 ) ≤ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ↔ 𝑠 ≼ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) |
96 |
94 88 95
|
syl2anc |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) → ( ( ♯ ‘ 𝑠 ) ≤ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ↔ 𝑠 ≼ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) |
97 |
92 96
|
mpbird |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) → ( ♯ ‘ 𝑠 ) ≤ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) |
98 |
|
hashcl |
⊢ ( 𝑠 ∈ Fin → ( ♯ ‘ 𝑠 ) ∈ ℕ0 ) |
99 |
94 98
|
syl |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) → ( ♯ ‘ 𝑠 ) ∈ ℕ0 ) |
100 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
101 |
99 100
|
eleqtrdi |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) → ( ♯ ‘ 𝑠 ) ∈ ( ℤ≥ ‘ 0 ) ) |
102 |
|
hashcl |
⊢ ( { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∈ Fin → ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ∈ ℕ0 ) |
103 |
56 102
|
syl |
⊢ ( 𝑁 ∈ ℕ → ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ∈ ℕ0 ) |
104 |
103
|
adantr |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) → ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ∈ ℕ0 ) |
105 |
104
|
nn0zd |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) → ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ∈ ℤ ) |
106 |
|
elfz5 |
⊢ ( ( ( ♯ ‘ 𝑠 ) ∈ ( ℤ≥ ‘ 0 ) ∧ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ∈ ℤ ) → ( ( ♯ ‘ 𝑠 ) ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ↔ ( ♯ ‘ 𝑠 ) ≤ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ) |
107 |
101 105 106
|
syl2anc |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) → ( ( ♯ ‘ 𝑠 ) ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ↔ ( ♯ ‘ 𝑠 ) ≤ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ) |
108 |
97 107
|
mpbird |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) → ( ♯ ‘ 𝑠 ) ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ) |
109 |
|
eqidd |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) → ( ♯ ‘ 𝑠 ) = ( ♯ ‘ 𝑠 ) ) |
110 |
|
eqeq2 |
⊢ ( 𝑧 = ( ♯ ‘ 𝑠 ) → ( ( ♯ ‘ 𝑠 ) = 𝑧 ↔ ( ♯ ‘ 𝑠 ) = ( ♯ ‘ 𝑠 ) ) ) |
111 |
110
|
rspcev |
⊢ ( ( ( ♯ ‘ 𝑠 ) ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ∧ ( ♯ ‘ 𝑠 ) = ( ♯ ‘ 𝑠 ) ) → ∃ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ( ♯ ‘ 𝑠 ) = 𝑧 ) |
112 |
108 109 111
|
syl2anc |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) → ∃ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ( ♯ ‘ 𝑠 ) = 𝑧 ) |
113 |
112
|
ralrimiva |
⊢ ( 𝑁 ∈ ℕ → ∀ 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∃ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ( ♯ ‘ 𝑠 ) = 𝑧 ) |
114 |
|
rabid2 |
⊢ ( 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 ) |
115 |
113 114
|
sylibr |
⊢ ( 𝑁 ∈ ℕ → 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } = { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ∃ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ( ♯ ‘ 𝑠 ) = 𝑧 } ) |
116 |
87 115
|
eqtr4id |
⊢ ( 𝑁 ∈ ℕ → ∪ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } = 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) |
117 |
116
|
sumeq1d |
⊢ ( 𝑁 ∈ ℕ → Σ 𝑥 ∈ ∪ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = Σ 𝑥 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) ) |
118 |
|
elfznn0 |
⊢ ( 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) → 𝑧 ∈ ℕ0 ) |
119 |
118
|
adantl |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ) → 𝑧 ∈ ℕ0 ) |
120 |
|
expcl |
⊢ ( ( - 1 ∈ ℂ ∧ 𝑧 ∈ ℕ0 ) → ( - 1 ↑ 𝑧 ) ∈ ℂ ) |
121 |
55 119 120
|
sylancr |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ) → ( - 1 ↑ 𝑧 ) ∈ ℂ ) |
122 |
|
fsumconst |
⊢ ( ( { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ∈ Fin ∧ ( - 1 ↑ 𝑧 ) ∈ ℂ ) → Σ 𝑥 ∈ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ( - 1 ↑ 𝑧 ) = ( ( ♯ ‘ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ) · ( - 1 ↑ 𝑧 ) ) ) |
