Step |
Hyp |
Ref |
Expression |
1 |
|
rpvmasum.z |
⊢ 𝑍 = ( ℤ/nℤ ‘ 𝑁 ) |
2 |
|
rpvmasum.l |
⊢ 𝐿 = ( ℤRHom ‘ 𝑍 ) |
3 |
|
rpvmasum.a |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
4 |
|
rpvmasum2.g |
⊢ 𝐺 = ( DChr ‘ 𝑁 ) |
5 |
|
rpvmasum2.d |
⊢ 𝐷 = ( Base ‘ 𝐺 ) |
6 |
|
rpvmasum2.1 |
⊢ 1 = ( 0g ‘ 𝐺 ) |
7 |
|
dchrisum0f.f |
⊢ 𝐹 = ( 𝑏 ∈ ℕ ↦ Σ 𝑣 ∈ { 𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏 } ( 𝑋 ‘ ( 𝐿 ‘ 𝑣 ) ) ) |
8 |
|
dchrisum0f.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐷 ) |
9 |
|
dchrisum0flb.r |
⊢ ( 𝜑 → 𝑋 : ( Base ‘ 𝑍 ) ⟶ ℝ ) |
10 |
|
dchrisum0flb.1 |
⊢ ( 𝜑 → 𝐴 ∈ ( ℤ≥ ‘ 2 ) ) |
11 |
|
dchrisum0flb.2 |
⊢ ( 𝜑 → 𝑃 ∈ ℙ ) |
12 |
|
dchrisum0flb.3 |
⊢ ( 𝜑 → 𝑃 ∥ 𝐴 ) |
13 |
|
dchrisum0flb.4 |
⊢ ( 𝜑 → ∀ 𝑦 ∈ ( 1 ..^ 𝐴 ) if ( ( √ ‘ 𝑦 ) ∈ ℕ , 1 , 0 ) ≤ ( 𝐹 ‘ 𝑦 ) ) |
14 |
|
breq1 |
⊢ ( 1 = if ( ( √ ‘ 𝐴 ) ∈ ℕ , 1 , 0 ) → ( 1 ≤ ( ( 𝐹 ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) · ( 𝐹 ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) ↔ if ( ( √ ‘ 𝐴 ) ∈ ℕ , 1 , 0 ) ≤ ( ( 𝐹 ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) · ( 𝐹 ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) ) ) |
15 |
|
breq1 |
⊢ ( 0 = if ( ( √ ‘ 𝐴 ) ∈ ℕ , 1 , 0 ) → ( 0 ≤ ( ( 𝐹 ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) · ( 𝐹 ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) ↔ if ( ( √ ‘ 𝐴 ) ∈ ℕ , 1 , 0 ) ≤ ( ( 𝐹 ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) · ( 𝐹 ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) ) ) |
16 |
|
1t1e1 |
⊢ ( 1 · 1 ) = 1 |
17 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → 𝑃 ∈ ℙ ) |
18 |
|
nnq |
⊢ ( ( √ ‘ 𝐴 ) ∈ ℕ → ( √ ‘ 𝐴 ) ∈ ℚ ) |
19 |
18
|
adantl |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( √ ‘ 𝐴 ) ∈ ℚ ) |
20 |
|
nnne0 |
⊢ ( ( √ ‘ 𝐴 ) ∈ ℕ → ( √ ‘ 𝐴 ) ≠ 0 ) |
21 |
20
|
adantl |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( √ ‘ 𝐴 ) ≠ 0 ) |
22 |
|
2z |
⊢ 2 ∈ ℤ |
23 |
22
|
a1i |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → 2 ∈ ℤ ) |
24 |
|
pcexp |
⊢ ( ( 𝑃 ∈ ℙ ∧ ( ( √ ‘ 𝐴 ) ∈ ℚ ∧ ( √ ‘ 𝐴 ) ≠ 0 ) ∧ 2 ∈ ℤ ) → ( 𝑃 pCnt ( ( √ ‘ 𝐴 ) ↑ 2 ) ) = ( 2 · ( 𝑃 pCnt ( √ ‘ 𝐴 ) ) ) ) |
25 |
17 19 21 23 24
|
syl121anc |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( 𝑃 pCnt ( ( √ ‘ 𝐴 ) ↑ 2 ) ) = ( 2 · ( 𝑃 pCnt ( √ ‘ 𝐴 ) ) ) ) |
26 |
|
eluz2nn |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → 𝐴 ∈ ℕ ) |
27 |
10 26
|
syl |
⊢ ( 𝜑 → 𝐴 ∈ ℕ ) |
28 |
27
|
nncnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
29 |
28
|
adantr |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → 𝐴 ∈ ℂ ) |
30 |
29
|
sqsqrtd |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 ) |
31 |
30
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( 𝑃 pCnt ( ( √ ‘ 𝐴 ) ↑ 2 ) ) = ( 𝑃 pCnt 𝐴 ) ) |
32 |
|
2cnd |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → 2 ∈ ℂ ) |
33 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( √ ‘ 𝐴 ) ∈ ℕ ) |
34 |
17 33
|
pccld |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( 𝑃 pCnt ( √ ‘ 𝐴 ) ) ∈ ℕ0 ) |
35 |
34
|
nn0cnd |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( 𝑃 pCnt ( √ ‘ 𝐴 ) ) ∈ ℂ ) |
36 |
32 35
|
mulcomd |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( 2 · ( 𝑃 pCnt ( √ ‘ 𝐴 ) ) ) = ( ( 𝑃 pCnt ( √ ‘ 𝐴 ) ) · 2 ) ) |
37 |
25 31 36
|
3eqtr3rd |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( ( 𝑃 pCnt ( √ ‘ 𝐴 ) ) · 2 ) = ( 𝑃 pCnt 𝐴 ) ) |
38 |
37
