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Theorem lgsdirprm 14006
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 14005 . . . 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 5881 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =  2 )  ->  ( ( A  x.  B )  /L P )  =  ( ( A  x.  B )  /L 2 ) )
75oveq2d 5881 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =  2 )  ->  ( A  /L P )  =  ( A  /L 2 ) )
85oveq2d 5881 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =  2 )  ->  ( B  /L P )  =  ( B  /L 2 ) )
97, 8oveq12d 5883 . . 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 2218 . 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 9354 . . . . 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 12078 . . . . . 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 13986 . . . . 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 9349 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  -> 
( ( A  x.  B )  /L
P )  e.  CC )
20 lgscl 13986 . . . . . 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 13986 . . . . . 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 9354 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  -> 
( ( A  /L P )  x.  ( B  /L
P ) )  e.  ZZ )
2524zcnd 9349 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  -> 
( ( A  /L P )  x.  ( B  /L
P ) )  e.  CC )
2619, 25subcld 8242 . . . 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 9353 . . . . . . 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 11063 . . . . . . 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 9599 . . . . . . 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 12077 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  NN )
32 nnq 9606 . . . . . . 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 11161 . . . . . 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 11158 . . . . . . 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 8962 . . . . . . . 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 8905 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  ->  P  e.  RR )
4019abscld 11158 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  -> 
( abs `  (
( A  x.  B
)  /L P ) )  e.  RR )
4125abscld 11158 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  -> 
( abs `  (
( A  /L
P )  x.  ( B  /L P ) ) )  e.  RR )
4240, 41readdcld 7961 . . . . . . . 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 11175 . . . . . . . 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 7947 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  -> 
1  e.  RR )
45 lgsle1 13987 . . . . . . . . . . 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 2175 . . . . . . . . . . . . . 14  |-  { x  e.  ZZ  |  ( abs `  x )  <_  1 }  =  { x  e.  ZZ  |  ( abs `  x )  <_  1 }
4847lgscl2 13984 . . . . . . . . . . . . 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 13984 . . . . . . . . . . . . 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 13974 . . . . . . . . . . . 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 5507 . . . . . . . . . . . . . 14  |-  ( x  =  ( ( A  /L P )  x.  ( B  /L P ) )  ->  ( abs `  x
)  =  ( abs `  ( ( A  /L P )  x.  ( B  /L
P ) ) ) )
5554breq1d 4008 . . . . . . . . . . . . 13  |-  ( x  =  ( ( A  /L P )  x.  ( B  /L P ) )  ->  ( ( abs `  x )  <_  1  <->  ( abs `  ( ( A  /L P )  x.  ( B  /L P ) ) )  <_  1
) )
5655elrab 2891 . . . . . . . . . . . 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 8494 . . . . . . . . 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 8951 . . . . . . . . 9  |-  2  =  ( 1  +  1 )
6159, 60breqtrrdi 4040 . . . . . . . 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 8055 . . . . . . 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 12098 . . . . . . . . 9  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
64 eluzle 9513 . . . . . . . . 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 9254 . . . . . . . . 9  |-  2  e.  ZZ
68 zltlen 9304 . . . . . . . . 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 8056 . . . . . 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 10319 . . . . . 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 9349 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  ->  A  e.  CC )
7512zcnd 9349 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  ->  B  e.  CC )
76 eldifsn 3716 . . . . . . . . . . . . . . 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 12226 . . . . . . . . . . . . . 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 9202 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  -> 
( ( P  - 
1 )  /  2
)  e.  NN0 )
8174, 75, 80mulexpd 10638 . . . . . . . . . . 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 10505 . . . . . . . . . . . . . 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 9349 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  -> 
( A ^ (
( P  -  1 )  /  2 ) )  e.  CC )
85 zexpcl 10505 . . . . . . . . . . . . . 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 9349 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  -> 
( B ^ (
( P  -  1 )  /  2 ) )  e.  CC )
8884, 87mulcomd 7953 . . . . . . . . . . 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 2208 . . . . . . . . . 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 5880 . . . . . . . . 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 13991 . . . . . . . . . 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 9599 . . . . . . . . . . . 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 9599 . . . . . . . . . . . 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 8936 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  -> 
0  <  P )
98 lgsvalmod 13991 . . . . . . . . . . . 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 10347 . . . . . . . . . 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 9349 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  P  e.  Prime )  /\  P  =/=  2 )  -> 
( B  /L
P )  e.  CC )
10284, 101mulcomd 7953 . . . . . . . . . . 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 5880 . . . . . . . . . 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 9599 . . . . . . . . . . . 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 9599 . . . . . . . . . . . 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 13991 . . . . . . . . . . . 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 10347 . . . . . . . . . 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 2212 . . . . . . . . 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 2218 . . . . . . . 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 11774 . . . . . . . . 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 11785 . . . . . . . 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 11768 . . . . . 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 2210 . . . 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 11163 . . 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 8250 . 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 9301 . . . 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 2356 . . 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 2146    =/= wne 2345   {crab 2457    \ cdif 3124   {csn 3589   class class class wbr 3998   ` cfv 5208  (class class class)co 5865   RRcr 7785   0cc0 7786   1c1 7787    + caddc 7789    x. cmul 7791    < clt 7966    <_ cle 7967    - cmin 8102    / cdiv 8602   NNcn 8892   2c2 8943   NN0cn0 9149   ZZcz 9226   ZZ>=cuz 9501   QQcq 9592    mod cmo 10292   ^cexp 10489   abscabs 10974    || cdvds 11762   Primecprime 12074    /Lclgs 13969
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906
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 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-isom 5217  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-frec 6382  df-1o 6407  df-2o 6408  df-oadd 6411  df-er 6525  df-en 6731  df-dom 6732  df-fin 6733  df-sup 6973  df-inf 6974  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8603  df-inn 8893  df-2 8951  df-3 8952  df-4 8953  df-5 8954  df-6 8955  df-7 8956  df-8 8957  df-9 8958  df-n0 9150  df-z 9227  df-uz 9502  df-q 9593  df-rp 9625  df-fz 9980  df-fzo 10113  df-fl 10240  df-mod 10293  df-seqfrec 10416  df-exp 10490  df-ihash 10724  df-cj 10819  df-re 10820  df-im 10821  df-rsqrt 10975  df-abs 10976  df-clim 11255  df-proddc 11527  df-dvds 11763  df-gcd 11911  df-prm 12075  df-phi 12178  df-pc 12252  df-lgs 13970
This theorem is referenced by:  lgsdir  14007
  Copyright terms: Public domain W3C validator