Step |
Hyp |
Ref |
Expression |
1 |
|
bpos.1 |
|- ( ph -> N e. ( ZZ>= ` 5 ) ) |
2 |
|
bpos.2 |
|- ( ph -> -. E. p e. Prime ( N < p /\ p <_ ( 2 x. N ) ) ) |
3 |
|
bpos.3 |
|- F = ( n e. NN |-> if ( n e. Prime , ( n ^ ( n pCnt ( ( 2 x. N ) _C N ) ) ) , 1 ) ) |
4 |
|
bpos.4 |
|- K = ( |_ ` ( ( 2 x. N ) / 3 ) ) |
5 |
|
bpos.5 |
|- M = ( |_ ` ( sqrt ` ( 2 x. N ) ) ) |
6 |
|
2nn |
|- 2 e. NN |
7 |
|
5nn |
|- 5 e. NN |
8 |
|
eluznn |
|- ( ( 5 e. NN /\ N e. ( ZZ>= ` 5 ) ) -> N e. NN ) |
9 |
7 1 8
|
sylancr |
|- ( ph -> N e. NN ) |
10 |
|
nnmulcl |
|- ( ( 2 e. NN /\ N e. NN ) -> ( 2 x. N ) e. NN ) |
11 |
6 9 10
|
sylancr |
|- ( ph -> ( 2 x. N ) e. NN ) |
12 |
11
|
nnred |
|- ( ph -> ( 2 x. N ) e. RR ) |
13 |
11
|
nnrpd |
|- ( ph -> ( 2 x. N ) e. RR+ ) |
14 |
13
|
rpge0d |
|- ( ph -> 0 <_ ( 2 x. N ) ) |
15 |
12 14
|
resqrtcld |
|- ( ph -> ( sqrt ` ( 2 x. N ) ) e. RR ) |
16 |
15
|
flcld |
|- ( ph -> ( |_ ` ( sqrt ` ( 2 x. N ) ) ) e. ZZ ) |
17 |
|
sqrt9 |
|- ( sqrt ` 9 ) = 3 |
18 |
|
9re |
|- 9 e. RR |
19 |
18
|
a1i |
|- ( ph -> 9 e. RR ) |
20 |
|
10re |
|- ; 1 0 e. RR |
21 |
20
|
a1i |
|- ( ph -> ; 1 0 e. RR ) |
22 |
|
lep1 |
|- ( 9 e. RR -> 9 <_ ( 9 + 1 ) ) |
23 |
18 22
|
ax-mp |
|- 9 <_ ( 9 + 1 ) |
24 |
|
9p1e10 |
|- ( 9 + 1 ) = ; 1 0 |
25 |
23 24
|
breqtri |
|- 9 <_ ; 1 0 |
26 |
25
|
a1i |
|- ( ph -> 9 <_ ; 1 0 ) |
27 |
|
5cn |
|- 5 e. CC |
28 |
|
2cn |
|- 2 e. CC |
29 |
|
5t2e10 |
|- ( 5 x. 2 ) = ; 1 0 |
30 |
27 28 29
|
mulcomli |
|- ( 2 x. 5 ) = ; 1 0 |
31 |
|
eluzle |
|- ( N e. ( ZZ>= ` 5 ) -> 5 <_ N ) |
32 |
1 31
|
syl |
|- ( ph -> 5 <_ N ) |
33 |
9
|
nnred |
|- ( ph -> N e. RR ) |
34 |
|
5re |
|- 5 e. RR |
35 |
|
2re |
|- 2 e. RR |
36 |
|
2pos |
|- 0 < 2 |
37 |
35 36
|
pm3.2i |
|- ( 2 e. RR /\ 0 < 2 ) |
38 |
|
lemul2 |
|- ( ( 5 e. RR /\ N e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( 5 <_ N <-> ( 2 x. 5 ) <_ ( 2 x. N ) ) ) |
39 |
34 37 38
|
mp3an13 |
|- ( N e. RR -> ( 5 <_ N <-> ( 2 x. 5 ) <_ ( 2 x. N ) ) ) |
40 |
33 39
|
syl |