123 |
71 121 122
|
syl2anc |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ) → Σ 𝑥 ∈ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ( - 1 ↑ 𝑧 ) = ( ( ♯ ‘ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ) · ( - 1 ↑ 𝑧 ) ) ) |
124 |
75
|
adantl |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ) ∧ 𝑥 ∈ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ) → ( ♯ ‘ 𝑥 ) = 𝑧 ) |
125 |
124
|
oveq2d |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ) ∧ 𝑥 ∈ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ) → ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = ( - 1 ↑ 𝑧 ) ) |
126 |
125
|
sumeq2dv |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ) → Σ 𝑥 ∈ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = Σ 𝑥 ∈ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ( - 1 ↑ 𝑧 ) ) |
127 |
|
elfzelz |
⊢ ( 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) → 𝑧 ∈ ℤ ) |
128 |
|
hashbc |
⊢ ( ( { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∈ Fin ∧ 𝑧 ∈ ℤ ) → ( ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) C 𝑧 ) = ( ♯ ‘ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ) ) |
129 |
56 127 128
|
syl2an |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ) → ( ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) C 𝑧 ) = ( ♯ ‘ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ) ) |
130 |
129
|
oveq1d |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ) → ( ( ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) C 𝑧 ) · ( - 1 ↑ 𝑧 ) ) = ( ( ♯ ‘ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ) · ( - 1 ↑ 𝑧 ) ) ) |
131 |
123 126 130
|
3eqtr4d |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ) → Σ 𝑥 ∈ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = ( ( ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) C 𝑧 ) · ( - 1 ↑ 𝑧 ) ) ) |
132 |
131
|
sumeq2dv |
⊢ ( 𝑁 ∈ ℕ → Σ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) Σ 𝑥 ∈ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = Σ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ( ( ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) C 𝑧 ) · ( - 1 ↑ 𝑧 ) ) ) |
133 |
|
1pneg1e0 |
⊢ ( 1 + - 1 ) = 0 |
134 |
133
|
oveq1i |
⊢ ( ( 1 + - 1 ) ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) = ( 0 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) |
135 |
|
binom1p |
⊢ ( ( - 1 ∈ ℂ ∧ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ∈ ℕ0 ) → ( ( 1 + - 1 ) ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) = Σ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ( ( ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) C 𝑧 ) · ( - 1 ↑ 𝑧 ) ) ) |
136 |
55 103 135
|
sylancr |
⊢ ( 𝑁 ∈ ℕ → ( ( 1 + - 1 ) ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) = Σ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ( ( ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) C 𝑧 ) · ( - 1 ↑ 𝑧 ) ) ) |
137 |
134 136
|
eqtr3id |
⊢ ( 𝑁 ∈ ℕ → ( 0 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) = Σ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) ( ( ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) C 𝑧 ) · ( - 1 ↑ 𝑧 ) ) ) |
138 |
|
eqeq2 |
⊢ ( 1 = if ( 𝑁 = 1 , 1 , 0 ) → ( ( 0 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) = 1 ↔ ( 0 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) = if ( 𝑁 = 1 , 1 , 0 ) ) ) |
139 |
|
eqeq2 |
⊢ ( 0 = if ( 𝑁 = 1 , 1 , 0 ) → ( ( 0 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) = 0 ↔ ( 0 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) = if ( 𝑁 = 1 , 1 , 0 ) ) ) |
140 |
|
nprmdvds1 |
⊢ ( 𝑝 ∈ ℙ → ¬ 𝑝 ∥ 1 ) |
141 |
|
simpr |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑁 = 1 ) → 𝑁 = 1 ) |
142 |
141
|
breq2d |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑁 = 1 ) → ( 𝑝 ∥ 𝑁 ↔ 𝑝 ∥ 1 ) ) |
143 |
