ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  lgsdirprm Unicode version
Theorem lgsdirprm 15275
Description: The Legendre symbol is completely multiplicative at the primes. See theorem 9.3 in [ApostolNT] p. 180. (Contributed by Mario Carneiro, 4-Feb-2015.) (Proof shortened by AV, 18-Mar-2022.)
Assertion
Ref Expression
lgsdirprm |- ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) -> ( ( A x. B ) /L P ) = ( ( A /L P ) x. ( B /L P ) ) )
Proof of Theorem lgsdirprm
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 simpl1 1002 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> A e. ZZ )
2 simpl2 1003 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> B e. ZZ )
3 lgsdir2 15274 . . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A x. B ) /L 2 ) = ( ( A /L 2 ) x. ( B /L 2 ) ) )
41, 2, 3syl2anc 411 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> ( ( A x. B ) /L 2 ) = ( ( A /L 2 ) x. ( B /L 2 ) ) )
5 simpr 110 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> P = 2 )
65oveq2d 5938 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> ( ( A x. B ) /L P ) = ( ( A x. B ) /L 2 ) )
75oveq2d 5938 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> ( A /L P ) = ( A /L 2 ) )
85oveq2d 5938 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> ( B /L P ) = ( B /L 2 ) )
97, 8oveq12d 5940 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> ( ( A /L P ) x. ( B /L P ) ) = ( ( A /L 2 ) x. ( B /L 2 ) ) )
104, 6, 93eqtr4d 2239 . 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> ( ( A x. B ) /L P ) = ( ( A /L P ) x. ( B /L P ) ) )
11 simpl1 1002 . . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> A e. ZZ )
12 simpl2 1003 . . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> B e. ZZ )
1311, 12zmulcld 9454 . . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( A x. B ) e. ZZ )
14 simpl3 1004 . . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> P e. Prime )
15 prmz 12279 . . . . . 6 |- ( P e. Prime -> P e. ZZ )
1614, 15syl 14 . . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> P e. ZZ )
17 lgscl 15255 . . . . 5 |- ( ( ( A x. B ) e. ZZ /\ P e. ZZ ) -> ( ( A x. B ) /L P ) e. ZZ )
1813, 16, 17syl2anc 411 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( A x. B ) /L P ) e. ZZ )
1918zcnd 9449 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( A x. B ) /L P ) e. CC )
20 lgscl 15255 . . . . . 6 |- ( ( A e. ZZ /\ P e. ZZ ) -> ( A /L P ) e. ZZ )
2111, 16, 20syl2anc 411 . . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( A /L P ) e. ZZ )
22 lgscl 15255 . . . . . 6 |- ( ( B e. ZZ /\ P e. ZZ ) -> ( B /L P ) e. ZZ )
2312, 16, 22syl2anc 411 . . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( B /L P ) e. ZZ )
2421, 23zmulcld 9454 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( A /L P ) x. ( B /L P ) ) e. ZZ )
2524zcnd 9449 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( A /L P ) x. ( B /L P ) ) e. CC )
2619, 25subcld 8337 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) e. CC )
2718, 24zsubcld 9453 . . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) e. ZZ )
28 zabscl 11251 . . . . . . 7 |- ( ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) e. ZZ -> ( abs ` ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) e. ZZ )
29 zq 9700 . . . . . . 7 |- ( ( abs ` ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) e. ZZ -> ( abs ` ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) e. QQ )
3027, 28, 293syl 17 . . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( abs ` ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) e. QQ )
