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Theorem lgsdirprm 15586
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 1003 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> A e. ZZ )
2 simpl2 1004 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> B e. ZZ )
3 lgsdir2 15585 . . . 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 5973 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> ( ( A x. B ) /L P ) = ( ( A x. B ) /L 2 ) )
75oveq2d 5973 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> ( A /L P ) = ( A /L 2 ) )
85oveq2d 5973 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P = 2 ) -> ( B /L P ) = ( B /L 2 ) )
97, 8oveq12d 5975 . . 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 2249 . 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 1003 . . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> A e. ZZ )
12 simpl2 1004 . . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> B e. ZZ )
1311, 12zmulcld 9521 . . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( A x. B ) e. ZZ )
14 simpl3 1005 . . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> P e. Prime )
15 prmz 12508 . . . . . 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 15566 . . . . 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 9516 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( A x. B ) /L P ) e. CC )
20 lgscl 15566 . . . . . 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 15566 . . . . . 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 9521 . . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( A /L P ) x. ( B /L P ) ) e. ZZ )
2524zcnd 9516 . . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( A /L P ) x. ( B /L P ) ) e. CC )
2619, 25subcld 8403 . . . 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 9520 . . . . . . 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 11472 . . . . . . 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 9767 . . . . . . 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 12507 . . . . . . 7 |- ( P e. Prime -> P e. NN )
32 nnq 9774 . . . . . . 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 11570 . . . . . 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 11567 . . . . . . 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 9126 . . . . . . . 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 9069 . . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> P e. RR )
4019abscld 11567 . . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( abs ` ( ( A x. B ) /L P ) ) e. RR )
4125abscld 11567 . . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( abs ` ( ( A /L P ) x. ( B /L P ) ) ) e. RR )
4240, 41readdcld 8122 . . . . . . . 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 11584 . . . . . . . 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 8107 . . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> 1 e. RR )
45 lgsle1 15567 . . . . . . . . . . 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 2206 . . . . . . . . . . . . . 14 |- { x e. ZZ | ( abs ` x ) <_ 1 } = { x e. ZZ | ( abs ` x ) <_ 1 }
4847lgscl2 15564 . . . . . . . . . . . . 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 15564 . . . . . . . . . . . . 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 15554 . . . . . . . . . . . 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 5589 . . . . . . . . . . . . . 14 |- ( x = ( ( A /L P ) x. ( B /L P ) ) -> ( abs ` x ) = ( abs ` ( ( A /L P ) x. ( B /L P ) ) ) )
5554breq1d 4061 . . . . . . . . . . . . 13 |- ( x = ( ( A /L P ) x. ( B /L P ) ) -> ( ( abs ` x ) <_ 1 <-> ( abs ` ( ( A /L P ) x. ( B /L P ) ) ) <_ 1 ) )
5655elrab 2933 . . . . . . . . . . . 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 8656 . . . . . . . . 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 9115 . . . . . . . . 9 |- 2 = ( 1 + 1 )
6159, 60breqtrrdi 4093 . . . . . . . 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 8216 . . . . . . 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 12528 . . . . . . . . 9 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
64 eluzle 9680 . . . . . . . . 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 9420 . . . . . . . . 9 |- 2 e. ZZ
68 zltlen 9471 . . . . . . . . 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 947 . . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> 2 < P )
7135, 37, 39, 62, 70lelttrd 8217 . . . . . 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 10516 . . . . . 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 1251 . . . . 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 9516 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> A e. CC )
7512zcnd 9516 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> B e. CC )
76 eldifsn 3766 . . . . . . . . . . . . . . 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 12657 . . . . . . . . . . . . . 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 9368 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( ( P - 1 ) / 2 ) e. NN0 )
8174, 75, 80mulexpd 10855 . . . . . . . . . . 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 10721 . . . . . . . . . . . . . 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 9516 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. CC )
85 zexpcl 10721 . . . . . . . . . . . . . 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 9516 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( B ^ ( ( P - 1 ) / 2 ) ) e. CC )
8884, 87mulcomd 8114 . . . . . . . . . . 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 2239 . . . . . . . . . 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 5972 . . . . . . . . 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 15571 . . . . . . . . . 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 9767 . . . . . . . . . . . 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 9767 . . . . . . . . . . . 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 9100 . . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> 0 < P )
98 lgsvalmod 15571 . . . . . . . . . . . 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 10544 . . . . . . . . . 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 9516 . . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ P e. Prime ) /\ P =/= 2 ) -> ( B /L P ) e. CC )
10284, 101mulcomd 8114 . . . . . . . . . . 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 5972 . . . . . . . . . 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 9767 . . . . . . . . . . . 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 9767 . . . . . . . . . . . 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 15571 . . . . . . . . . . . 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 10544 . . . . . . . . . 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 2243 . . . . . . . . 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 2249 . . . . . . . 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 12185 . . . . . . . . 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 1250 . . . . . . . 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 12196 . . . . . . . 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 12179 . . . . . 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 2241 . . . 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 11572 . . 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 8411 . 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 1023 . . . 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 9468 . . . 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 2388 . . 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 800 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 710  DECID wdc 836   /\ w3a 981   = wceq 1373   e. wcel 2177   =/= wne 2377   {crab 2489   \ cdif 3167   {csn 3638   class class class wbr 4051   ` cfv 5280  (class class class)co 5957   RRcr 7944   0cc0 7945   1c1 7946   + caddc 7948   x. cmul 7950   < clt 8127   <_ cle 8128   - cmin 8263   / cdiv 8765   NNcn 9056   2c2 9107   NN0cn0 9315   ZZcz 9392   ZZ>=cuz 9668   QQcq 9760   mod cmo 10489   ^cexp 10705   abscabs 11383   || cdvds 12173   Primecprime 12504   /Lclgs 15549
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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-mulrcl 8044  ax-addcom 8045  ax-mulcom 8046  ax-addass 8047  ax-mulass 8048  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-1rid 8052  ax-0id 8053  ax-rnegex 8054  ax-precex 8055  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-apti 8060  ax-pre-ltadd 8061  ax-pre-mulgt0 8062  ax-pre-mulext 8063  ax-arch 8064  ax-caucvg 8065
This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-xor 1396  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-po 4351  df-iso 4352  df-iord 4421  df-on 4423  df-ilim 4424  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-isom 5289  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-irdg 6469  df-frec 6490  df-1o 6515  df-2o 6516  df-oadd 6519  df-er 6633  df-en 6841  df-dom 6842  df-fin 6843  df-sup 7101  df-inf 7102  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-reap 8668  df-ap 8675  df-div 8766  df-inn 9057  df-2 9115  df-3 9116  df-4 9117  df-5 9118  df-6 9119  df-7 9120  df-8 9121  df-9 9122  df-n0 9316  df-z 9393  df-uz 9669  df-q 9761  df-rp 9796  df-fz 10151  df-fzo 10285  df-fl 10435  df-mod 10490  df-seqfrec 10615  df-exp 10706  df-ihash 10943  df-cj 11228  df-re 11229  df-im 11230  df-rsqrt 11384  df-abs 11385  df-clim 11665  df-proddc 11937  df-dvds 12174  df-gcd 12350  df-prm 12505  df-phi 12608  df-pc 12683  df-lgs 15550
This theorem is referenced by:  lgsdir  15587
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