Step |
Hyp |
Ref |
Expression |
1 |
|
fveq2 |
β’ ( π₯ = 0 β ( β€β₯ β π₯ ) = ( β€β₯ β 0 ) ) |
2 |
|
fveq2 |
β’ ( π₯ = 0 β ( ! β π₯ ) = ( ! β 0 ) ) |
3 |
2
|
oveq2d |
β’ ( π₯ = 0 β ( π pCnt ( ! β π₯ ) ) = ( π pCnt ( ! β 0 ) ) ) |
4 |
|
fvoveq1 |
β’ ( π₯ = 0 β ( β β ( π₯ / ( π β π ) ) ) = ( β β ( 0 / ( π β π ) ) ) ) |
5 |
4
|
sumeq2sdv |
β’ ( π₯ = 0 β Ξ£ π β ( 1 ... π ) ( β β ( π₯ / ( π β π ) ) ) = Ξ£ π β ( 1 ... π ) ( β β ( 0 / ( π β π ) ) ) ) |
6 |
3 5
|
eqeq12d |
β’ ( π₯ = 0 β ( ( π pCnt ( ! β π₯ ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π₯ / ( π β π ) ) ) β ( π pCnt ( ! β 0 ) ) = Ξ£ π β ( 1 ... π ) ( β β ( 0 / ( π β π ) ) ) ) ) |
7 |
1 6
|
raleqbidv |
β’ ( π₯ = 0 β ( β π β ( β€β₯ β π₯ ) ( π pCnt ( ! β π₯ ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π₯ / ( π β π ) ) ) β β π β ( β€β₯ β 0 ) ( π pCnt ( ! β 0 ) ) = Ξ£ π β ( 1 ... π ) ( β β ( 0 / ( π β π ) ) ) ) ) |
8 |
7
|
imbi2d |
β’ ( π₯ = 0 β ( ( π β β β β π β ( β€β₯ β π₯ ) ( π pCnt ( ! β π₯ ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π₯ / ( π β π ) ) ) ) β ( π β β β β π β ( β€β₯ β 0 ) ( π pCnt ( ! β 0 ) ) = Ξ£ π β ( 1 ... π ) ( β β ( 0 / ( π β π ) ) ) ) ) ) |
9 |
|
fveq2 |
β’ ( π₯ = π β ( β€β₯ β π₯ ) = ( β€β₯ β π ) ) |
10 |
|
fveq2 |
β’ ( π₯ = π β ( ! β π₯ ) = ( ! β π ) ) |
11 |
10
|
oveq2d |
β’ ( π₯ = π β ( π pCnt ( ! β π₯ ) ) = ( π pCnt ( ! β π ) ) ) |
12 |
|
fvoveq1 |
β’ ( π₯ = π β ( β β ( π₯ / ( π β π ) ) ) = ( β β ( π / ( π β π ) ) ) ) |
13 |
12
|
sumeq2sdv |
β’ ( π₯ = π β Ξ£ π β ( 1 ... π ) ( β β ( π₯ / ( π β π ) ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) ) |
14 |
11 13
|
eqeq12d |
β’ ( π₯ = π β ( ( π pCnt ( ! β π₯ ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π₯ / ( π β π ) ) ) β ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) ) ) |
15 |
9 14
|
raleqbidv |
β’ ( π₯ = π β ( β π β ( β€β₯ β π₯ ) ( π pCnt ( ! β π₯ ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π₯ / ( π β π ) ) ) β β π β ( β€β₯ β π ) ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) ) ) |
16 |
15
|
imbi2d |
β’ ( π₯ = π β ( ( π β β β β π β ( β€β₯ β π₯ ) ( π pCnt ( ! β π₯ ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π₯ / ( π β π ) ) ) ) β ( π β β β β π β ( β€β₯ β π ) ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) ) ) ) |
17 |
|
fveq2 |
β’ ( π₯ = ( π + 1 ) β ( β€β₯ β π₯ ) = ( β€β₯ β ( π + 1 ) ) ) |
18 |
|
fveq2 |
β’ ( π₯ = ( π + 1 ) β ( ! β π₯ ) = ( ! β ( π + 1 ) ) ) |
19 |
18
|
oveq2d |
β’ ( π₯ = ( π + 1 ) β ( π pCnt ( ! β π₯ ) ) = ( π pCnt ( ! β ( π + 1 ) ) ) ) |
20 |
|
fvoveq1 |
β’ ( π₯ = ( π + 1 ) β ( β β ( π₯ / ( π β π ) ) ) = ( β β ( ( π + 1 ) / ( π β π ) ) ) ) |
21 |
20
|
sumeq2sdv |
β’ ( π₯ = ( π + 1 ) β Ξ£ π β ( 1 ... π ) ( β β ( π₯ / ( π β π ) ) ) = Ξ£ π β ( 1 ... π ) ( β β ( ( π + 1 ) / ( π β π ) ) ) ) |
22 |
19 21
|
eqeq12d |
