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Theorem lgsdirprm 15762
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 1026 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> A e. ZZ )
2 simpl2 1027 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> B e. ZZ )
3 lgsdir2 15761 . . . 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 6033 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> ( ( A x. B ) /L P ) = ( ( A x. B ) /L 2 ) )
75oveq2d 6033 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> ( A /L P ) = ( A /L 2 ) )
85oveq2d 6033 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> ( B /L P ) = ( B /L 2 ) )
97, 8oveq12d 6035 . . 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 2274 . 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 1026 . . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> A e. ZZ )
12 simpl2 1027 . . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> B e. ZZ )
1311, 12zmulcld 9607 . . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( A x. B ) e. ZZ )
14 simpl3 1028 . . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> P e. Prime )
15 prmz 12682 . . . . . 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 15742 . . . . 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 9602 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( A x. B ) /L P ) e. CC )
20 lgscl 15742 . . . . . 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 15742 . . . . . 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 9607 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( A /L P ) x. ( B /L P ) ) e. ZZ )
2524zcnd 9602 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( A /L P ) x. ( B /L P ) ) e. CC )
2619, 25subcld 8489 . . . 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 9606 . . . . . . 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 11646 . . . . . . 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 9859 . . . . . . 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 12681 . . . . . . 7 |- ( P e. Prime -> P e. NN )
32 nnq 9866 . . . . . . 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 11744 . . . . . 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 11741 . . . . . . 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 9212 . . . . . . . 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 9155 . . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> P e. RR )
4019abscld 11741 . . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( abs ` ( ( A x. B ) /L P ) ) e. RR )
4125abscld 11741 . . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( abs ` ( ( A /L P ) x. ( B /L P ) ) ) e. RR )
4240, 41readdcld 8208 . . . . . . . 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 11758 . . . . . . . 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 8193 . . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> 1 e. RR )
45 lgsle1 15743 . . . . . . . . . . 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 2231 . . . . . . . . . . . . . 14 |- { x e. ZZ | ( abs ` x ) <_ 1 } = { x e. ZZ | ( abs ` x ) <_ 1 }
4847lgscl2 15740 . . . . . . . . . . . . 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 15740 . . . . . . . . . . . . 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 15730 . . . . . . . . . . . 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 5639 . . . . . . . . . . . . . 14 |- ( x = ( ( A /L P ) x. ( B /L P ) ) -> ( abs ` x ) = ( abs ` ( ( A /L P ) x. ( B /L P ) ) ) )
5554breq1d 4098 . . . . . . . . . . . . 13 |- ( x = ( ( A /L P ) x. ( B /L P ) ) -> ( ( abs ` x ) <_ 1 <-> ( abs ` ( ( A /L P ) x. ( B /L P ) ) ) <_ 1 ) )
5655elrab 2962 . . . . . . . . . . . 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 8742 . . . . . . . . 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 9201 . . . . . . . . 9 |- 2 = ( 1 + 1 )
6159, 60breqtrrdi 4130 . . . . . . . 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 8302 . . . . . . 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 12702 . . . . . . . . 9 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
64 eluzle 9767 . . . . . . . . 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 9506 . . . . . . . . 9 |- 2 e. ZZ
68 zltlen 9557 . . . . . . . . 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 952 . . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> 2 < P )
7135, 37, 39, 62, 70lelttrd 8303 . . . . . 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 10610 . . . . . 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 1274 . . . . 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 9602 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> A e. CC )
7512zcnd 9602 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> B e. CC )
76 eldifsn 3800 . . . . . . . . . . . . . . 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 12831 . . . . . . . . . . . . . 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 9454 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( P - 1 ) / 2 ) e. NN0 )
8174, 75, 80mulexpd 10949 . . . . . . . . . . 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 10815 . . . . . . . . . . . . . 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 9602 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. CC )
85 zexpcl 10815 . . . . . . . . . . . . . 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 9602 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( B ^ ( ( P - 1 ) / 2 ) ) e. CC )
8884, 87mulcomd 8200 . . . . . . . . . . 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 2264 . . . . . . . . . 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 6032 . . . . . . . . 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 15747 . . . . . . . . . 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 9859 . . . . . . . . . . . 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 9859 . . . . . . . . . . . 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 9186 . . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> 0 < P )
98 lgsvalmod 15747 . . . . . . . . . . . 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 10638 . . . . . . . . . 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 9602 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( B /L P ) e. CC )
10284, 101mulcomd 8200 . . . . . . . . . . 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 6032 . . . . . . . . . 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 9859 . . . . . . . . . . . 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 9859 . . . . . . . . . . . 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 15747 . . . . . . . . . . . 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 10638 . . . . . . . . . 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 2268 . . . . . . . . 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 2274 . . . . . . . 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 12359 . . . . . . . . 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 1273 . . . . . . . 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 12370 . . . . . . . 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 12353 . . . . . 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 2266 . . . 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 11746 . . 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 8497 . 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 1046 . . . 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 9554 . . . 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 2413 . . 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 805 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 715  DECID wdc 841   /\ w3a 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   {crab 2514   \ cdif 3197   {csn 3669   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   RRcr 8030   0cc0 8031   1c1 8032   + caddc 8034   x. cmul 8036   < clt 8213   <_ cle 8214   - cmin 8349   / cdiv 8851   NNcn 9142   2c2 9193   NN0cn0 9401   ZZcz 9478   ZZ>=cuz 9754   QQcq 9852   mod cmo 10583   ^cexp 10799   abscabs 11557   || cdvds 12347   Primecprime 12678   /Lclgs 15725
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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151
This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-xor 1420  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-frec 6556  df-1o 6581  df-2o 6582  df-oadd 6585  df-er 6701  df-en 6909  df-dom 6910  df-fin 6911  df-sup 7182  df-inf 7183  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-9 9208  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fzo 10377  df-fl 10529  df-mod 10584  df-seqfrec 10709  df-exp 10800  df-ihash 11037  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-clim 11839  df-proddc 12111  df-dvds 12348  df-gcd 12524  df-prm 12679  df-phi 12782  df-pc 12857  df-lgs 15726
This theorem is referenced by:  lgsdir  15763
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