31 prmnn 12278 . . . . . . 7 |- ( P e. Prime -> P e. NN )
32 nnq 9707 . . . . . . 7 |- ( P e. NN -> P e. QQ )
3314, 31, 323syl 17 . . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> P e. QQ )
3426absge0d 11349 . . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> 0 <_ ( abs ` ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) )
3526abscld 11346 . . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( abs ` ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) e. RR )
36 2re 9060 . . . . . . . 8 |- 2 e. RR
3736a1i 9 . . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> 2 e. RR )
3814, 31syl 14 . . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> P e. NN )
3938nnred 9003 . . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> P e. RR )
4019abscld 11346 . . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( abs ` ( ( A x. B ) /L P ) ) e. RR )
4125abscld 11346 . . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( abs ` ( ( A /L P ) x. ( B /L P ) ) ) e. RR )
4240, 41readdcld 8056 . . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( abs ` ( ( A x. B ) /L P ) ) + ( abs ` ( ( A /L P ) x. ( B /L P ) ) ) ) e. RR )
4319, 25abs2dif2d 11363 . . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( abs ` ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) <_ ( ( abs ` ( ( A x. B ) /L P ) ) + ( abs ` ( ( A /L P ) x. ( B /L P ) ) ) ) )
44 1red 8041 . . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> 1 e. RR )
45 lgsle1 15256 . . . . . . . . . . 11 |- ( ( ( A x. B ) e. ZZ /\ P e. ZZ ) -> ( abs ` ( ( A x. B ) /L P ) ) <_ 1 )
4613, 16, 45syl2anc 411 . . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( abs ` ( ( A x. B ) /L P ) ) <_ 1 )
47 eqid 2196 . . . . . . . . . . . . . 14 |- { x e. ZZ | ( abs ` x ) <_ 1 } = { x e. ZZ | ( abs ` x ) <_ 1 }
4847lgscl2 15253 . . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ P e. ZZ ) -> ( A /L P ) e. { x e. ZZ | ( abs ` x ) <_ 1 } )
4911, 16, 48syl2anc 411 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( A /L P ) e. { x e. ZZ | ( abs ` x ) <_ 1 } )
5047lgscl2 15253 . . . . . . . . . . . . 13 |- ( ( B e. ZZ /\ P e. ZZ ) -> ( B /L P ) e. { x e. ZZ | ( abs ` x ) <_ 1 } )
5112, 16, 50syl2anc 411 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( B /L P ) e. { x e. ZZ | ( abs ` x ) <_ 1 } )
5247lgslem3 15243 . . . . . . . . . . . 12 |- ( ( ( A /L P ) e. { x e. ZZ | ( abs ` x ) <_ 1 } /\ ( B /L P ) e. { x e. ZZ | ( abs ` x ) <_ 1 } ) -> ( ( A /L P ) x. ( B /L P ) ) e. { x e. ZZ | ( abs ` x ) <_ 1 } )
5349, 51, 52syl2anc 411 . . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( A /L P ) x. ( B /L P ) ) e. { x e. ZZ | ( abs ` x ) <_ 1 } )
54 fveq2 5558 . . . . . . . . . . . . . 14 |- ( x = ( ( A /L P ) x. ( B /L P ) ) -> ( abs ` x ) = ( abs ` ( ( A /L P ) x. ( B /L P ) ) ) )
5554breq1d 4043 . . . . . . . . . . . . 13 |- ( x = ( ( A /L P ) x. ( B /L P ) ) -> ( ( abs ` x ) <_ 1 <-> ( abs ` ( ( A /L P ) x. ( B /L P ) ) ) <_ 1 ) )
5655elrab 2920 . . . . . . . . . . . 12 |- ( ( ( A /L P ) x. ( B /L P ) ) e. { x e. ZZ | ( abs ` x ) <_ 1 } <-> ( ( ( A /L P ) x. ( B /L P ) ) e. ZZ /\ ( abs ` ( ( A /L P ) x. ( B /L P ) ) ) <_ 1 ) )
5756simprbi 275 . . . . . . . . . . 11 |- ( ( ( A /L P ) x. ( B /L P ) ) e. { x e. ZZ | ( abs ` x ) <_ 1 } -> ( abs ` ( ( A /L P ) x. ( B /L P ) ) ) <_ 1 )
5853, 57syl 14 . . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( abs ` ( ( A /L P ) x. ( B /L P ) ) ) <_ 1 )
5940, 41, 44, 44, 46, 58le2addd 8590 . . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( abs ` ( ( A x. B ) /L P ) ) + ( abs ` ( ( A /L P ) x. ( B /L P ) ) ) ) <_ ( 1 + 1 ) )