β’ ( π₯ = ( π + 1 ) β ( ( π pCnt ( ! β π₯ ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π₯ / ( π β π ) ) ) β ( π pCnt ( ! β ( π + 1 ) ) ) = Ξ£ π β ( 1 ... π ) ( β β ( ( π + 1 ) / ( π β π ) ) ) ) ) |
23 |
17 22
|
raleqbidv |
β’ ( π₯ = ( π + 1 ) β ( β π β ( β€β₯ β π₯ ) ( π pCnt ( ! β π₯ ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π₯ / ( π β π ) ) ) β β π β ( β€β₯ β ( π + 1 ) ) ( π pCnt ( ! β ( π + 1 ) ) ) = Ξ£ π β ( 1 ... π ) ( β β ( ( π + 1 ) / ( π β π ) ) ) ) ) |
24 |
23
|
imbi2d |
β’ ( π₯ = ( π + 1 ) β ( ( π β β β β π β ( β€β₯ β π₯ ) ( π pCnt ( ! β π₯ ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π₯ / ( π β π ) ) ) ) β ( π β β β β π β ( β€β₯ β ( π + 1 ) ) ( π pCnt ( ! β ( π + 1 ) ) ) = Ξ£ π β ( 1 ... π ) ( β β ( ( π + 1 ) / ( π β π ) ) ) ) ) ) |
25 |
|
fveq2 |
β’ ( π₯ = π β ( β€β₯ β π₯ ) = ( β€β₯ β π ) ) |
26 |
|
fveq2 |
β’ ( π₯ = π β ( ! β π₯ ) = ( ! β π ) ) |
27 |
26
|
oveq2d |
β’ ( π₯ = π β ( π pCnt ( ! β π₯ ) ) = ( π pCnt ( ! β π ) ) ) |
28 |
|
fvoveq1 |
β’ ( π₯ = π β ( β β ( π₯ / ( π β π ) ) ) = ( β β ( π / ( π β π ) ) ) ) |
29 |
28
|
sumeq2sdv |
β’ ( π₯ = π β Ξ£ π β ( 1 ... π ) ( β β ( π₯ / ( π β π ) ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) ) |
30 |
27 29
|
eqeq12d |
β’ ( π₯ = π β ( ( π pCnt ( ! β π₯ ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π₯ / ( π β π ) ) ) β ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) ) ) |
31 |
25 30
|
raleqbidv |
β’ ( π₯ = π β ( β π β ( β€β₯ β π₯ ) ( π pCnt ( ! β π₯ ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π₯ / ( π β π ) ) ) β β π β ( β€β₯ β π ) ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) ) ) |
32 |
31
|
imbi2d |
β’ ( π₯ = π β ( ( π β β β β π β ( β€β₯ β π₯ ) ( π pCnt ( ! β π₯ ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π₯ / ( π β π ) ) ) ) β ( π β β β β π β ( β€β₯ β π ) ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) ) ) ) |
33 |
|
fzfid |
β’ ( ( π β β β§ π β ( β€β₯ β 0 ) ) β ( 1 ... π ) β Fin ) |
34 |
|
sumz |
β’ ( ( ( 1 ... π ) β ( β€β₯ β 1 ) β¨ ( 1 ... π ) β Fin ) β Ξ£ π β ( 1 ... π ) 0 = 0 ) |
35 |
34
|
olcs |
β’ ( ( 1 ... π ) β Fin β Ξ£ π β ( 1 ... π ) 0 = 0 ) |
36 |
33 35
|
syl |
β’ ( ( π β β β§ π β ( β€β₯ β 0 ) ) β Ξ£ π β ( 1 ... π ) 0 = 0 ) |
37 |
|
0nn0 |
β’ 0 β β0 |
38 |
|
elfznn |
β’ ( π β ( 1 ... π ) β π β β ) |
39 |
38
|
nnnn0d |
β’ ( π β ( 1 ... π ) β π β β0 ) |
40 |
|
nn0uz |
β’ β0 = ( β€β₯ β 0 ) |
41 |
39 40
|
eleqtrdi |
β’ ( π β ( 1 ... π ) β π β ( β€β₯ β 0 ) ) |
42 |
41
|
adantl |
β’ ( ( ( π β β β§ π β ( β€β₯ β 0 ) ) β§ π β ( 1 ... π ) ) β π β ( β€β₯ β 0 ) ) |
43 |
|
simpll |
β’ ( ( ( π β β β§ π β ( β€β₯ β 0 ) ) β§ π β ( 1 ... π ) ) β π β β ) |
44 |
|
pcfaclem |
β’ ( ( 0 β β0 β§ π β ( β€β₯ β 0 ) β§ π β β ) β ( β β ( 0 / ( π β π ) ) ) = 0 ) |
45 |
37 42 43 44
|
mp3an2i |
