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Theorem lgsdirprm 15191
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 15190 . . . 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 5935 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> ( ( A x. B ) /L P ) = ( ( A x. B ) /L 2 ) )
75oveq2d 5935 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> ( A /L P ) = ( A /L 2 ) )
85oveq2d 5935 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> ( B /L P ) = ( B /L 2 ) )
97, 8oveq12d 5937 . . 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 2236 . 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 9448 . . . . 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 12252 . . . . . 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 15171 . . . . 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 9443 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( A x. B ) /L P ) e. CC )
20 lgscl 15171 . . . . . 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 15171 . . . . . 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 9448 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( A /L P ) x. ( B /L P ) ) e. ZZ )
2524zcnd 9443 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( A /L P ) x. ( B /L P ) ) e. CC )
2619, 25subcld 8332 . . . 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 9447 . . . . . . 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 11233 . . . . . . 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 9694 . . . . . . 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 12251 . . . . . . 7 |- ( P e. Prime -> P e. NN )
32 nnq 9701 . . . . . . 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 11331 . . . . . 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 11328 . . . . . . 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 9054 . . . . . . . 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 8997 . . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> P e. RR )
4019abscld 11328 . . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( abs ` ( ( A x. B ) /L P ) ) e. RR )
4125abscld 11328 . . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( abs ` ( ( A /L P ) x. ( B /L P ) ) ) e. RR )
4240, 41readdcld 8051 . . . . . . . 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 11345 . . . . . . . 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 8036 . . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> 1 e. RR )
45 lgsle1 15172 . . . . . . . . . . 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 2193 . . . . . . . . . . . . . 14 |- { x e. ZZ | ( abs ` x ) <_ 1 } = { x e. ZZ | ( abs ` x ) <_ 1 }
4847lgscl2 15169 . . . . . . . . . . . . 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 15169 . . . . . . . . . . . . 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 15159 . . . . . . . . . . . 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 5555 . . . . . . . . . . . . . 14 |- ( x = ( ( A /L P ) x. ( B /L P ) ) -> ( abs ` x ) = ( abs ` ( ( A /L P ) x. ( B /L P ) ) ) )
5554breq1d 4040 . . . . . . . . . . . . 13 |- ( x = ( ( A /L P ) x. ( B /L P ) ) -> ( ( abs ` x ) <_ 1 <-> ( abs ` ( ( A /L P ) x. ( B /L P ) ) ) <_ 1 ) )
5655elrab 2917 . . . . . . . . . . . 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 8584 . . . . . . . . 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 9043 . . . . . . . . 9 |- 2 = ( 1 + 1 )
6159, 60breqtrrdi 4072 . . . . . . . 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 8145 . . . . . . 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 12272 . . . . . . . . 9 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
64 eluzle 9607 . . . . . . . . 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 9348 . . . . . . . . 9 |- 2 e. ZZ
68 zltlen 9398 . . . . . . . . 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 8146 . . . . . 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 10423 . . . . . 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 9443 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> A e. CC )
7512zcnd 9443 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> B e. CC )
76 eldifsn 3746 . . . . . . . . . . . . . . 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 12400 . . . . . . . . . . . . . 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 9296 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( P - 1 ) / 2 ) e. NN0 )
8174, 75, 80mulexpd 10762 . . . . . . . . . . 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 10628 . . . . . . . . . . . . . 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 9443 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. CC )
85 zexpcl 10628 . . . . . . . . . . . . . 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 9443 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( B ^ ( ( P - 1 ) / 2 ) ) e. CC )
8884, 87mulcomd 8043 . . . . . . . . . . 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 2226 . . . . . . . . . 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 5934 . . . . . . . . 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 15176 . . . . . . . . . 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 9694 . . . . . . . . . . . 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 9694 . . . . . . . . . . . 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 9028 . . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> 0 < P )
98 lgsvalmod 15176 . . . . . . . . . . . 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 10451 . . . . . . . . . 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 9443 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( B /L P ) e. CC )
10284, 101mulcomd 8043 . . . . . . . . . . 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 5934 . . . . . . . . . 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 9694 . . . . . . . . . . . 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 9694 . . . . . . . . . . . 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 15176 . . . . . . . . . . . 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 10451 . . . . . . . . . 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 2230 . . . . . . . . 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 2236 . . . . . . . 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 11945 . . . . . . . . 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 11956 . . . . . . . 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 11939 . . . . . 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 2228 . . . 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 11333 . . 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 8340 . 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 9395 . . . 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 2375 . . 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 2164   =/= wne 2364   {crab 2476   \ cdif 3151   {csn 3619   class class class wbr 4030   ` cfv 5255  (class class class)co 5919   RRcr 7873   0cc0 7874   1c1 7875   + caddc 7877   x. cmul 7879   < clt 8056   <_ cle 8057   - cmin 8192   / cdiv 8693   NNcn 8984   2c2 9035   NN0cn0 9243   ZZcz 9320   ZZ>=cuz 9595   QQcq 9687   mod cmo 10396   ^cexp 10612   abscabs 11144   || cdvds 11933   Primecprime 12248   /Lclgs 15154
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994
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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-frec 6446  df-1o 6471  df-2o 6472  df-oadd 6475  df-er 6589  df-en 6797  df-dom 6798  df-fin 6799  df-sup 7045  df-inf 7046  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-5 9046  df-6 9047  df-7 9048  df-8 9049  df-9 9050  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-fz 10078  df-fzo 10212  df-fl 10342  df-mod 10397  df-seqfrec 10522  df-exp 10613  df-ihash 10850  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-clim 11425  df-proddc 11697  df-dvds 11934  df-gcd 12083  df-prm 12249  df-phi 12352  df-pc 12426  df-lgs 15155
This theorem is referenced by:  lgsdir  15192
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