60 df-2 9049 . . . . . . . . 9 |- 2 = ( 1 + 1 )
6159, 60breqtrrdi 4075 . . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( abs ` ( ( A x. B ) /L P ) ) + ( abs ` ( ( A /L P ) x. ( B /L P ) ) ) ) <_ 2 )
6235, 42, 37, 43, 61letrd 8150 . . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( abs ` ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) <_ 2 )
63 prmuz2 12299 . . . . . . . . 9 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
64 eluzle 9613 . . . . . . . . 9 |- ( P e. ( ZZ>= ` 2 ) -> 2 <_ P )
6514, 63, 643syl 17 . . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> 2 <_ P )
66 simpr 110 . . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> P =/= 2 )
67 2z 9354 . . . . . . . . 9 |- 2 e. ZZ
68 zltlen 9404 . . . . . . . . 9 |- ( ( 2 e. ZZ /\ P e. ZZ ) -> ( 2 < P <-> ( 2 <_ P /\ P =/= 2 ) ) )
6967, 16, 68sylancr 414 . . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( 2 < P <-> ( 2 <_ P /\ P =/= 2 ) ) )
7065, 66, 69mpbir2and 946 . . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> 2 < P )
7135, 37, 39, 62, 70lelttrd 8151 . . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( abs ` ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) < P )
72 modqid 10441 . . . . . 6 |- ( ( ( ( abs ` ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) e. QQ /\ P e. QQ ) /\ ( 0 <_ ( abs ` ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) /\ ( abs ` ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) < P ) ) -> ( ( abs ` ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) mod P ) = ( abs ` ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) )
7330, 33, 34, 71, 72syl22anc 1250 . . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( abs ` ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) mod P ) = ( abs ` ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) )
7411zcnd 9449 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> A e. CC )
7512zcnd 9449 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> B e. CC )
76 eldifsn 3749 . . . . . . . . . . . . . . 15 |- ( P e. ( Prime \ { 2 } ) <-> ( P e. Prime /\ P =/= 2 ) )
7714, 66, 76sylanbrc 417 . . . . . . . . . . . . . 14 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> P e. ( Prime \ { 2 } ) )
78 oddprm 12428 . . . . . . . . . . . . . 14 |- ( P e. ( Prime \ { 2 } ) -> ( ( P - 1 ) / 2 ) e. NN )
7977, 78syl 14 . . . . . . . . . . . . 13 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( P - 1 ) / 2 ) e. NN )
8079nnnn0d 9302 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( P - 1 ) / 2 ) e. NN0 )
8174, 75, 80mulexpd 10780 . . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( A x. B ) ^ ( ( P - 1 ) / 2 ) ) = ( ( A ^ ( ( P - 1 ) / 2 ) ) x. ( B ^ ( ( P - 1 ) / 2 ) ) ) )
82 zexpcl 10646 . . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ ( ( P - 1 ) / 2 ) e. NN0 ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. ZZ )
8311, 80, 82syl2anc 411 . . . . . . . . . . . . 13 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. ZZ )
8483zcnd 9449 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. CC )
85 zexpcl 10646 . . . . . . . . . . . . . 14 |- ( ( B e. ZZ /\ ( ( P - 1 ) / 2 ) e. NN0 ) -> ( B ^ ( ( P - 1 ) / 2 ) ) e. ZZ )
8612, 80, 85syl2anc 411 . . . . . . . . . . . . 13 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( B ^ ( ( P - 1 ) / 2 ) ) e. ZZ )
8786zcnd 9449 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( B ^ ( ( P - 1 ) / 2 ) ) e. CC )
8884, 87mulcomd 8048 . . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) x. ( B ^ ( ( P - 1 ) / 2 ) ) ) = ( ( B ^ ( ( P - 1 ) / 2 ) ) x. ( A ^ ( ( P - 1 ) / 2 ) ) ) )