|- ( ph -> ( 5 <_ N <-> ( 2 x. 5 ) <_ ( 2 x. N ) ) ) |
41 |
32 40
|
mpbid |
|- ( ph -> ( 2 x. 5 ) <_ ( 2 x. N ) ) |
42 |
30 41
|
eqbrtrrid |
|- ( ph -> ; 1 0 <_ ( 2 x. N ) ) |
43 |
19 21 12 26 42
|
letrd |
|- ( ph -> 9 <_ ( 2 x. N ) ) |
44 |
|
0re |
|- 0 e. RR |
45 |
|
9pos |
|- 0 < 9 |
46 |
44 18 45
|
ltleii |
|- 0 <_ 9 |
47 |
18 46
|
pm3.2i |
|- ( 9 e. RR /\ 0 <_ 9 ) |
48 |
13
|
rprege0d |
|- ( ph -> ( ( 2 x. N ) e. RR /\ 0 <_ ( 2 x. N ) ) ) |
49 |
|
sqrtle |
|- ( ( ( 9 e. RR /\ 0 <_ 9 ) /\ ( ( 2 x. N ) e. RR /\ 0 <_ ( 2 x. N ) ) ) -> ( 9 <_ ( 2 x. N ) <-> ( sqrt ` 9 ) <_ ( sqrt ` ( 2 x. N ) ) ) ) |
50 |
47 48 49
|
sylancr |
|- ( ph -> ( 9 <_ ( 2 x. N ) <-> ( sqrt ` 9 ) <_ ( sqrt ` ( 2 x. N ) ) ) ) |
51 |
43 50
|
mpbid |
|- ( ph -> ( sqrt ` 9 ) <_ ( sqrt ` ( 2 x. N ) ) ) |
52 |
17 51
|
eqbrtrrid |
|- ( ph -> 3 <_ ( sqrt ` ( 2 x. N ) ) ) |
53 |
|
3z |
|- 3 e. ZZ |
54 |
|
flge |
|- ( ( ( sqrt ` ( 2 x. N ) ) e. RR /\ 3 e. ZZ ) -> ( 3 <_ ( sqrt ` ( 2 x. N ) ) <-> 3 <_ ( |_ ` ( sqrt ` ( 2 x. N ) ) ) ) ) |
55 |
15 53 54
|
sylancl |
|- ( ph -> ( 3 <_ ( sqrt ` ( 2 x. N ) ) <-> 3 <_ ( |_ ` ( sqrt ` ( 2 x. N ) ) ) ) ) |
56 |
52 55
|
mpbid |
|- ( ph -> 3 <_ ( |_ ` ( sqrt ` ( 2 x. N ) ) ) ) |
57 |
53
|
eluz1i |
|- ( ( |_ ` ( sqrt ` ( 2 x. N ) ) ) e. ( ZZ>= ` 3 ) <-> ( ( |_ ` ( sqrt ` ( 2 x. N ) ) ) e. ZZ /\ 3 <_ ( |_ ` ( sqrt ` ( 2 x. N ) ) ) ) ) |
58 |
16 56 57
|
sylanbrc |
|- ( ph -> ( |_ ` ( sqrt ` ( 2 x. N ) ) ) e. ( ZZ>= ` 3 ) ) |
59 |
|
3nn |
|- 3 e. NN |
60 |
|
nndivre |
|- ( ( ( 2 x. N ) e. RR /\ 3 e. NN ) -> ( ( 2 x. N ) / 3 ) e. RR ) |
61 |
12 59 60
|
sylancl |
|- ( ph -> ( ( 2 x. N ) / 3 ) e. RR ) |
62 |
|
3re |
|- 3 e. RR |
63 |
62
|
a1i |
|- ( ph -> 3 e. RR ) |
64 |
13
|
sqrtgt0d |
|- ( ph -> 0 < ( sqrt ` ( 2 x. N ) ) ) |
65 |
|
lemul2 |
|- ( ( 3 e. RR /\ ( sqrt ` ( 2 x. N ) ) e. RR /\ ( ( sqrt ` ( 2 x. N ) ) e. RR /\ 0 < ( sqrt ` ( 2 x. N ) ) ) ) -> ( 3 <_ ( sqrt ` ( 2 x. N ) ) <-> ( ( sqrt ` ( 2 x. N ) ) x. 3 ) <_ ( ( sqrt ` ( 2 x. N ) ) x. ( sqrt ` ( 2 x. N ) ) ) ) ) |