β’ ( ( ( π β β β§ π β ( β€β₯ β 0 ) ) β§ π β ( 1 ... π ) ) β ( β β ( 0 / ( π β π ) ) ) = 0 ) |
46 |
45
|
sumeq2dv |
β’ ( ( π β β β§ π β ( β€β₯ β 0 ) ) β Ξ£ π β ( 1 ... π ) ( β β ( 0 / ( π β π ) ) ) = Ξ£ π β ( 1 ... π ) 0 ) |
47 |
|
fac0 |
β’ ( ! β 0 ) = 1 |
48 |
47
|
oveq2i |
β’ ( π pCnt ( ! β 0 ) ) = ( π pCnt 1 ) |
49 |
|
pc1 |
β’ ( π β β β ( π pCnt 1 ) = 0 ) |
50 |
48 49
|
eqtrid |
β’ ( π β β β ( π pCnt ( ! β 0 ) ) = 0 ) |
51 |
50
|
adantr |
β’ ( ( π β β β§ π β ( β€β₯ β 0 ) ) β ( π pCnt ( ! β 0 ) ) = 0 ) |
52 |
36 46 51
|
3eqtr4rd |
β’ ( ( π β β β§ π β ( β€β₯ β 0 ) ) β ( π pCnt ( ! β 0 ) ) = Ξ£ π β ( 1 ... π ) ( β β ( 0 / ( π β π ) ) ) ) |
53 |
52
|
ralrimiva |
β’ ( π β β β β π β ( β€β₯ β 0 ) ( π pCnt ( ! β 0 ) ) = Ξ£ π β ( 1 ... π ) ( β β ( 0 / ( π β π ) ) ) ) |
54 |
|
nn0z |
β’ ( π β β0 β π β β€ ) |
55 |
54
|
adantr |
β’ ( ( π β β0 β§ π β β ) β π β β€ ) |
56 |
|
uzid |
β’ ( π β β€ β π β ( β€β₯ β π ) ) |
57 |
|
peano2uz |
β’ ( π β ( β€β₯ β π ) β ( π + 1 ) β ( β€β₯ β π ) ) |
58 |
55 56 57
|
3syl |
β’ ( ( π β β0 β§ π β β ) β ( π + 1 ) β ( β€β₯ β π ) ) |
59 |
|
uzss |
β’ ( ( π + 1 ) β ( β€β₯ β π ) β ( β€β₯ β ( π + 1 ) ) β ( β€β₯ β π ) ) |
60 |
|
ssralv |
β’ ( ( β€β₯ β ( π + 1 ) ) β ( β€β₯ β π ) β ( β π β ( β€β₯ β π ) ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) β β π β ( β€β₯ β ( π + 1 ) ) ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) ) ) |
61 |
58 59 60
|
3syl |
β’ ( ( π β β0 β§ π β β ) β ( β π β ( β€β₯ β π ) ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) β β π β ( β€β₯ β ( π + 1 ) ) ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) ) ) |
62 |
|
oveq1 |
β’ ( ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) β ( ( π pCnt ( ! β π ) ) + ( π pCnt ( π + 1 ) ) ) = ( Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) + ( π pCnt ( π + 1 ) ) ) ) |
63 |
|
simpll |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β π β β0 ) |
64 |
|
facp1 |
β’ ( π β β0 β ( ! β ( π + 1 ) ) = ( ( ! β π ) Β· ( π + 1 ) ) ) |
65 |
63 64
|
syl |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( ! β ( π + 1 ) ) = ( ( ! β π ) Β· ( π + 1 ) ) ) |
66 |
65
|
oveq2d |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( π pCnt ( ! β ( π + 1 ) ) ) = ( π pCnt ( ( ! β π ) Β· ( π + 1 ) ) ) ) |
67 |
|
simplr |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β π β β ) |
68 |
|
faccl |
β’ ( π β β0 β ( ! β π ) β β ) |
69 |
|
nnz |
β’ ( ( ! β π ) β β β ( ! β π ) β β€ ) |
70 |
|
nnne0 |
β’ ( ( ! β π ) β β β ( ! β π ) β 0 ) |
71 |
69 70
|
jca |
β’ ( ( ! β π ) β β β ( ( ! β π ) β β€ β§ ( ! β π ) β 0 ) ) |
72 |
63 68 71
|
3syl |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( ( ! β π ) β β€ β§ ( ! β π ) β 0 ) ) |
73 |
|
nn0p1nn |
β’ ( π β β0 β ( π + 1 ) β β ) |
74 |
|
nnz |
β’ ( ( π + 1 ) β β β ( π + 1 ) β β€ ) |
75 |
|
nnne0 |
β’ ( ( π + 1 ) β β β ( π + 1 ) β 0 ) |
