Step |
Hyp |
Ref |
Expression |
1 |
|
2nn |
|- 2 e. NN |
2 |
|
fpprel |
|- ( 2 e. NN -> ( X e. ( FPPr ` 2 ) <-> ( X e. ( ZZ>= ` 4 ) /\ X e/ Prime /\ ( ( 2 ^ ( X - 1 ) ) mod X ) = 1 ) ) ) |
3 |
1 2
|
mp1i |
|- ( X e. ( FPPr ` 2 ) -> ( X e. ( FPPr ` 2 ) <-> ( X e. ( ZZ>= ` 4 ) /\ X e/ Prime /\ ( ( 2 ^ ( X - 1 ) ) mod X ) = 1 ) ) ) |
4 |
|
eluz4eluz2 |
|- ( X e. ( ZZ>= ` 4 ) -> X e. ( ZZ>= ` 2 ) ) |
5 |
4
|
3ad2ant1 |
|- ( ( X e. ( ZZ>= ` 4 ) /\ X e/ Prime /\ ( ( 2 ^ ( X - 1 ) ) mod X ) = 1 ) -> X e. ( ZZ>= ` 2 ) ) |
6 |
5
|
adantl |
|- ( ( X e. ( FPPr ` 2 ) /\ ( X e. ( ZZ>= ` 4 ) /\ X e/ Prime /\ ( ( 2 ^ ( X - 1 ) ) mod X ) = 1 ) ) -> X e. ( ZZ>= ` 2 ) ) |
7 |
|
fppr2odd |
|- ( X e. ( FPPr ` 2 ) -> X e. Odd ) |
8 |
7
|
adantr |
|- ( ( X e. ( FPPr ` 2 ) /\ ( X e. ( ZZ>= ` 4 ) /\ X e/ Prime /\ ( ( 2 ^ ( X - 1 ) ) mod X ) = 1 ) ) -> X e. Odd ) |
9 |
|
simpr2 |
|- ( ( X e. ( FPPr ` 2 ) /\ ( X e. ( ZZ>= ` 4 ) /\ X e/ Prime /\ ( ( 2 ^ ( X - 1 ) ) mod X ) = 1 ) ) -> X e/ Prime ) |
10 |
6 8 9
|
3jca |
|- ( ( X e. ( FPPr ` 2 ) /\ ( X e. ( ZZ>= ` 4 ) /\ X e/ Prime /\ ( ( 2 ^ ( X - 1 ) ) mod X ) = 1 ) ) -> ( X e. ( ZZ>= ` 2 ) /\ X e. Odd /\ X e/ Prime ) ) |
11 |
|
fpprwppr |
|- ( X e. ( FPPr ` 2 ) -> ( ( 2 ^ X ) mod X ) = ( 2 mod X ) ) |
12 |
|
2re |
|- 2 e. RR |
13 |
12
|
a1i |
|- ( X e. ( ZZ>= ` 4 ) -> 2 e. RR ) |
14 |
|
eluz4nn |
|- ( X e. ( ZZ>= ` 4 ) -> X e. NN ) |
15 |
14
|
nnrpd |
|- ( X e. ( ZZ>= ` 4 ) -> X e. RR+ ) |
16 |
|
0le2 |
|- 0 <_ 2 |
17 |
16
|
a1i |
|- ( X e. ( ZZ>= ` 4 ) -> 0 <_ 2 ) |
18 |
|
eluz2 |
|- ( X e. ( ZZ>= ` 4 ) <-> ( 4 e. ZZ /\ X e. ZZ /\ 4 <_ X ) ) |
19 |
|
4z |
|- 4 e. ZZ |
20 |
|
zlem1lt |
|- ( ( 4 e. ZZ /\ X e. ZZ ) -> ( 4 <_ X <-> ( 4 - 1 ) < X ) ) |
21 |
19 20
|
mpan |
|- ( X e. ZZ -> ( 4 <_ X <-> ( 4 - 1 ) < X ) ) |
22 |
|
4m1e3 |
|- ( 4 - 1 ) = 3 |
23 |
22
|
breq1i |
|- ( ( 4 - 1 ) < X <-> 3 < X ) |
24 |
12
|
a1i |
|- ( ( X e. ZZ /\ 3 < X ) -> 2 e. RR ) |
25 |
|
3re |
|- 3 e. RR |
26 |
25
|
a1i |
|- ( ( X e. ZZ /\ 3 < X ) -> 3 e. RR ) |
27 |
|
zre |
|- ( X e. ZZ -> X e. RR ) |
28 |
27
|
adantr |
|- ( ( X e. ZZ /\ 3 < X ) -> X e. RR ) |
29 |
|
2lt3 |
|- 2 < 3 |
30 |
29
|
a1i |
|- ( ( X e. ZZ /\ 3 < X ) -> 2 < 3 ) |
