Metamath Proof Explorer

Theorem fmtnoprmfac1

Description: Divisor of Fermat number (special form of Euler's result, see fmtnofac1 ): Let F_n be a Fermat number. Let p be a prime divisor of F_n. Then p is in the form: k*2^(n+1)+1 where k is a positive integer. (Contributed by AV, 25-Jul-2021)

Ref Expression
Assertion fmtnoprmfac1 
|- ( ( N e. NN /\ P e. Prime /\ P || ( FermatNo ` N ) ) -> E. k e. NN P = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) )

Proof

Step Hyp Ref Expression
1 breq1 
 |-  ( P = 2 -> ( P || ( FermatNo ` N ) <-> 2 || ( FermatNo ` N ) ) )
2 1 adantr 
 |-  ( ( P = 2 /\ N e. NN ) -> ( P || ( FermatNo ` N ) <-> 2 || ( FermatNo ` N ) ) )
3 nnnn0 
 |-  ( N e. NN -> N e. NN0 )
4 fmtnoodd 
 |-  ( N e. NN0 -> -. 2 || ( FermatNo ` N ) )
5 3 4 syl 
 |-  ( N e. NN -> -. 2 || ( FermatNo ` N ) )
6 5 adantl 
 |-  ( ( P = 2 /\ N e. NN ) -> -. 2 || ( FermatNo ` N ) )
7 6 pm2.21d 
 |-  ( ( P = 2 /\ N e. NN ) -> ( 2 || ( FermatNo ` N ) -> E. k e. NN P = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) )
8 2 7 sylbid 
 |-  ( ( P = 2 /\ N e. NN ) -> ( P || ( FermatNo ` N ) -> E. k e. NN P = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) )
9 8 a1d 
 |-  ( ( P = 2 /\ N e. NN ) -> ( P e. Prime -> ( P || ( FermatNo ` N ) -> E. k e. NN P = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) ) )
10 9 ex 
 |-  ( P = 2 -> ( N e. NN -> ( P e. Prime -> ( P || ( FermatNo ` N ) -> E. k e. NN P = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) ) ) )
11 10 3impd 
 |-  ( P = 2 -> ( ( N e. NN /\ P e. Prime /\ P || ( FermatNo ` N ) ) -> E. k e. NN P = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) )
12 simpr1 
 |-  ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime /\ P || ( FermatNo ` N ) ) ) -> N e. NN )
13 neqne 
 |-  ( -. P = 2 -> P =/= 2 )
14 13 anim2i 
 |-  ( ( P e. Prime /\ -. P = 2 ) -> ( P e. Prime /\ P =/= 2 ) )
15 eldifsn 
 |-  ( P e. ( Prime \ { 2 } ) <-> ( P e. Prime /\ P =/= 2 ) )
16 14 15 sylibr 
 |-  ( ( P e. Prime /\ -. P = 2 ) -> P e. ( Prime \ { 2 } ) )
17 16 ex 
 |-  ( P e. Prime -> ( -. P = 2 -> P e. ( Prime \ { 2 } ) ) )
18 17 3ad2ant2 
 |-  ( ( N e. NN /\ P e. Prime /\ P || ( FermatNo ` N ) ) -> ( -. P = 2 -> P e. ( Prime \ { 2 } ) ) )
19 18 impcom 
 |-  ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime /\ P || ( FermatNo ` N ) ) ) -> P e. ( Prime \ { 2 } ) )
20 simpr3 
 |-  ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime /\ P || ( FermatNo ` N ) ) ) -> P || ( FermatNo ` N ) )
21 fmtnoprmfac1lem 
 |-  ( ( N e. NN /\ P e. ( Prime \ { 2 } ) /\ P || ( FermatNo ` N ) ) -> ( ( odZ ` P ) ` 2 ) = ( 2 ^ ( N + 1 ) ) )
22 12 19 20 21 syl3anc 
 |-  ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime /\ P || ( FermatNo ` N ) ) ) -> ( ( odZ ` P ) ` 2 ) = ( 2 ^ ( N + 1 ) ) )
23 prmnn 
 |-  ( P e. Prime -> P e. NN )
24 23 ad2antll 
 |-  ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) -> P e. NN )
25 2z 
 |-  2 e. ZZ
26 25 a1i 
 |-  ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) -> 2 e. ZZ )
27 13 necomd 
 |-  ( -. P = 2 -> 2 =/= P )
28 27 adantr 
 |-  ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) -> 2 =/= P )
29 2prm 
 |-  2 e. Prime
30 29 a1i 
 |-  ( N e. NN -> 2 e. Prime )
31 30 anim1i 
 |-  ( ( N e. NN /\ P e. Prime ) -> ( 2 e. Prime /\ P e. Prime ) )
32 31 adantl 
 |-  ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) -> ( 2 e. Prime /\ P e. Prime ) )
33 prmrp 
 |-  ( ( 2 e. Prime /\ P e. Prime ) -> ( ( 2 gcd P ) = 1 <-> 2 =/= P ) )