76 |
74 75
|
jca |
β’ ( ( π + 1 ) β β β ( ( π + 1 ) β β€ β§ ( π + 1 ) β 0 ) ) |
77 |
63 73 76
|
3syl |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( ( π + 1 ) β β€ β§ ( π + 1 ) β 0 ) ) |
78 |
|
pcmul |
β’ ( ( π β β β§ ( ( ! β π ) β β€ β§ ( ! β π ) β 0 ) β§ ( ( π + 1 ) β β€ β§ ( π + 1 ) β 0 ) ) β ( π pCnt ( ( ! β π ) Β· ( π + 1 ) ) ) = ( ( π pCnt ( ! β π ) ) + ( π pCnt ( π + 1 ) ) ) ) |
79 |
67 72 77 78
|
syl3anc |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( π pCnt ( ( ! β π ) Β· ( π + 1 ) ) ) = ( ( π pCnt ( ! β π ) ) + ( π pCnt ( π + 1 ) ) ) ) |
80 |
66 79
|
eqtr2d |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( ( π pCnt ( ! β π ) ) + ( π pCnt ( π + 1 ) ) ) = ( π pCnt ( ! β ( π + 1 ) ) ) ) |
81 |
63
|
adantr |
β’ ( ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β§ π β ( 1 ... π ) ) β π β β0 ) |
82 |
81
|
nn0zd |
β’ ( ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β§ π β ( 1 ... π ) ) β π β β€ ) |
83 |
|
prmnn |
β’ ( π β β β π β β ) |
84 |
83
|
ad2antlr |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β π β β ) |
85 |
|
nnexpcl |
β’ ( ( π β β β§ π β β0 ) β ( π β π ) β β ) |
86 |
84 39 85
|
syl2an |
β’ ( ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β§ π β ( 1 ... π ) ) β ( π β π ) β β ) |
87 |
|
fldivp1 |
β’ ( ( π β β€ β§ ( π β π ) β β ) β ( ( β β ( ( π + 1 ) / ( π β π ) ) ) β ( β β ( π / ( π β π ) ) ) ) = if ( ( π β π ) β₯ ( π + 1 ) , 1 , 0 ) ) |
88 |
82 86 87
|
syl2anc |
β’ ( ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β§ π β ( 1 ... π ) ) β ( ( β β ( ( π + 1 ) / ( π β π ) ) ) β ( β β ( π / ( π β π ) ) ) ) = if ( ( π β π ) β₯ ( π + 1 ) , 1 , 0 ) ) |
89 |
|
elfzuz |
β’ ( π β ( 1 ... π ) β π β ( β€β₯ β 1 ) ) |
90 |
63 73
|
syl |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( π + 1 ) β β ) |
91 |
67 90
|
pccld |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( π pCnt ( π + 1 ) ) β β0 ) |
92 |
91
|
nn0zd |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( π pCnt ( π + 1 ) ) β β€ ) |
93 |
|
elfz5 |
β’ ( ( π β ( β€β₯ β 1 ) β§ ( π pCnt ( π + 1 ) ) β β€ ) β ( π β ( 1 ... ( π pCnt ( π + 1 ) ) ) β π β€ ( π pCnt ( π + 1 ) ) ) ) |
94 |
89 92 93
|
syl2anr |
β’ ( ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β§ π β ( 1 ... π ) ) β ( π β ( 1 ... ( π pCnt ( π + 1 ) ) ) β π β€ ( π pCnt ( π + 1 ) ) ) ) |
95 |
|
simpllr |
β’ ( ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β§ π β ( 1 ... π ) ) β π β β ) |
96 |
81 73
|
syl |
β’ ( ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β§ π β ( 1 ... π ) ) β ( π + 1 ) β β ) |
97 |
96
|
nnzd |
β’ ( ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β§ π β ( 1 ... π ) ) β ( π + 1 ) β β€ ) |
98 |
39
|
adantl |
β’ ( ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β§ π β ( 1 ... π ) ) β π β β0 ) |