66 |
63 15 15 64 65
|
syl112anc |
|- ( ph -> ( 3 <_ ( sqrt ` ( 2 x. N ) ) <-> ( ( sqrt ` ( 2 x. N ) ) x. 3 ) <_ ( ( sqrt ` ( 2 x. N ) ) x. ( sqrt ` ( 2 x. N ) ) ) ) ) |
67 |
52 66
|
mpbid |
|- ( ph -> ( ( sqrt ` ( 2 x. N ) ) x. 3 ) <_ ( ( sqrt ` ( 2 x. N ) ) x. ( sqrt ` ( 2 x. N ) ) ) ) |
68 |
|
remsqsqrt |
|- ( ( ( 2 x. N ) e. RR /\ 0 <_ ( 2 x. N ) ) -> ( ( sqrt ` ( 2 x. N ) ) x. ( sqrt ` ( 2 x. N ) ) ) = ( 2 x. N ) ) |
69 |
12 14 68
|
syl2anc |
|- ( ph -> ( ( sqrt ` ( 2 x. N ) ) x. ( sqrt ` ( 2 x. N ) ) ) = ( 2 x. N ) ) |
70 |
67 69
|
breqtrd |
|- ( ph -> ( ( sqrt ` ( 2 x. N ) ) x. 3 ) <_ ( 2 x. N ) ) |
71 |
|
3pos |
|- 0 < 3 |
72 |
62 71
|
pm3.2i |
|- ( 3 e. RR /\ 0 < 3 ) |
73 |
72
|
a1i |
|- ( ph -> ( 3 e. RR /\ 0 < 3 ) ) |
74 |
|
lemuldiv |
|- ( ( ( sqrt ` ( 2 x. N ) ) e. RR /\ ( 2 x. N ) e. RR /\ ( 3 e. RR /\ 0 < 3 ) ) -> ( ( ( sqrt ` ( 2 x. N ) ) x. 3 ) <_ ( 2 x. N ) <-> ( sqrt ` ( 2 x. N ) ) <_ ( ( 2 x. N ) / 3 ) ) ) |
75 |
15 12 73 74
|
syl3anc |
|- ( ph -> ( ( ( sqrt ` ( 2 x. N ) ) x. 3 ) <_ ( 2 x. N ) <-> ( sqrt ` ( 2 x. N ) ) <_ ( ( 2 x. N ) / 3 ) ) ) |
76 |
70 75
|
mpbid |
|- ( ph -> ( sqrt ` ( 2 x. N ) ) <_ ( ( 2 x. N ) / 3 ) ) |
77 |
|
flword2 |
|- ( ( ( sqrt ` ( 2 x. N ) ) e. RR /\ ( ( 2 x. N ) / 3 ) e. RR /\ ( sqrt ` ( 2 x. N ) ) <_ ( ( 2 x. N ) / 3 ) ) -> ( |_ ` ( ( 2 x. N ) / 3 ) ) e. ( ZZ>= ` ( |_ ` ( sqrt ` ( 2 x. N ) ) ) ) ) |
78 |
15 61 76 77
|
syl3anc |
|- ( ph -> ( |_ ` ( ( 2 x. N ) / 3 ) ) e. ( ZZ>= ` ( |_ ` ( sqrt ` ( 2 x. N ) ) ) ) ) |
79 |
|
elfzuzb |
|- ( ( |_ ` ( sqrt ` ( 2 x. N ) ) ) e. ( 3 ... ( |_ ` ( ( 2 x. N ) / 3 ) ) ) <-> ( ( |_ ` ( sqrt ` ( 2 x. N ) ) ) e. ( ZZ>= ` 3 ) /\ ( |_ ` ( ( 2 x. N ) / 3 ) ) e. ( ZZ>= ` ( |_ ` ( sqrt ` ( 2 x. N ) ) ) ) ) ) |
80 |
58 78 79
|
sylanbrc |
|- ( ph -> ( |_ ` ( sqrt ` ( 2 x. N ) ) ) e. ( 3 ... ( |_ ` ( ( 2 x. N ) / 3 ) ) ) ) |
81 |
4
|
oveq2i |
|- ( 3 ... K ) = ( 3 ... ( |_ ` ( ( 2 x. N ) / 3 ) ) ) |
82 |
80 5 81
|
3eltr4g |
|- ( ph -> M e. ( 3 ... K ) ) |