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Theorem lgsqr 22628
Description: The Legendre symbol for odd primes is  1 iff the number is not a multiple of the prime (in which case it is  0, see lgsne0 22615) and the number is a quadratic residue  mod  P (it is  -u 1 for nonresidues by the process of elimination from lgsabs1 22616). Given our definition of the Legendre symbol, this theorem is equivalent to Euler's criterion. (Contributed by Mario Carneiro, 15-Jun-2015.)
Assertion
Ref Expression
lgsqr  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  /L P )  =  1  <->  ( -.  P  ||  A  /\  E. x  e.  ZZ  P  ||  ( ( x ^
2 )  -  A
) ) ) )
Distinct variable groups:    x, A    x, P

Proof of Theorem lgsqr
StepHypRef Expression
1 eldifi 3475 . . . . . . . . . . 11  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  e.  Prime )
21adantl 463 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  P  e.  Prime )
3 prmz 13763 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  ZZ )
42, 3syl 16 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  P  e.  ZZ )
5 simpl 454 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  A  e.  ZZ )
6 gcdcom 13700 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  A  e.  ZZ )  ->  ( P  gcd  A
)  =  ( A  gcd  P ) )
74, 5, 6syl2anc 656 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( P  gcd  A )  =  ( A  gcd  P ) )
87eqeq1d 2449 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( P  gcd  A )  =  1  <->  ( A  gcd  P )  =  1 ) )
9 coprm 13782 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
102, 5, 9syl2anc 656 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
11 lgsne0 22615 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ZZ )  ->  ( ( A  /L P )  =/=  0  <->  ( A  gcd  P )  =  1 ) )
125, 4, 11syl2anc 656 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  /L P )  =/=  0  <->  ( A  gcd  P )  =  1 ) )
138, 10, 123bitr4d 285 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( -.  P  ||  A  <->  ( A  /L P )  =/=  0 ) )
1413necon4bbid 2674 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( P  ||  A  <->  ( A  /L P )  =  0 ) )
15 0ne1 10385 . . . . . 6  |-  0  =/=  1
16 neeq1 2614 . . . . . 6  |-  ( ( A  /L P )  =  0  -> 
( ( A  /L P )  =/=  1  <->  0  =/=  1
) )
1715, 16mpbiri 233 . . . . 5  |-  ( ( A  /L P )  =  0  -> 
( A  /L
P )  =/=  1
)
1814, 17syl6bi 228 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( P  ||  A  ->  ( A  /L P )  =/=  1 ) )
1918necon2bd 2658 . . 3  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  /L P )  =  1  ->  -.  P  ||  A ) )
20 lgsqrlem5 22627 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  ( A  /L P )  =  1 )  ->  E. x  e.  ZZ  P  ||  (
( x ^ 2 )  -  A ) )
21203expia 1184 . . 3  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  /L P )  =  1  ->  E. x  e.  ZZ  P  ||  (
( x ^ 2 )  -  A ) ) )
2219, 21jcad 530 . 2  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  /L P )  =  1  ->  ( -.  P  ||  A  /\  E. x  e.  ZZ  P  ||  ( ( x ^
2 )  -  A
) ) ) )
23 simprl 750 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  x  e.  ZZ )
2423zred 10743 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  x  e.  RR )
25 absresq 12787 . . . . . . 7  |-  ( x  e.  RR  ->  (
( abs `  x
) ^ 2 )  =  ( x ^
2 ) )
2624, 25syl 16 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( ( abs `  x
) ^ 2 )  =  ( x ^
2 ) )
2726oveq1d 6105 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( ( ( abs `  x ) ^ 2 )  /L P )  =  ( ( x ^ 2 )  /L P ) )
28 simplr 749 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  -.  P  ||  A )
291ad3antlr 725 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  P  e.  Prime )
3029, 3syl 16 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  P  e.  ZZ )
31 zsqcl 11932 . . . . . . . . . . . 12  |-  ( x  e.  ZZ  ->  (
x ^ 2 )  e.  ZZ )
3223, 31syl 16 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( x ^ 2 )  e.  ZZ )
33 simplll 752 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  A  e.  ZZ )
34 simprr 751 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  P  ||  ( ( x ^ 2 )  -  A ) )