31 |
|
simpr |
|- ( ( X e. ZZ /\ 3 < X ) -> 3 < X ) |
32 |
24 26 28 30 31
|
lttrd |
|- ( ( X e. ZZ /\ 3 < X ) -> 2 < X ) |
33 |
32
|
ex |
|- ( X e. ZZ -> ( 3 < X -> 2 < X ) ) |
34 |
23 33
|
syl5bi |
|- ( X e. ZZ -> ( ( 4 - 1 ) < X -> 2 < X ) ) |
35 |
21 34
|
sylbid |
|- ( X e. ZZ -> ( 4 <_ X -> 2 < X ) ) |
36 |
35
|
a1i |
|- ( 4 e. ZZ -> ( X e. ZZ -> ( 4 <_ X -> 2 < X ) ) ) |
37 |
36
|
3imp |
|- ( ( 4 e. ZZ /\ X e. ZZ /\ 4 <_ X ) -> 2 < X ) |
38 |
18 37
|
sylbi |
|- ( X e. ( ZZ>= ` 4 ) -> 2 < X ) |
39 |
|
modid |
|- ( ( ( 2 e. RR /\ X e. RR+ ) /\ ( 0 <_ 2 /\ 2 < X ) ) -> ( 2 mod X ) = 2 ) |
40 |
13 15 17 38 39
|
syl22anc |
|- ( X e. ( ZZ>= ` 4 ) -> ( 2 mod X ) = 2 ) |
41 |
40
|
3ad2ant1 |
|- ( ( X e. ( ZZ>= ` 4 ) /\ X e/ Prime /\ ( ( 2 ^ ( X - 1 ) ) mod X ) = 1 ) -> ( 2 mod X ) = 2 ) |
42 |
11 41
|
sylan9eq |
|- ( ( X e. ( FPPr ` 2 ) /\ ( X e. ( ZZ>= ` 4 ) /\ X e/ Prime /\ ( ( 2 ^ ( X - 1 ) ) mod X ) = 1 ) ) -> ( ( 2 ^ X ) mod X ) = 2 ) |
43 |
10 42
|
jca |
|- ( ( X e. ( FPPr ` 2 ) /\ ( X e. ( ZZ>= ` 4 ) /\ X e/ Prime /\ ( ( 2 ^ ( X - 1 ) ) mod X ) = 1 ) ) -> ( ( X e. ( ZZ>= ` 2 ) /\ X e. Odd /\ X e/ Prime ) /\ ( ( 2 ^ X ) mod X ) = 2 ) ) |
44 |
43
|
ex |
|- ( X e. ( FPPr ` 2 ) -> ( ( X e. ( ZZ>= ` 4 ) /\ X e/ Prime /\ ( ( 2 ^ ( X - 1 ) ) mod X ) = 1 ) -> ( ( X e. ( ZZ>= ` 2 ) /\ X e. Odd /\ X e/ Prime ) /\ ( ( 2 ^ X ) mod X ) = 2 ) ) ) |
45 |
3 44
|
sylbid |
|- ( X e. ( FPPr ` 2 ) -> ( X e. ( FPPr ` 2 ) -> ( ( X e. ( ZZ>= ` 2 ) /\ X e. Odd /\ X e/ Prime ) /\ ( ( 2 ^ X ) mod X ) = 2 ) ) ) |
46 |
45
|
pm2.43i |
|- ( X e. ( FPPr ` 2 ) -> ( ( X e. ( ZZ>= ` 2 ) /\ X e. Odd /\ X e/ Prime ) /\ ( ( 2 ^ X ) mod X ) = 2 ) ) |
47 |
|
ge2nprmge4 |
|- ( ( X e. ( ZZ>= ` 2 ) /\ X e/ Prime ) -> X e. ( ZZ>= ` 4 ) ) |
48 |
47
|
3adant2 |
|- ( ( X e. ( ZZ>= ` 2 ) /\ X e. Odd /\ X e/ Prime ) -> X e. ( ZZ>= ` 4 ) ) |
49 |
|
simp3 |
|- ( ( X e. ( ZZ>= ` 2 ) /\ X e. Odd /\ X e/ Prime ) -> X e/ Prime ) |
50 |
48 49
|
jca |
|- ( ( X e. ( ZZ>= ` 2 ) /\ X e. Odd /\ X e/ Prime ) -> ( X e. ( ZZ>= ` 4 ) /\ X e/ Prime ) ) |
51 |
50
|
adantr |
|- ( ( ( X e. ( ZZ>= ` 2 ) /\ X e. Odd /\ X e/ Prime ) /\ ( ( 2 ^ X ) mod X ) = 2 ) -> ( X e. ( ZZ>= ` 4 ) /\ X e/ Prime ) ) |