142
|
notbid |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑁 = 1 ) → ( ¬ 𝑝 ∥ 𝑁 ↔ ¬ 𝑝 ∥ 1 ) ) |
144 |
140 143
|
syl5ibr |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑁 = 1 ) → ( 𝑝 ∈ ℙ → ¬ 𝑝 ∥ 𝑁 ) ) |
145 |
144
|
ralrimiv |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑁 = 1 ) → ∀ 𝑝 ∈ ℙ ¬ 𝑝 ∥ 𝑁 ) |
146 |
|
rabeq0 |
⊢ ( { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } = ∅ ↔ ∀ 𝑝 ∈ ℙ ¬ 𝑝 ∥ 𝑁 ) |
147 |
145 146
|
sylibr |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑁 = 1 ) → { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } = ∅ ) |
148 |
147
|
fveq2d |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑁 = 1 ) → ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) = ( ♯ ‘ ∅ ) ) |
149 |
|
hash0 |
⊢ ( ♯ ‘ ∅ ) = 0 |
150 |
148 149
|
eqtrdi |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑁 = 1 ) → ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) = 0 ) |
151 |
150
|
oveq2d |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑁 = 1 ) → ( 0 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) = ( 0 ↑ 0 ) ) |
152 |
|
0exp0e1 |
⊢ ( 0 ↑ 0 ) = 1 |
153 |
151 152
|
eqtrdi |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑁 = 1 ) → ( 0 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) = 1 ) |
154 |
|
df-ne |
⊢ ( 𝑁 ≠ 1 ↔ ¬ 𝑁 = 1 ) |
155 |
|
eluz2b3 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 2 ) ↔ ( 𝑁 ∈ ℕ ∧ 𝑁 ≠ 1 ) ) |
156 |
155
|
biimpri |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑁 ≠ 1 ) → 𝑁 ∈ ( ℤ≥ ‘ 2 ) ) |
157 |
154 156
|
sylan2br |
⊢ ( ( 𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1 ) → 𝑁 ∈ ( ℤ≥ ‘ 2 ) ) |
158 |
|
exprmfct |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 2 ) → ∃ 𝑝 ∈ ℙ 𝑝 ∥ 𝑁 ) |
159 |
157 158
|
syl |
⊢ ( ( 𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1 ) → ∃ 𝑝 ∈ ℙ 𝑝 ∥ 𝑁 ) |
160 |
|
rabn0 |
⊢ ( { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ≠ ∅ ↔ ∃ 𝑝 ∈ ℙ 𝑝 ∥ 𝑁 ) |
161 |
159 160
|
sylibr |
⊢ ( ( 𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1 ) → { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ≠ ∅ ) |
162 |
56
|
adantr |
⊢ ( ( 𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1 ) → { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∈ Fin ) |
163 |
|
hashnncl |
⊢ ( { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∈ Fin → ( ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ∈ ℕ ↔ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ≠ ∅ ) ) |
164 |
162 163
|
syl |
⊢ ( ( 𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1 ) → ( ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ∈ ℕ ↔ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ≠ ∅ ) ) |
165 |
161 164
|
mpbird |
⊢ ( ( 𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1 ) → ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ∈ ℕ ) |
166 |
165
|
0expd |
⊢ ( ( 𝑁 ∈ ℕ ∧ ¬ 𝑁 = 1 ) → ( 0 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) = 0 ) |
167 |
138 139 153 166
|
ifbothda |
⊢ ( 𝑁 ∈ ℕ → ( 0 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) = if ( 𝑁 = 1 , 1 , 0 ) ) |
168 |
132 137 167
|
3eqtr2d |
⊢ ( 𝑁 ∈ ℕ → Σ 𝑧 ∈ ( 0 ... ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ) ) Σ 𝑥 ∈ { 𝑠 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ∣ ( ♯ ‘ 𝑠 ) = 𝑧 } ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = if ( 𝑁 = 1 , 1 , 0 ) ) |
169 |
86 117 168
|
3eqtr3d |
⊢ ( 𝑁 ∈ ℕ → Σ 𝑥 ∈ 𝒫 { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑁 } ( - 1 ↑ ( ♯ ‘ 𝑥 ) ) = if ( 𝑁 = 1 , 1 , 0 ) ) |
170 |
64 169
|
eqtr3d |
⊢ ( 𝑁 ∈ ℕ → Σ 𝑘 ∈ { 𝑛 ∈ ℕ ∣ ( ( μ ‘ 𝑛 ) ≠ 0 ∧ 𝑛 ∥ 𝑁 ) } ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑘 } ) ) = if ( 𝑁 = 1 , 1 , 0 ) ) |
171 |
10 36 170
|
3eqtr3d |
⊢ ( 𝑁 ∈ ℕ → Σ 𝑘 ∈ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁 } ( μ ‘ 𝑘 ) = if ( 𝑁 = 1 , 1 , 0 ) ) |