99 |
|
pcdvdsb |
β’ ( ( π β β β§ ( π + 1 ) β β€ β§ π β β0 ) β ( π β€ ( π pCnt ( π + 1 ) ) β ( π β π ) β₯ ( π + 1 ) ) ) |
100 |
95 97 98 99
|
syl3anc |
β’ ( ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β§ π β ( 1 ... π ) ) β ( π β€ ( π pCnt ( π + 1 ) ) β ( π β π ) β₯ ( π + 1 ) ) ) |
101 |
94 100
|
bitr2d |
β’ ( ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β§ π β ( 1 ... π ) ) β ( ( π β π ) β₯ ( π + 1 ) β π β ( 1 ... ( π pCnt ( π + 1 ) ) ) ) ) |
102 |
101
|
ifbid |
β’ ( ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β§ π β ( 1 ... π ) ) β if ( ( π β π ) β₯ ( π + 1 ) , 1 , 0 ) = if ( π β ( 1 ... ( π pCnt ( π + 1 ) ) ) , 1 , 0 ) ) |
103 |
88 102
|
eqtrd |
β’ ( ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β§ π β ( 1 ... π ) ) β ( ( β β ( ( π + 1 ) / ( π β π ) ) ) β ( β β ( π / ( π β π ) ) ) ) = if ( π β ( 1 ... ( π pCnt ( π + 1 ) ) ) , 1 , 0 ) ) |
104 |
103
|
sumeq2dv |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β Ξ£ π β ( 1 ... π ) ( ( β β ( ( π + 1 ) / ( π β π ) ) ) β ( β β ( π / ( π β π ) ) ) ) = Ξ£ π β ( 1 ... π ) if ( π β ( 1 ... ( π pCnt ( π + 1 ) ) ) , 1 , 0 ) ) |
105 |
|
fzfid |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( 1 ... π ) β Fin ) |
106 |
63
|
nn0red |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β π β β ) |
107 |
|
peano2re |
β’ ( π β β β ( π + 1 ) β β ) |
108 |
106 107
|
syl |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( π + 1 ) β β ) |
109 |
108
|
adantr |
β’ ( ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β§ π β ( 1 ... π ) ) β ( π + 1 ) β β ) |
110 |
109 86
|
nndivred |
β’ ( ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β§ π β ( 1 ... π ) ) β ( ( π + 1 ) / ( π β π ) ) β β ) |
111 |
110
|
flcld |
β’ ( ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β§ π β ( 1 ... π ) ) β ( β β ( ( π + 1 ) / ( π β π ) ) ) β β€ ) |
112 |
111
|
zcnd |
β’ ( ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β§ π β ( 1 ... π ) ) β ( β β ( ( π + 1 ) / ( π β π ) ) ) β β ) |
113 |
106
|
adantr |
β’ ( ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β§ π β ( 1 ... π ) ) β π β β ) |
114 |
113 86
|
nndivred |
β’ ( ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β§ π β ( 1 ... π ) ) β ( π / ( π β π ) ) β β ) |
115 |
114
|
flcld |
β’ ( ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β§ π β ( 1 ... π ) ) β ( β β ( π / ( π β π ) ) ) β β€ ) |
116 |
115
|
zcnd |
β’ ( ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β§ π β ( 1 ... π ) ) β ( β β ( π / ( π β π ) ) ) β β ) |
117 |
105 112 116
|
fsumsub |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β Ξ£ π β ( 1 ... π ) ( ( β β ( ( π + 1 ) / ( π β π ) ) ) β ( β β ( π / ( π β π ) ) ) ) = ( Ξ£ π β ( 1 ... π ) ( β β ( ( π + 1 ) / ( π β π ) ) ) β Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) ) ) |