8981, 88eqtrd 2229 . . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( A x. B ) ^ ( ( P - 1 ) / 2 ) ) = ( ( B ^ ( ( P - 1 ) / 2 ) ) x. ( A ^ ( ( P - 1 ) / 2 ) ) ) )
9089oveq1d 5937 . . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( ( A x. B ) ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( ( ( B ^ ( ( P - 1 ) / 2 ) ) x. ( A ^ ( ( P - 1 ) / 2 ) ) ) mod P ) )
91 lgsvalmod 15260 . . . . . . . . . 10 |- ( ( ( A x. B ) e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A x. B ) /L P ) mod P ) = ( ( ( A x. B ) ^ ( ( P - 1 ) / 2 ) ) mod P ) )
9213, 77, 91syl2anc 411 . . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( ( A x. B ) /L P ) mod P ) = ( ( ( A x. B ) ^ ( ( P - 1 ) / 2 ) ) mod P ) )
93 zq 9700 . . . . . . . . . . . 12 |- ( ( A /L P ) e. ZZ -> ( A /L P ) e. QQ )
9421, 93syl 14 . . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( A /L P ) e. QQ )
95 zq 9700 . . . . . . . . . . . 12 |- ( ( A ^ ( ( P - 1 ) / 2 ) ) e. ZZ -> ( A ^ ( ( P - 1 ) / 2 ) ) e. QQ )
9683, 95syl 14 . . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. QQ )
9738nngt0d 9034 . . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> 0 < P )
98 lgsvalmod 15260 . . . . . . . . . . . 12 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) mod P ) = ( ( A ^ ( ( P - 1 ) / 2 ) ) mod P ) )
9911, 77, 98syl2anc 411 . . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( A /L P ) mod P ) = ( ( A ^ ( ( P - 1 ) / 2 ) ) mod P ) )
10094, 96, 23, 33, 97, 99modqmul1 10469 . . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( ( A /L P ) x. ( B /L P ) ) mod P ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) x. ( B /L P ) ) mod P ) )
10123zcnd 9449 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( B /L P ) e. CC )
10284, 101mulcomd 8048 . . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) x. ( B /L P ) ) = ( ( B /L P ) x. ( A ^ ( ( P - 1 ) / 2 ) ) ) )
103102oveq1d 5937 . . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) x. ( B /L P ) ) mod P ) = ( ( ( B /L P ) x. ( A ^ ( ( P - 1 ) / 2 ) ) ) mod P ) )
104 zq 9700 . . . . . . . . . . . 12 |- ( ( B /L P ) e. ZZ -> ( B /L P ) e. QQ )
10523, 104syl 14 . . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( B /L P ) e. QQ )
106 zq 9700 . . . . . . . . . . . 12 |- ( ( B ^ ( ( P - 1 ) / 2 ) ) e. ZZ -> ( B ^ ( ( P - 1 ) / 2 ) ) e. QQ )
10786, 106syl 14 . . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( B ^ ( ( P - 1 ) / 2 ) ) e. QQ )
108 lgsvalmod 15260 . . . . . . . . . . . 12 |- ( ( B e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( B /L P ) mod P ) = ( ( B ^ ( ( P - 1 ) / 2 ) ) mod P ) )
10912, 77, 108syl2anc 411 . . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( B /L P ) mod P ) = ( ( B ^ ( ( P - 1 ) / 2 ) ) mod P ) )
110105, 107, 83, 33, 97, 109modqmul1 10469 . . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( ( B /L P ) x. ( A ^ ( ( P - 1 ) / 2 ) ) ) mod P ) = ( ( ( B ^ ( ( P - 1 ) / 2 ) ) x. ( A ^ ( ( P - 1 ) / 2 ) ) ) mod P ) )
111100, 103, 1103eqtrd 2233 . . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( ( A /L P ) x. ( B /L P ) ) mod P ) = ( ( ( B ^ ( ( P - 1 ) / 2 ) ) x. ( A ^ ( ( P - 1 ) / 2 ) ) ) mod P ) )
11290, 92, 1113eqtr4d 2239 . . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( ( A x. B ) /L P ) mod P ) = ( ( ( A /L P ) x. ( B /L P ) ) mod P ) )
113 moddvds 11964 . . . . . . . . 9 |- ( ( P e. NN /\ ( ( A x. B ) /L P ) e. ZZ /\ ( ( A /L P ) x. ( B /L P ) ) e. ZZ ) -> ( ( ( ( A x. B ) /L P ) mod P ) = ( ( ( A /L P ) x. ( B /L P ) ) mod P ) <-> P || ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) )