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( 𝑃 ↑ ( ( 𝑃 pCnt ( √ ‘ 𝐴 ) ) · 2 ) ) = ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) |
39 |
|
prmnn |
⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ℕ ) |
40 |
17 39
|
syl |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → 𝑃 ∈ ℕ ) |
41 |
40
|
nncnd |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → 𝑃 ∈ ℂ ) |
42 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
43 |
42
|
a1i |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → 2 ∈ ℕ0 ) |
44 |
41 43 34
|
expmuld |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( 𝑃 ↑ ( ( 𝑃 pCnt ( √ ‘ 𝐴 ) ) · 2 ) ) = ( ( 𝑃 ↑ ( 𝑃 pCnt ( √ ‘ 𝐴 ) ) ) ↑ 2 ) ) |
45 |
38 44
|
eqtr3d |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) = ( ( 𝑃 ↑ ( 𝑃 pCnt ( √ ‘ 𝐴 ) ) ) ↑ 2 ) ) |
46 |
45
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( √ ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) = ( √ ‘ ( ( 𝑃 ↑ ( 𝑃 pCnt ( √ ‘ 𝐴 ) ) ) ↑ 2 ) ) ) |
47 |
40 34
|
nnexpcld |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( 𝑃 ↑ ( 𝑃 pCnt ( √ ‘ 𝐴 ) ) ) ∈ ℕ ) |
48 |
47
|
nnrpd |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( 𝑃 ↑ ( 𝑃 pCnt ( √ ‘ 𝐴 ) ) ) ∈ ℝ+ ) |
49 |
48
|
rprege0d |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( ( 𝑃 ↑ ( 𝑃 pCnt ( √ ‘ 𝐴 ) ) ) ∈ ℝ ∧ 0 ≤ ( 𝑃 ↑ ( 𝑃 pCnt ( √ ‘ 𝐴 ) ) ) ) ) |
50 |
|
sqrtsq |
⊢ ( ( ( 𝑃 ↑ ( 𝑃 pCnt ( √ ‘ 𝐴 ) ) ) ∈ ℝ ∧ 0 ≤ ( 𝑃 ↑ ( 𝑃 pCnt ( √ ‘ 𝐴 ) ) ) ) → ( √ ‘ ( ( 𝑃 ↑ ( 𝑃 pCnt ( √ ‘ 𝐴 ) ) ) ↑ 2 ) ) = ( 𝑃 ↑ ( 𝑃 pCnt ( √ ‘ 𝐴 ) ) ) ) |
51 |
49 50
|
syl |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( √ ‘ ( ( 𝑃 ↑ ( 𝑃 pCnt ( √ ‘ 𝐴 ) ) ) ↑ 2 ) ) = ( 𝑃 ↑ ( 𝑃 pCnt ( √ ‘ 𝐴 ) ) ) ) |
52 |
46 51
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( √ ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) = ( 𝑃 ↑ ( 𝑃 pCnt ( √ ‘ 𝐴 ) ) ) ) |
53 |
52 47
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( √ ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℕ ) |
54 |
53
|
iftrued |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → if ( ( √ ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℕ , 1 , 0 ) = 1 ) |
55 |
11 27
|
pccld |
⊢ ( 𝜑 → ( 𝑃 pCnt 𝐴 ) ∈ ℕ0 ) |
56 |
1 2 3 4 5 6 7 8 9 11 55
|
dchrisum0flblem1 |
⊢ ( 𝜑 → if ( ( √ ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℕ , 1 , 0 ) ≤ ( 𝐹 ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) |
57 |
56
|
adantr |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → if ( ( √ ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℕ , 1 , 0 ) ≤ ( 𝐹 ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) |
58 |
54 57
|
eqbrtrrd |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → 1 ≤ ( 𝐹 ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) |
59 |
|
pcdvds |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ ) → ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ∥ 𝐴 ) |
60 |
11 27 59
|
syl2anc |
⊢ ( 𝜑 → ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ∥ 𝐴 ) |
61 |
11 39
|
syl |
⊢ ( 𝜑 → 𝑃 ∈ ℕ ) |
62 |
61 55
|
nnexpcld |
⊢ ( 𝜑 → ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ∈ ℕ ) |
63 |
|
nndivdvds |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ∈ ℕ ) → ( ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ∥ 𝐴 ↔ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℕ ) ) |
64 |
27 62 63