118 |
|
fzfi |
β’ ( 1 ... π ) β Fin |
119 |
91
|
nn0red |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( π pCnt ( π + 1 ) ) β β ) |
120 |
|
eluzelz |
β’ ( π β ( β€β₯ β ( π + 1 ) ) β π β β€ ) |
121 |
120
|
adantl |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β π β β€ ) |
122 |
121
|
zred |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β π β β ) |
123 |
|
prmuz2 |
β’ ( π β β β π β ( β€β₯ β 2 ) ) |
124 |
123
|
ad2antlr |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β π β ( β€β₯ β 2 ) ) |
125 |
90
|
nnnn0d |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( π + 1 ) β β0 ) |
126 |
|
bernneq3 |
β’ ( ( π β ( β€β₯ β 2 ) β§ ( π + 1 ) β β0 ) β ( π + 1 ) < ( π β ( π + 1 ) ) ) |
127 |
124 125 126
|
syl2anc |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( π + 1 ) < ( π β ( π + 1 ) ) ) |
128 |
119 108
|
letrid |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( ( π pCnt ( π + 1 ) ) β€ ( π + 1 ) β¨ ( π + 1 ) β€ ( π pCnt ( π + 1 ) ) ) ) |
129 |
128
|
ord |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( Β¬ ( π pCnt ( π + 1 ) ) β€ ( π + 1 ) β ( π + 1 ) β€ ( π pCnt ( π + 1 ) ) ) ) |
130 |
90
|
nnzd |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( π + 1 ) β β€ ) |
131 |
|
pcdvdsb |
β’ ( ( π β β β§ ( π + 1 ) β β€ β§ ( π + 1 ) β β0 ) β ( ( π + 1 ) β€ ( π pCnt ( π + 1 ) ) β ( π β ( π + 1 ) ) β₯ ( π + 1 ) ) ) |
132 |
67 130 125 131
|
syl3anc |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( ( π + 1 ) β€ ( π pCnt ( π + 1 ) ) β ( π β ( π + 1 ) ) β₯ ( π + 1 ) ) ) |
133 |
84 125
|
nnexpcld |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( π β ( π + 1 ) ) β β ) |
134 |
133
|
nnzd |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( π β ( π + 1 ) ) β β€ ) |
135 |
|
dvdsle |
β’ ( ( ( π β ( π + 1 ) ) β β€ β§ ( π + 1 ) β β ) β ( ( π β ( π + 1 ) ) β₯ ( π + 1 ) β ( π β ( π + 1 ) ) β€ ( π + 1 ) ) ) |
136 |
134 90 135
|
syl2anc |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( ( π β ( π + 1 ) ) β₯ ( π + 1 ) β ( π β ( π + 1 ) ) β€ ( π + 1 ) ) ) |
137 |
133
|
nnred |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( π β ( π + 1 ) ) β β ) |
138 |
137 108
|
lenltd |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( ( π β ( π + 1 ) ) β€ ( π + 1 ) β Β¬ ( π + 1 ) < ( π β ( π + 1 ) ) ) ) |
139 |
136 138
|
sylibd |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( ( π β ( π + 1 ) ) β₯ ( π + 1 ) β Β¬ ( π + 1 ) < ( π β ( π + 1 ) ) ) ) |
140 |
132 139
|
sylbid |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( ( π + 1 ) β€ ( π pCnt ( π + 1 ) ) β Β¬ ( π + 1 ) < ( π β ( π + 1 ) ) ) ) |
141 |
129 140
|
syld |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( Β¬ ( π pCnt ( π + 1 ) ) β€ ( π + 1 ) β Β¬ ( π + 1 ) < ( π β ( π + 1 ) ) ) ) |
142 |
127 141
|