52 |
1
|
a1i |
|- ( ( ( X e. ( ZZ>= ` 2 ) /\ X e. Odd /\ X e/ Prime ) /\ ( ( 2 ^ X ) mod X ) = 2 ) -> 2 e. NN ) |
53 |
12
|
a1i |
|- ( ( X e. ( ZZ>= ` 2 ) /\ X e. Odd /\ X e/ Prime ) -> 2 e. RR ) |
54 |
|
eluz2nn |
|- ( X e. ( ZZ>= ` 2 ) -> X e. NN ) |
55 |
54
|
nnrpd |
|- ( X e. ( ZZ>= ` 2 ) -> X e. RR+ ) |
56 |
55
|
3ad2ant1 |
|- ( ( X e. ( ZZ>= ` 2 ) /\ X e. Odd /\ X e/ Prime ) -> X e. RR+ ) |
57 |
16
|
a1i |
|- ( ( X e. ( ZZ>= ` 2 ) /\ X e. Odd /\ X e/ Prime ) -> 0 <_ 2 ) |
58 |
48 38
|
syl |
|- ( ( X e. ( ZZ>= ` 2 ) /\ X e. Odd /\ X e/ Prime ) -> 2 < X ) |
59 |
53 56 57 58 39
|
syl22anc |
|- ( ( X e. ( ZZ>= ` 2 ) /\ X e. Odd /\ X e/ Prime ) -> ( 2 mod X ) = 2 ) |
60 |
59
|
eqcomd |
|- ( ( X e. ( ZZ>= ` 2 ) /\ X e. Odd /\ X e/ Prime ) -> 2 = ( 2 mod X ) ) |
61 |
60
|
eqeq2d |
|- ( ( X e. ( ZZ>= ` 2 ) /\ X e. Odd /\ X e/ Prime ) -> ( ( ( 2 ^ X ) mod X ) = 2 <-> ( ( 2 ^ X ) mod X ) = ( 2 mod X ) ) ) |
62 |
61
|
biimpa |
|- ( ( ( X e. ( ZZ>= ` 2 ) /\ X e. Odd /\ X e/ Prime ) /\ ( ( 2 ^ X ) mod X ) = 2 ) -> ( ( 2 ^ X ) mod X ) = ( 2 mod X ) ) |
63 |
52 62
|
jca |
|- ( ( ( X e. ( ZZ>= ` 2 ) /\ X e. Odd /\ X e/ Prime ) /\ ( ( 2 ^ X ) mod X ) = 2 ) -> ( 2 e. NN /\ ( ( 2 ^ X ) mod X ) = ( 2 mod X ) ) ) |
64 |
|
gcd2odd1 |
|- ( X e. Odd -> ( X gcd 2 ) = 1 ) |
65 |
64
|
3ad2ant2 |
|- ( ( X e. ( ZZ>= ` 2 ) /\ X e. Odd /\ X e/ Prime ) -> ( X gcd 2 ) = 1 ) |
66 |
65
|
adantr |
|- ( ( ( X e. ( ZZ>= ` 2 ) /\ X e. Odd /\ X e/ Prime ) /\ ( ( 2 ^ X ) mod X ) = 2 ) -> ( X gcd 2 ) = 1 ) |
67 |
|
fpprwpprb |
|- ( ( X gcd 2 ) = 1 -> ( X e. ( FPPr ` 2 ) <-> ( ( X e. ( ZZ>= ` 4 ) /\ X e/ Prime ) /\ ( 2 e. NN /\ ( ( 2 ^ X ) mod X ) = ( 2 mod X ) ) ) ) ) |
68 |
66 67
|
syl |
|- ( ( ( X e. ( ZZ>= ` 2 ) /\ X e. Odd /\ X e/ Prime ) /\ ( ( 2 ^ X ) mod X ) = 2 ) -> ( X e. ( FPPr ` 2 ) <-> ( ( X e. ( ZZ>= ` 4 ) /\ X e/ Prime ) /\ ( 2 e. NN /\ ( ( 2 ^ X ) mod X ) = ( 2 mod X ) ) ) ) ) |
69 |
51 63 68
|
mpbir2and |
|- ( ( ( X e. ( ZZ>= ` 2 ) /\ X e. Odd /\ X e/ Prime ) /\ ( ( 2 ^ X ) mod X ) = 2 ) -> X e. ( FPPr ` 2 ) ) |
70 |
46 69
|
impbii |
|- ( X e. ( FPPr ` 2 ) <-> ( ( X e. ( ZZ>= ` 2 ) /\ X e. Odd /\ X e/ Prime ) /\ ( ( 2 ^ X ) mod X ) = 2 ) ) |