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ∥ 𝐴 ↔ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℕ ) ) |
65 |
60 64
|
mpbid |
⊢ ( 𝜑 → ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℕ ) |
66 |
65
|
nnzd |
⊢ ( 𝜑 → ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℤ ) |
67 |
66
|
adantr |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℤ ) |
68 |
27
|
adantr |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → 𝐴 ∈ ℕ ) |
69 |
68
|
nnrpd |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → 𝐴 ∈ ℝ+ ) |
70 |
69
|
rprege0d |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ) |
71 |
62
|
adantr |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ∈ ℕ ) |
72 |
71
|
nnrpd |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ∈ ℝ+ ) |
73 |
|
sqrtdiv |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ∈ ℝ+ ) → ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) = ( ( √ ‘ 𝐴 ) / ( √ ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) |
74 |
70 72 73
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) = ( ( √ ‘ 𝐴 ) / ( √ ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) |
75 |
|
nnz |
⊢ ( ( √ ‘ 𝐴 ) ∈ ℕ → ( √ ‘ 𝐴 ) ∈ ℤ ) |
76 |
|
znq |
⊢ ( ( ( √ ‘ 𝐴 ) ∈ ℤ ∧ ( √ ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℕ ) → ( ( √ ‘ 𝐴 ) / ( √ ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℚ ) |
77 |
75 53 76
|
syl2an2 |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( ( √ ‘ 𝐴 ) / ( √ ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℚ ) |
78 |
74 77
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℚ ) |
79 |
|
zsqrtelqelz |
⊢ ( ( ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℤ ∧ ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℚ ) → ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℤ ) |
80 |
67 78 79
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℤ ) |
81 |
65
|
adantr |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℕ ) |
82 |
81
|
nnrpd |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℝ+ ) |
83 |
82
|
sqrtgt0d |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → 0 < ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) |
84 |
|
elnnz |
⊢ ( ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℕ ↔ ( ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℤ ∧ 0 < ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) ) |
85 |
80 83 84
|
sylanbrc |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℕ ) |
86 |
85
|
iftrued |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → if ( ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℕ , 1 , 0 ) = 1 ) |
87 |
|
fveq2 |
⊢ ( 𝑦 = ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) → ( √ ‘ 𝑦 ) = ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) |
88 |
87
|
eleq1d |
⊢ ( 𝑦 = ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) → ( ( √ ‘ 𝑦 ) ∈ ℕ ↔ ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℕ ) ) |
89 |
88
|
ifbid |
⊢ ( 𝑦 = ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) → if ( ( √ ‘ 𝑦 ) ∈ ℕ , 1 , 0 ) = if ( ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℕ , 1 , 0 ) ) |
90 |
|
fveq2 |
⊢ ( 𝑦 = ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) |
91 |
89 90
|
breq12d |
⊢ ( 𝑦 = ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) → ( if ( ( √ ‘ 𝑦 ) ∈ ℕ , 1 , 0 ) ≤ ( 𝐹 ‘ 𝑦 ) ↔ if ( ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℕ , 1 , 0 ) ≤ ( 𝐹 ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) ) |
