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Theorem lgsdirprm 15753
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 1024 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> A e. ZZ )
2 simpl2 1025 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> B e. ZZ )
3 lgsdir2 15752 . . . 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 6029 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> ( ( A x. B ) /L P ) = ( ( A x. B ) /L 2 ) )
75oveq2d 6029 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> ( A /L P ) = ( A /L 2 ) )
85oveq2d 6029 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> ( B /L P ) = ( B /L 2 ) )
97, 8oveq12d 6031 . . 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 2272 . 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 1024 . . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> A e. ZZ )
12 simpl2 1025 . . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> B e. ZZ )
1311, 12zmulcld 9598 . . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( A x. B ) e. ZZ )
14 simpl3 1026 . . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> P e. Prime )
15 prmz 12673 . . . . . 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 15733 . . . . 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 9593 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( A x. B ) /L P ) e. CC )
20 lgscl 15733 . . . . . 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 15733 . . . . . 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 9598 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( A /L P ) x. ( B /L P ) ) e. ZZ )
2524zcnd 9593 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( A /L P ) x. ( B /L P ) ) e. CC )
2619, 25subcld 8480 . . . 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 9597 . . . . . . 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 11637 . . . . . . 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 9850 . . . . . . 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 12672 . . . . . . 7 |- ( P e. Prime -> P e. NN )
32 nnq 9857 . . . . . . 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 11735 . . . . . 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 11732 . . . . . . 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 9203 . . . . . . . 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 9146 . . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> P e. RR )
4019abscld 11732 . . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( abs ` ( ( A x. B ) /L P ) ) e. RR )
4125abscld 11732 . . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( abs ` ( ( A /L P ) x. ( B /L P ) ) ) e. RR )
4240, 41readdcld 8199 . . . . . . . 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 11749 . . . . . . . 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 8184 . . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> 1 e. RR )
45 lgsle1 15734 . . . . . . . . . . 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 2229 . . . . . . . . . . . . . 14 |- { x e. ZZ | ( abs ` x ) <_ 1 } = { x e. ZZ | ( abs ` x ) <_ 1 }
4847lgscl2 15731 . . . . . . . . . . . . 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 15731 . . . . . . . . . . . . 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 15721 . . . . . . . . . . . 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 5635 . . . . . . . . . . . . . 14 |- ( x = ( ( A /L P ) x. ( B /L P ) ) -> ( abs ` x ) = ( abs ` ( ( A /L P ) x. ( B /L P ) ) ) )
5554breq1d 4096 . . . . . . . . . . . . 13 |- ( x = ( ( A /L P ) x. ( B /L P ) ) -> ( ( abs ` x ) <_ 1 <-> ( abs ` ( ( A /L P ) x. ( B /L P ) ) ) <_ 1 ) )
5655elrab 2960 . . . . . . . . . . . 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 8733 . . . . . . . . 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 9192 . . . . . . . . 9 |- 2 = ( 1 + 1 )
6159, 60breqtrrdi 4128 . . . . . . . 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 8293 . . . . . . 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 12693 . . . . . . . . 9 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
64 eluzle 9758 . . . . . . . . 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 9497 . . . . . . . . 9 |- 2 e. ZZ
68 zltlen 9548 . . . . . . . . 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 950 . . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> 2 < P )
7135, 37, 39, 62, 70lelttrd 8294 . . . . . 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 10601 . . . . . 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 1272 . . . . 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 9593 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> A e. CC )
7512zcnd 9593 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> B e. CC )
76 eldifsn 3798 . . . . . . . . . . . . . . 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 12822 . . . . . . . . . . . . . 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 9445 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( P - 1 ) / 2 ) e. NN0 )
8174, 75, 80mulexpd 10940 . . . . . . . . . . 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 10806 . . . . . . . . . . . . . 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 9593 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. CC )
85 zexpcl 10806 . . . . . . . . . . . . . 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 9593 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( B ^ ( ( P - 1 ) / 2 ) ) e. CC )
8884, 87mulcomd 8191 . . . . . . . . . . 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 2262 . . . . . . . . . 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 6028 . . . . . . . . 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 15738 . . . . . . . . . 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 9850 . . . . . . . . . . . 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 9850 . . . . . . . . . . . 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 9177 . . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> 0 < P )
98 lgsvalmod 15738 . . . . . . . . . . . 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 10629 . . . . . . . . . 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 9593 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( B /L P ) e. CC )
10284, 101mulcomd 8191 . . . . . . . . . . 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 6028 . . . . . . . . . 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 9850 . . . . . . . . . . . 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 9850 . . . . . . . . . . . 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 15738 . . . . . . . . . . . 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 10629 . . . . . . . . . 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 2266 . . . . . . . . 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 2272 . . . . . . . 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 12350 . . . . . . . . 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 1271 . . . . . . . 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 12361 . . . . . . . 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 12344 . . . . . 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 2264 . . . 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 11737 . . 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 8488 . 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 1044 . . . 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 9545 . . . 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 2411 . . 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 803 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 713  DECID wdc 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   {crab 2512   \ cdif 3195   {csn 3667   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   RRcr 8021   0cc0 8022   1c1 8023   + caddc 8025   x. cmul 8027   < clt 8204   <_ cle 8205   - cmin 8340   / cdiv 8842   NNcn 9133   2c2 9184   NN0cn0 9392   ZZcz 9469   ZZ>=cuz 9745   QQcq 9843   mod cmo 10574   ^cexp 10790   abscabs 11548   || cdvds 12338   Primecprime 12669   /Lclgs 15716
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142
This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-xor 1418  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-frec 6552  df-1o 6577  df-2o 6578  df-oadd 6581  df-er 6697  df-en 6905  df-dom 6906  df-fin 6907  df-sup 7174  df-inf 7175  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-5 9195  df-6 9196  df-7 9197  df-8 9198  df-9 9199  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-fz 10234  df-fzo 10368  df-fl 10520  df-mod 10575  df-seqfrec 10700  df-exp 10791  df-ihash 11028  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-clim 11830  df-proddc 12102  df-dvds 12339  df-gcd 12515  df-prm 12670  df-phi 12773  df-pc 12848  df-lgs 15717
This theorem is referenced by:  lgsdir  15754
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