mt4d |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( π pCnt ( π + 1 ) ) β€ ( π + 1 ) ) |
143 |
|
eluzle |
β’ ( π β ( β€β₯ β ( π + 1 ) ) β ( π + 1 ) β€ π ) |
144 |
143
|
adantl |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( π + 1 ) β€ π ) |
145 |
119 108 122 142 144
|
letrd |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( π pCnt ( π + 1 ) ) β€ π ) |
146 |
|
eluz |
β’ ( ( ( π pCnt ( π + 1 ) ) β β€ β§ π β β€ ) β ( π β ( β€β₯ β ( π pCnt ( π + 1 ) ) ) β ( π pCnt ( π + 1 ) ) β€ π ) ) |
147 |
92 121 146
|
syl2anc |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( π β ( β€β₯ β ( π pCnt ( π + 1 ) ) ) β ( π pCnt ( π + 1 ) ) β€ π ) ) |
148 |
145 147
|
mpbird |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β π β ( β€β₯ β ( π pCnt ( π + 1 ) ) ) ) |
149 |
|
fzss2 |
β’ ( π β ( β€β₯ β ( π pCnt ( π + 1 ) ) ) β ( 1 ... ( π pCnt ( π + 1 ) ) ) β ( 1 ... π ) ) |
150 |
148 149
|
syl |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( 1 ... ( π pCnt ( π + 1 ) ) ) β ( 1 ... π ) ) |
151 |
|
sumhash |
β’ ( ( ( 1 ... π ) β Fin β§ ( 1 ... ( π pCnt ( π + 1 ) ) ) β ( 1 ... π ) ) β Ξ£ π β ( 1 ... π ) if ( π β ( 1 ... ( π pCnt ( π + 1 ) ) ) , 1 , 0 ) = ( β― β ( 1 ... ( π pCnt ( π + 1 ) ) ) ) ) |
152 |
118 150 151
|
sylancr |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β Ξ£ π β ( 1 ... π ) if ( π β ( 1 ... ( π pCnt ( π + 1 ) ) ) , 1 , 0 ) = ( β― β ( 1 ... ( π pCnt ( π + 1 ) ) ) ) ) |
153 |
|
hashfz1 |
β’ ( ( π pCnt ( π + 1 ) ) β β0 β ( β― β ( 1 ... ( π pCnt ( π + 1 ) ) ) ) = ( π pCnt ( π + 1 ) ) ) |
154 |
91 153
|
syl |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( β― β ( 1 ... ( π pCnt ( π + 1 ) ) ) ) = ( π pCnt ( π + 1 ) ) ) |
155 |
152 154
|
eqtrd |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β Ξ£ π β ( 1 ... π ) if ( π β ( 1 ... ( π pCnt ( π + 1 ) ) ) , 1 , 0 ) = ( π pCnt ( π + 1 ) ) ) |
156 |
104 117 155
|
3eqtr3d |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( Ξ£ π β ( 1 ... π ) ( β β ( ( π + 1 ) / ( π β π ) ) ) β Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) ) = ( π pCnt ( π + 1 ) ) ) |
157 |
105 112
|
fsumcl |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β Ξ£ π β ( 1 ... π ) ( β β ( ( π + 1 ) / ( π β π ) ) ) β β ) |
158 |
105 116
|
fsumcl |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) β β ) |
159 |
119
|
recnd |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( π pCnt ( π + 1 ) ) β β ) |
160 |
157 158 159
|
subaddd |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( ( Ξ£ π β ( 1 ... π ) ( β β ( ( π + 1 ) / ( π β π ) ) ) β Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) ) = ( π pCnt ( π + 1 ) ) β ( Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) + ( π pCnt ( π + 1 ) ) ) = Ξ£ π β ( 1 ... π ) ( β β ( ( π + 1 ) / ( π β π ) ) ) ) ) |
161 |
156 160
|