34 32 33 syl 
 |-  ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) -> ( ( 2 gcd P ) = 1 <-> 2 =/= P ) )
35 28 34 mpbird 
 |-  ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) -> ( 2 gcd P ) = 1 )
36 odzphi 
 |-  ( ( P e. NN /\ 2 e. ZZ /\ ( 2 gcd P ) = 1 ) -> ( ( odZ ` P ) ` 2 ) || ( phi ` P ) )
37 24 26 35 36 syl3anc 
 |-  ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) -> ( ( odZ ` P ) ` 2 ) || ( phi ` P ) )
38 phiprm 
 |-  ( P e. Prime -> ( phi ` P ) = ( P - 1 ) )
39 38 ad2antll 
 |-  ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) -> ( phi ` P ) = ( P - 1 ) )
40 39 breq2d 
 |-  ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) -> ( ( ( odZ ` P ) ` 2 ) || ( phi ` P ) <-> ( ( odZ ` P ) ` 2 ) || ( P - 1 ) ) )
41 breq1 
 |-  ( ( ( odZ ` P ) ` 2 ) = ( 2 ^ ( N + 1 ) ) -> ( ( ( odZ ` P ) ` 2 ) || ( P - 1 ) <-> ( 2 ^ ( N + 1 ) ) || ( P - 1 ) ) )
42 41 adantl 
 |-  ( ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) /\ ( ( odZ ` P ) ` 2 ) = ( 2 ^ ( N + 1 ) ) ) -> ( ( ( odZ ` P ) ` 2 ) || ( P - 1 ) <-> ( 2 ^ ( N + 1 ) ) || ( P - 1 ) ) )
43 2nn 
 |-  2 e. NN
44 43 a1i 
 |-  ( N e. NN -> 2 e. NN )
45 peano2nn 
 |-  ( N e. NN -> ( N + 1 ) e. NN )
46 45 nnnn0d 
 |-  ( N e. NN -> ( N + 1 ) e. NN0 )
47 44 46 nnexpcld 
 |-  ( N e. NN -> ( 2 ^ ( N + 1 ) ) e. NN )
48 23 nnnn0d 
 |-  ( P e. Prime -> P e. NN0 )
49 prmuz2 
 |-  ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
50 eluzle 
 |-  ( P e. ( ZZ>= ` 2 ) -> 2 <_ P )
51 49 50 syl 
 |-  ( P e. Prime -> 2 <_ P )
52 nn0ge2m1nn 
 |-  ( ( P e. NN0 /\ 2 <_ P ) -> ( P - 1 ) e. NN )
53 48 51 52 syl2anc 
 |-  ( P e. Prime -> ( P - 1 ) e. NN )
54 47 53 anim12i 
 |-  ( ( N e. NN /\ P e. Prime ) -> ( ( 2 ^ ( N + 1 ) ) e. NN /\ ( P - 1 ) e. NN ) )
55 54 adantl 
 |-  ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) -> ( ( 2 ^ ( N + 1 ) ) e. NN /\ ( P - 1 ) e. NN ) )
56 nndivides 
 |-  ( ( ( 2 ^ ( N + 1 ) ) e. NN /\ ( P - 1 ) e. NN ) -> ( ( 2 ^ ( N + 1 ) ) || ( P - 1 ) <-> E. k e. NN ( k x. ( 2 ^ ( N + 1 ) ) ) = ( P - 1 ) ) )
57 55 56 syl 
 |-  ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) -> ( ( 2 ^ ( N + 1 ) ) || ( P - 1 ) <-> E. k e. NN ( k x. ( 2 ^ ( N + 1 ) ) ) = ( P - 1 ) ) )
58 eqcom 
 |-  ( ( k x. ( 2 ^ ( N + 1 ) ) ) = ( P - 1 ) <-> ( P - 1 ) = ( k x. ( 2 ^ ( N + 1 ) ) ) )
59 58 a1i 
 |-  ( ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) /\ k e. NN ) -> ( ( k x. ( 2 ^ ( N + 1 ) ) ) = ( P - 1 ) <-> ( P - 1 ) = ( k x. ( 2 ^ ( N + 1 ) ) ) ) )
60 23 nncnd 
 |-  ( P e. Prime -> P e. CC )
61 60 adantl 
 |-  ( ( N e. NN /\ P e. Prime ) -> P e. CC )
62 61 adantr 
 |-  ( ( ( N e. NN /\ P e. Prime ) /\ k e. NN ) -> P e. CC )
63 1cnd 
 |-  ( ( ( N e. NN /\ P e. Prime ) /\ k e. NN ) -> 1 e. CC )
64 nncn 
 |-  ( k e. NN -> k e. CC )
65 64 adantl 
 |-  ( ( ( N e. NN /\ P e. Prime ) /\ k e. NN ) -> k e. CC )
66 peano2nn0 
 |-  ( N e. NN0 -> ( N + 1 ) e. NN0 )
67 3 66 syl 
 |-  ( N e. NN -> ( N + 1 ) e. NN0 )
68 44 67 nnexpcld 
 |-  ( N e. NN -> ( 2 ^ ( N + 1 ) ) e. NN )
69 68 nncnd 
 |-  ( N e. NN -> ( 2 ^ ( N + 1 ) ) e. CC )
70 69 adantr 
 |-  ( ( N e. NN /\ P e. Prime ) -> ( 2 ^ ( N + 1 ) ) e. CC )
71 70 adantr 
 |-  ( ( ( N e. NN /\ P e. Prime ) /\ k e. NN ) -> ( 2 ^ ( N + 1 ) ) e. CC )
72 65 71 mulcld 
 |-  ( ( ( N e. NN /\ P e. Prime ) /\ k e. NN ) -> ( k x. ( 2 ^ ( N + 1 ) ) ) e. CC )
73 62 63 72 subadd2d 
 |-  ( ( ( N e. NN /\ P e. Prime ) /\ k e. NN ) -> ( ( P - 1 ) = ( k x. ( 2 ^ ( N + 1 ) ) ) <-> ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) = P ) )
