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Theorem lgsdirprm 14102
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 1000 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =  2 )  ->  A  e.  ZZ )
2 simpl2 1001 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =  2 )  ->  B  e.  ZZ )
3 lgsdir2 14101 . . . 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 5885 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =  2 )  ->  ( ( A  x.  B )  /L P )  =  ( ( A  x.  B )  /L 2 ) )
75oveq2d 5885 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =  2 )  ->  ( A  /L P )  =  ( A  /L 2 ) )
85oveq2d 5885 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =  2 )  ->  ( B  /L P )  =  ( B  /L 2 ) )
97, 8oveq12d 5887 . . 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 2220 . 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 1000 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  ->  A  e.  ZZ )
12 simpl2 1001 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  ->  B  e.  ZZ )
1311, 12zmulcld 9370 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  -> 
( A  x.  B
)  e.  ZZ )
14 simpl3 1002 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  ->  P  e.  Prime )
15 prmz 12094 . . . . . 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 14082 . . . . 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 9365 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  -> 
( ( A  x.  B )  /L
P )  e.  CC )
20 lgscl 14082 . . . . . 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 14082 . . . . . 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 9370 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  -> 
( ( A  /L P )  x.  ( B  /L
P ) )  e.  ZZ )
2524zcnd 9365 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  -> 
( ( A  /L P )  x.  ( B  /L
P ) )  e.  CC )
2619, 25subcld 8258 . . . 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 9369 . . . . . . 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 11079 . . . . . . 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 9615 . . . . . . 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 12093 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  NN )
32 nnq 9622 . . . . . . 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 11177 . . . . . 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 11174 . . . . . . 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 8978 . . . . . . . 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 8921 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  ->  P  e.  RR )
4019abscld 11174 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  -> 
( abs `  (
( A  x.  B
)  /L P ) )  e.  RR )
4125abscld 11174 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  -> 
( abs `  (
( A  /L
P )  x.  ( B  /L P ) ) )  e.  RR )
4240, 41readdcld 7977 . . . . . . . 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 11191 . . . . . . . 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 7963 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  -> 
1  e.  RR )
45 lgsle1 14083 . . . . . . . . . . 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 2177 . . . . . . . . . . . . . 14  |-  { x  e.  ZZ  |  ( abs `  x )  <_  1 }  =  { x  e.  ZZ  |  ( abs `  x )  <_  1 }
4847lgscl2 14080 . . . . . . . . . . . . 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 14080 . . . . . . . . . . . . 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 14070 . . . . . . . . . . . 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 5511 . . . . . . . . . . . . . 14  |-  ( x  =  ( ( A  /L P )  x.  ( B  /L P ) )  ->  ( abs `  x
)  =  ( abs `  ( ( A  /L P )  x.  ( B  /L
P ) ) ) )
5554breq1d 4010 . . . . . . . . . . . . 13  |-  ( x  =  ( ( A  /L P )  x.  ( B  /L P ) )  ->  ( ( abs `  x )  <_  1  <->  ( abs `  ( ( A  /L P )  x.  ( B  /L P ) ) )  <_  1
) )
5655elrab 2893 . . . . . . . . . . . 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 8510 . . . . . . . . 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 8967 . . . . . . . . 9  |-  2  =  ( 1  +  1 )
6159, 60breqtrrdi 4042 . . . . . . . 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 8071 . . . . . . 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 12114 . . . . . . . . 9  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
64 eluzle 9529 . . . . . . . . 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 9270 . . . . . . . . 9  |-  2  e.  ZZ
68 zltlen 9320 . . . . . . . . 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 944 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  -> 
2  <  P )
7135, 37, 39, 62, 70lelttrd 8072 . . . . . 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 10335 . . . . . 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 1239 . . . . 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 9365 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  ->  A  e.  CC )
7512zcnd 9365 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  ->  B  e.  CC )
76 eldifsn 3718 . . . . . . . . . . . . . . 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 12242 . . . . . . . . . . . . . 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 9218 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  -> 
( ( P  - 
1 )  /  2
)  e.  NN0 )
8174, 75, 80mulexpd 10654 . . . . . . . . . . 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 10521 . . . . . . . . . . . . . 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 9365 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  -> 
( A ^ (
( P  -  1 )  /  2 ) )  e.  CC )
85 zexpcl 10521 . . . . . . . . . . . . . 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 9365 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  -> 
( B ^ (
( P  -  1 )  /  2 ) )  e.  CC )
8884, 87mulcomd 7969 . . . . . . . . . . 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 2210 . . . . . . . . . 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 5884 . . . . . . . . 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 14087 . . . . . . . . . 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 9615 . . . . . . . . . . . 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 9615 . . . . . . . . . . . 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 8952 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  -> 
0  <  P )
98 lgsvalmod 14087 . . . . . . . . . . . 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 10363 . . . . . . . . . 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 9365 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  -> 
( B  /L
P )  e.  CC )
10284, 101mulcomd 7969 . . . . . . . . . . 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 5884 . . . . . . . . . 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 9615 . . . . . . . . . . . 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 9615 . . . . . . . . . . . 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 14087 . . . . . . . . . . . 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 10363 . . . . . . . . . 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 2214 . . . . . . . . 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 2220 . . . . . . . 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 11790 . . . . . . . . 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 1238 . . . . . . . 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 11801 . . . . . . . 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 11784 . . . . . 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 2212 . . . 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 11179 . . 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 8266 . 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 1020 . . . 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 9317 . . . 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 2358 . . 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 798 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 708  DECID wdc 834    /\ w3a 978    = wceq 1353    e. wcel 2148    =/= wne 2347   {crab 2459    \ cdif 3126   {csn 3591   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   RRcr 7801   0cc0 7802   1c1 7803    + caddc 7805    x. cmul 7807    < clt 7982    <_ cle 7983    - cmin 8118    / cdiv 8618   NNcn 8908   2c2 8959   NN0cn0 9165   ZZcz 9242   ZZ>=cuz 9517   QQcq 9608    mod cmo 10308   ^cexp 10505   abscabs 10990    || cdvds 11778   Primecprime 12090    /Lclgs 14065
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-xor 1376  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-2o 6412  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-sup 6977  df-inf 6978  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-5 8970  df-6 8971  df-7 8972  df-8 8973  df-9 8974  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-ihash 10740  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-proddc 11543  df-dvds 11779  df-gcd 11927  df-prm 12091  df-phi 12194  df-pc 12268  df-lgs 14066
This theorem is referenced by:  lgsdir  14103
  Copyright terms: Public domain W3C validator