11438, 18, 24, 113syl3anc 1249 . . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( ( ( A x. B ) /L P ) mod P ) = ( ( ( A /L P ) x. ( B /L P ) ) mod P ) <-> P || ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) )
115112, 114mpbid 147 . . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> P || ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) )
116 dvdsabsb 11975 . . . . . . . 8 |- ( ( P e. ZZ /\ ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) e. ZZ ) -> ( P || ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) <-> P || ( abs ` ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) ) )
11716, 27, 116syl2anc 411 . . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( P || ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) <-> P || ( abs ` ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) ) )
118115, 117mpbid 147 . . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> P || ( abs ` ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) )
119 dvdsmod0 11958 . . . . . 6 |- ( ( P e. NN /\ P || ( abs ` ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) ) -> ( ( abs ` ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) mod P ) = 0 )
12038, 118, 119syl2anc 411 . . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( abs ` ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) mod P ) = 0 )
12173, 120eqtr3d 2231 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( abs ` ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) ) = 0 )
12226, 121abs00d 11351 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( ( A x. B ) /L P ) - ( ( A /L P ) x. ( B /L P ) ) ) = 0 )
12319, 25, 122subeq0d 8345 . 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( A x. B ) /L P ) = ( ( A /L P ) x. ( B /L P ) ) )
124153ad2ant3 1022 . . . 4 |- ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) -> P e. ZZ )
12567a1i 9 . . . 4 |- ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) -> 2 e. ZZ )
126 zdceq 9401 . . . 4 |- ( ( P e. ZZ /\ 2 e. ZZ ) -> DECID P = 2 )
127124, 125, 126syl2anc 411 . . 3 |- ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) -> DECID P = 2 )
128 dcne 2378 . . 3 |- (DECID P = 2 <-> ( P = 2 \/ P =/= 2 ) )
129127, 128sylib 122 . 2 |- ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) -> ( P = 2 \/ P =/= 2 ) )
13010, 123, 129mpjaodan 799 1 |- ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) -> ( ( A x. B ) /L P ) = ( ( A /L P ) x. ( B /L P ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   /\ w3a 980   = wceq 1364   e. wcel 2167   =/= wne 2367   {crab 2479   \ cdif 3154   {csn 3622   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   RRcr 7878   0cc0 7879   1c1 7880   + caddc 7882   x. cmul 7884   < clt 8061   <_ cle 8062   - cmin 8197   / cdiv 8699   NNcn 8990   2c2 9041   NN0cn0 9249   ZZcz 9326   ZZ>=cuz 9601   QQcq 9693   mod cmo 10414   ^cexp 10630   abscabs 11162   || cdvds 11952   Primecprime 12275   /Lclgs 15238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-xor 1387  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-frec 6449  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6592  df-en 6800  df-dom 6801  df-fin 6802  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-5 9052  df-6 9053  df-7 9054  df-8 9055  df-9 9056  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-fl 10360  df-mod 10415  df-seqfrec 10540  df-exp 10631  df-ihash 10868  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-clim 11444  df-proddc 11716  df-dvds 11953  df-gcd 12121  df-prm 12276  df-phi 12379  df-pc 12454  df-lgs 15239
This theorem is referenced by:  lgsdir  15276
  Copyright terms: Public domain W3C validator