mpbid |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) + ( π pCnt ( π + 1 ) ) ) = Ξ£ π β ( 1 ... π ) ( β β ( ( π + 1 ) / ( π β π ) ) ) ) |
162 |
80 161
|
eqeq12d |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( ( ( π pCnt ( ! β π ) ) + ( π pCnt ( π + 1 ) ) ) = ( Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) + ( π pCnt ( π + 1 ) ) ) β ( π pCnt ( ! β ( π + 1 ) ) ) = Ξ£ π β ( 1 ... π ) ( β β ( ( π + 1 ) / ( π β π ) ) ) ) ) |
163 |
62 162
|
syl5ib |
β’ ( ( ( π β β0 β§ π β β ) β§ π β ( β€β₯ β ( π + 1 ) ) ) β ( ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) β ( π pCnt ( ! β ( π + 1 ) ) ) = Ξ£ π β ( 1 ... π ) ( β β ( ( π + 1 ) / ( π β π ) ) ) ) ) |
164 |
163
|
ralimdva |
β’ ( ( π β β0 β§ π β β ) β ( β π β ( β€β₯ β ( π + 1 ) ) ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) β β π β ( β€β₯ β ( π + 1 ) ) ( π pCnt ( ! β ( π + 1 ) ) ) = Ξ£ π β ( 1 ... π ) ( β β ( ( π + 1 ) / ( π β π ) ) ) ) ) |
165 |
61 164
|
syld |
β’ ( ( π β β0 β§ π β β ) β ( β π β ( β€β₯ β π ) ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) β β π β ( β€β₯ β ( π + 1 ) ) ( π pCnt ( ! β ( π + 1 ) ) ) = Ξ£ π β ( 1 ... π ) ( β β ( ( π + 1 ) / ( π β π ) ) ) ) ) |
166 |
165
|
ex |
β’ ( π β β0 β ( π β β β ( β π β ( β€β₯ β π ) ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) β β π β ( β€β₯ β ( π + 1 ) ) ( π pCnt ( ! β ( π + 1 ) ) ) = Ξ£ π β ( 1 ... π ) ( β β ( ( π + 1 ) / ( π β π ) ) ) ) ) ) |
167 |
166
|
a2d |
β’ ( π β β0 β ( ( π β β β β π β ( β€β₯ β π ) ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) ) β ( π β β β β π β ( β€β₯ β ( π + 1 ) ) ( π pCnt ( ! β ( π + 1 ) ) ) = Ξ£ π β ( 1 ... π ) ( β β ( ( π + 1 ) / ( π β π ) ) ) ) ) ) |
168 |
8 16 24 32 53 167
|
nn0ind |
β’ ( π β β0 β ( π β β β β π β ( β€β₯ β π ) ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) ) ) |
169 |
168
|
imp |
β’ ( ( π β β0 β§ π β β ) β β π β ( β€β₯ β π ) ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) ) |
170 |
|
oveq2 |
β’ ( π = π β ( 1 ... π ) = ( 1 ... π ) ) |
171 |
170
|
sumeq1d |
β’ ( π = π β Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) ) |
172 |
171
|
eqeq2d |
β’ ( π = π β ( ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) β ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) ) ) |
173 |
172
|
rspcv |
β’ ( π β ( β€β₯ β π ) β ( β π β ( β€β₯ β π ) ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) β ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) ) ) |
174 |
169 173
|
syl5 |
β’ ( π β ( β€β₯ β π ) β ( ( π β β0 β§ π β β ) β ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) ) ) |
175 |
174
|
3impib |
β’ ( ( π β ( β€β₯ β π ) β§ π β β0 β§ π β β ) β ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) ) |
176 |
175
|
3com12 |
β’ ( ( π β β0 β§ π β ( β€β₯ β π ) β§ π β β ) β ( π pCnt ( ! β π ) ) = Ξ£ π β ( 1 ... π ) ( β β ( π / ( π β π ) ) ) ) |