92 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
93 |
65 92
|
eleqtrdi |
⊢ ( 𝜑 → ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ( ℤ≥ ‘ 1 ) ) |
94 |
27
|
nnzd |
⊢ ( 𝜑 → 𝐴 ∈ ℤ ) |
95 |
61
|
nnred |
⊢ ( 𝜑 → 𝑃 ∈ ℝ ) |
96 |
|
pcelnn |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ ) → ( ( 𝑃 pCnt 𝐴 ) ∈ ℕ ↔ 𝑃 ∥ 𝐴 ) ) |
97 |
11 27 96
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝑃 pCnt 𝐴 ) ∈ ℕ ↔ 𝑃 ∥ 𝐴 ) ) |
98 |
12 97
|
mpbird |
⊢ ( 𝜑 → ( 𝑃 pCnt 𝐴 ) ∈ ℕ ) |
99 |
|
prmuz2 |
⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ( ℤ≥ ‘ 2 ) ) |
100 |
|
eluz2gt1 |
⊢ ( 𝑃 ∈ ( ℤ≥ ‘ 2 ) → 1 < 𝑃 ) |
101 |
11 99 100
|
3syl |
⊢ ( 𝜑 → 1 < 𝑃 ) |
102 |
|
expgt1 |
⊢ ( ( 𝑃 ∈ ℝ ∧ ( 𝑃 pCnt 𝐴 ) ∈ ℕ ∧ 1 < 𝑃 ) → 1 < ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) |
103 |
95 98 101 102
|
syl3anc |
⊢ ( 𝜑 → 1 < ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) |
104 |
|
1red |
⊢ ( 𝜑 → 1 ∈ ℝ ) |
105 |
|
0lt1 |
⊢ 0 < 1 |
106 |
105
|
a1i |
⊢ ( 𝜑 → 0 < 1 ) |
107 |
62
|
nnred |
⊢ ( 𝜑 → ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ∈ ℝ ) |
108 |
62
|
nngt0d |
⊢ ( 𝜑 → 0 < ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) |
109 |
27
|
nnred |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
110 |
27
|
nngt0d |
⊢ ( 𝜑 → 0 < 𝐴 ) |
111 |
|
ltdiv2 |
⊢ ( ( ( 1 ∈ ℝ ∧ 0 < 1 ) ∧ ( ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ∈ ℝ ∧ 0 < ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∧ ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ) → ( 1 < ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ↔ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) < ( 𝐴 / 1 ) ) ) |
112 |
104 106 107 108 109 110 111
|
syl222anc |
⊢ ( 𝜑 → ( 1 < ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ↔ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) < ( 𝐴 / 1 ) ) ) |
113 |
103 112
|
mpbid |
⊢ ( 𝜑 → ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) < ( 𝐴 / 1 ) ) |
114 |
28
|
div1d |
⊢ ( 𝜑 → ( 𝐴 / 1 ) = 𝐴 ) |
115 |
113 114
|
breqtrd |
⊢ ( 𝜑 → ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) < 𝐴 ) |
116 |
|
elfzo2 |
⊢ ( ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ( 1 ..^ 𝐴 ) ↔ ( ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ( ℤ≥ ‘ 1 ) ∧ 𝐴 ∈ ℤ ∧ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) < 𝐴 ) ) |
117 |
93 94 115 116
|
syl3anbrc |
⊢ ( 𝜑 → ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ( 1 ..^ 𝐴 ) ) |
118 |
91 13 117
|
rspcdva |
⊢ ( 𝜑 → if ( ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℕ , 1 , 0 ) ≤ ( 𝐹 ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) |
119 |
118
|
adantr |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → if ( ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℕ , 1 , 0 ) ≤ ( 𝐹 ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) |
120 |
86 119
|
eqbrtrrd |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → 1 ≤ ( 𝐹 ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) |
121 |
|
1re |
⊢ 1 ∈ ℝ |
122 |
|
0le1 |
⊢ 0 ≤ 1 |
123 |
121 122
|
pm3.2i |
⊢ ( 1 ∈ ℝ ∧ 0 ≤ 1 ) |
124 |
123
|
a1i |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( 1 ∈ ℝ ∧ 0 ≤ 1 ) ) |
125 |
1 2 3 4 5 6 7 8 9
|
dchrisum0ff |
⊢ ( 𝜑 → 𝐹 : ℕ ⟶ ℝ ) |
126 |
125 62
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℝ ) |
127 |
126
|
adantr |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( 𝐹 ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℝ ) |