74 73 adantll 
 |-  ( ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) /\ k e. NN ) -> ( ( P - 1 ) = ( k x. ( 2 ^ ( N + 1 ) ) ) <-> ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) = P ) )
75 eqcom 
 |-  ( ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) = P <-> P = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) )
76 75 a1i 
 |-  ( ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) /\ k e. NN ) -> ( ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) = P <-> P = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) )
77 59 74 76 3bitrd 
 |-  ( ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) /\ k e. NN ) -> ( ( k x. ( 2 ^ ( N + 1 ) ) ) = ( P - 1 ) <-> P = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) )
78 77 rexbidva 
 |-  ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) -> ( E. k e. NN ( k x. ( 2 ^ ( N + 1 ) ) ) = ( P - 1 ) <-> E. k e. NN P = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) )
79 78 biimpd 
 |-  ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) -> ( E. k e. NN ( k x. ( 2 ^ ( N + 1 ) ) ) = ( P - 1 ) -> E. k e. NN P = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) )
80 57 79 sylbid 
 |-  ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) -> ( ( 2 ^ ( N + 1 ) ) || ( P - 1 ) -> E. k e. NN P = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) )
81 80 adantr 
 |-  ( ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) /\ ( ( odZ ` P ) ` 2 ) = ( 2 ^ ( N + 1 ) ) ) -> ( ( 2 ^ ( N + 1 ) ) || ( P - 1 ) -> E. k e. NN P = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) )
82 42 81 sylbid 
 |-  ( ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) /\ ( ( odZ ` P ) ` 2 ) = ( 2 ^ ( N + 1 ) ) ) -> ( ( ( odZ ` P ) ` 2 ) || ( P - 1 ) -> E. k e. NN P = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) )
83 82 ex 
 |-  ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) -> ( ( ( odZ ` P ) ` 2 ) = ( 2 ^ ( N + 1 ) ) -> ( ( ( odZ ` P ) ` 2 ) || ( P - 1 ) -> E. k e. NN P = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) ) )
84 83 com23 
 |-  ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) -> ( ( ( odZ ` P ) ` 2 ) || ( P - 1 ) -> ( ( ( odZ ` P ) ` 2 ) = ( 2 ^ ( N + 1 ) ) -> E. k e. NN P = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) ) )
85 40 84 sylbid 
 |-  ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) -> ( ( ( odZ ` P ) ` 2 ) || ( phi ` P ) -> ( ( ( odZ ` P ) ` 2 ) = ( 2 ^ ( N + 1 ) ) -> E. k e. NN P = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) ) )
86 37 85 mpd 
 |-  ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime ) ) -> ( ( ( odZ ` P ) ` 2 ) = ( 2 ^ ( N + 1 ) ) -> E. k e. NN P = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) )
87 86 3adantr3 
 |-  ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime /\ P || ( FermatNo ` N ) ) ) -> ( ( ( odZ ` P ) ` 2 ) = ( 2 ^ ( N + 1 ) ) -> E. k e. NN P = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) )
88 22 87 mpd 
 |-  ( ( -. P = 2 /\ ( N e. NN /\ P e. Prime /\ P || ( FermatNo ` N ) ) ) -> E. k e. NN P = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) )
89 88 ex 
 |-  ( -. P = 2 -> ( ( N e. NN /\ P e. Prime /\ P || ( FermatNo ` N ) ) -> E. k e. NN P = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) )
90 11 89 pm2.61i 
 |-  ( ( N e. NN /\ P e. Prime /\ P || ( FermatNo ` N ) ) -> E. k e. NN P = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) )