128 |
125 65
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℝ ) |
129 |
128
|
adantr |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( 𝐹 ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℝ ) |
130 |
|
lemul12a |
⊢ ( ( ( ( 1 ∈ ℝ ∧ 0 ≤ 1 ) ∧ ( 𝐹 ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℝ ) ∧ ( ( 1 ∈ ℝ ∧ 0 ≤ 1 ) ∧ ( 𝐹 ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℝ ) ) → ( ( 1 ≤ ( 𝐹 ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∧ 1 ≤ ( 𝐹 ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) → ( 1 · 1 ) ≤ ( ( 𝐹 ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) · ( 𝐹 ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) ) ) |
131 |
124 127 124 129 130
|
syl22anc |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( ( 1 ≤ ( 𝐹 ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∧ 1 ≤ ( 𝐹 ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) → ( 1 · 1 ) ≤ ( ( 𝐹 ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) · ( 𝐹 ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) ) ) |
132 |
58 120 131
|
mp2and |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → ( 1 · 1 ) ≤ ( ( 𝐹 ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) · ( 𝐹 ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) ) |
133 |
16 132
|
eqbrtrrid |
⊢ ( ( 𝜑 ∧ ( √ ‘ 𝐴 ) ∈ ℕ ) → 1 ≤ ( ( 𝐹 ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) · ( 𝐹 ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) ) |
134 |
|
0red |
⊢ ( 𝜑 → 0 ∈ ℝ ) |
135 |
|
0re |
⊢ 0 ∈ ℝ |
136 |
121 135
|
ifcli |
⊢ if ( ( √ ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℕ , 1 , 0 ) ∈ ℝ |
137 |
136
|
a1i |
⊢ ( 𝜑 → if ( ( √ ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℕ , 1 , 0 ) ∈ ℝ ) |
138 |
|
breq2 |
⊢ ( 1 = if ( ( √ ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℕ , 1 , 0 ) → ( 0 ≤ 1 ↔ 0 ≤ if ( ( √ ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℕ , 1 , 0 ) ) ) |
139 |
|
breq2 |
⊢ ( 0 = if ( ( √ ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℕ , 1 , 0 ) → ( 0 ≤ 0 ↔ 0 ≤ if ( ( √ ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℕ , 1 , 0 ) ) ) |
140 |
|
0le0 |
⊢ 0 ≤ 0 |
141 |
138 139 122 140
|
keephyp |
⊢ 0 ≤ if ( ( √ ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℕ , 1 , 0 ) |
142 |
141
|
a1i |
⊢ ( 𝜑 → 0 ≤ if ( ( √ ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℕ , 1 , 0 ) ) |
143 |
134 137 126 142 56
|
letrd |
⊢ ( 𝜑 → 0 ≤ ( 𝐹 ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) |
144 |
121 135
|
ifcli |
⊢ if ( ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℕ , 1 , 0 ) ∈ ℝ |
145 |
144
|
a1i |
⊢ ( 𝜑 → if ( ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℕ , 1 , 0 ) ∈ ℝ ) |
146 |
|
breq2 |
⊢ ( 1 = if ( ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℕ , 1 , 0 ) → ( 0 ≤ 1 ↔ 0 ≤ if ( ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℕ , 1 , 0 ) ) ) |
147 |
|
breq2 |
⊢ ( 0 = if ( ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℕ , 1 , 0 ) → ( 0 ≤ 0 ↔ 0 ≤ if ( ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℕ , 1 , 0 ) ) ) |
148 |
146 147 122 140
|
keephyp |
⊢ 0 ≤ if ( ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℕ , 1 , 0 ) |
149 |
148
|
a1i |
⊢ ( 𝜑 → 0 ≤ if ( ( √ ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ∈ ℕ , 1 , 0 ) ) |
150 |
134 145 128 149 118
|
letrd |
⊢ ( 𝜑 → 0 ≤ ( 𝐹 ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) |
151 |
126 128 143 150
|
mulge0d |
⊢ ( 𝜑 → 0 ≤ ( ( 𝐹 ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) · ( 𝐹 ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) ) |
152 |
151
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( √ ‘ 𝐴 ) ∈ ℕ ) → 0 ≤ ( ( 𝐹 ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) · ( 𝐹 ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) ) |
153 |
14 15 133 152
|
ifbothda |
⊢ ( 𝜑 → if ( ( √ ‘ 𝐴 ) ∈ ℕ , 1 , 0 ) ≤ ( ( 𝐹 ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) · ( 𝐹 ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) ) |
154 |
62
|
nncnd |
⊢ ( 𝜑 → ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ∈ ℂ ) |
155 |
62
|
nnne0d |
⊢ ( 𝜑 → ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ≠ 0 ) |
156 |
28 154 155
|
divcan2d |
⊢ ( 𝜑 → ( ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) · ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) = 𝐴 ) |
157 |
156
|
fveq2d |
⊢ ( 𝜑 → ( 𝐹 ‘ ( ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) · ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) = ( 𝐹 ‘ 𝐴 ) ) |
158 |
|
pcndvds2 |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ ) → ¬ 𝑃 ∥ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) |
159 |
11 27 158
|
syl2anc |
⊢ ( 𝜑 → ¬ 𝑃 ∥ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) |
160 |
|
coprm |
⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℤ ) → ( ¬ 𝑃 ∥ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ↔ ( 𝑃 gcd ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) = 1 ) ) |
161 |
11 66 160
|
syl2anc |
⊢ ( 𝜑 → ( ¬ 𝑃 ∥ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ↔ ( 𝑃 gcd ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) = 1 ) ) |
162 |
159 161
|
mpbid |
⊢ ( 𝜑 → ( 𝑃 gcd ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) = 1 ) |
163 |
|
prmz |
⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ℤ ) |
164 |
11 163
|
syl |
⊢ ( 𝜑 → 𝑃 ∈ ℤ ) |
165 |
|
rpexp1i |
⊢ ( ( 𝑃 ∈ ℤ ∧ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ∈ ℤ ∧ ( 𝑃 pCnt 𝐴 ) ∈ ℕ0 ) → ( ( 𝑃 gcd ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) = 1 → ( ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) gcd ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) = 1 ) ) |
166 |
164 66 55 165
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑃 gcd ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) = 1 → ( ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) gcd ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) = 1 ) ) |
167 |
162 166
|
mpd |
⊢ ( 𝜑 → ( ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) gcd ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) = 1 ) |
168 |
1 2 3 4 5 6 7 8 62 65 167
|
dchrisum0fmul |
⊢ ( 𝜑 → ( 𝐹 ‘ ( ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) · ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) = ( ( 𝐹 ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) · ( 𝐹 ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) ) |
169 |
157 168
|
eqtr3d |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) = ( ( 𝐹 ‘ ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) · ( 𝐹 ‘ ( 𝐴 / ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) ) ) ) |
170 |
153 169
|
breqtrrd |
⊢ ( 𝜑 → if ( ( √ ‘ 𝐴 ) ∈ ℕ , 1 , 0 ) ≤ ( 𝐹 ‘ 𝐴 ) ) |