35 dvdssub2 13566 . . . . . . . . . . 11  |-  ( ( ( P  e.  ZZ  /\  ( x ^ 2 )  e.  ZZ  /\  A  e.  ZZ )  /\  P  ||  ( ( x ^ 2 )  -  A ) )  ->  ( P  ||  ( x ^ 2 )  <->  P  ||  A ) )
3630, 32, 33, 34, 35syl31anc 1216 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( P  ||  (
x ^ 2 )  <-> 
P  ||  A )
)
3728, 36mtbird 301 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  -.  P  ||  ( x ^ 2 ) )
38 2nn 10475 . . . . . . . . . . 11  |-  2  e.  NN
3938a1i 11 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
2  e.  NN )
40 prmdvdsexp 13796 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  x  e.  ZZ  /\  2  e.  NN )  ->  ( P  ||  ( x ^
2 )  <->  P  ||  x
) )
4129, 23, 39, 40syl3anc 1213 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( P  ||  (
x ^ 2 )  <-> 
P  ||  x )
)
4237, 41mtbid 300 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  -.  P  ||  x )
43 dvds0 13544 . . . . . . . . . . 11  |-  ( P  e.  ZZ  ->  P  ||  0 )
4430, 43syl 16 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  P  ||  0 )
45 breq2 4293 . . . . . . . . . 10  |-  ( x  =  0  ->  ( P  ||  x  <->  P  ||  0
) )
4644, 45syl5ibrcom 222 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( x  =  0  ->  P  ||  x
) )
4746necon3bd 2643 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( -.  P  ||  x  ->  x  =/=  0
) )
4842, 47mpd 15 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  x  =/=  0 )
49 nnabscl 12809 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  x  =/=  0 )  -> 
( abs `  x
)  e.  NN )
5023, 48, 49syl2anc 656 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( abs `  x
)  e.  NN )
5150nnzd 10742 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( abs `  x
)  e.  ZZ )
52 gcdcom 13700 . . . . . . . 8  |-  ( ( ( abs `  x
)  e.  ZZ  /\  P  e.  ZZ )  ->  ( ( abs `  x
)  gcd  P )  =  ( P  gcd  ( abs `  x ) ) )
5351, 30, 52syl2anc 656 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( ( abs `  x
)  gcd  P )  =  ( P  gcd  ( abs `  x ) ) )
54 dvdsabsb 13548 . . . . . . . . . 10  |-  ( ( P  e.  ZZ  /\  x  e.  ZZ )  ->  ( P  ||  x  <->  P 
||  ( abs `  x
) ) )
5530, 23, 54syl2anc 656 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( P  ||  x  <->  P 
||  ( abs `  x
) ) )
5642, 55mtbid 300 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  -.  P  ||  ( abs `  x ) )
57 coprm 13782 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( abs `  x )  e.  ZZ )  ->  ( -.  P  ||  ( abs `  x )  <->  ( P  gcd  ( abs `  x
) )  =  1 ) )
5829, 51, 57syl2anc 656 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( -.  P  ||  ( abs `  x )  <-> 
( P  gcd  ( abs `  x ) )  =  1 ) )
5956, 58mpbid 210 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( P  gcd  ( abs `  x ) )  =  1 )
6053, 59eqtrd 2473 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( ( abs `  x
)  gcd  P )  =  1 )
61 lgssq 22617 . . . . . 6  |-  ( ( ( abs `  x
)  e.  NN  /\  P  e.  ZZ  /\  (
( abs `  x
)  gcd  P )  =  1 )  -> 
( ( ( abs `  x ) ^ 2 )  /L P )  =  1 )
6250, 30, 60, 61syl3anc 1213 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( ( ( abs `  x ) ^ 2 )  /L P )  =  1 )
63 prmnn 13762 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
6429, 63syl 16 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  P  e.  NN )
65 moddvds 13538 . . . . . . . . 9  |-  ( ( P  e.  NN  /\  ( x ^ 2 )  e.  ZZ  /\  A  e.  ZZ )  ->  ( ( ( x ^ 2 )  mod 
P )  =  ( A  mod  P )  <-> 
P  ||  ( (
x ^ 2 )  -  A ) ) )
6664, 32, 33, 65syl3anc 1213 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( ( ( x ^ 2 )  mod 
P )  =  ( A  mod  P )  <-> 
P  ||  ( (
x ^ 2 )  -  A ) ) )
6734, 66mpbird 232 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( ( x ^
2 )  mod  P
)  =  ( A  mod  P ) )
6867oveq1d 6105 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( ( ( x ^ 2 )  mod 
P )  /L
P )  =  ( ( A  mod  P
)  /L P ) )
69 eldifsni 3998 . . . . . . . . . 10  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  =/=  2 )
7069ad3antlr 725 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  P  =/=  2 )
7170necomd 2693 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
2  =/=  P )
72 2z 10674 . . . . . . . . . 10  |-  2  e.  ZZ
73 uzid 10871 . . . . . . . . . 10  |-  ( 2  e.  ZZ  ->  2  e.  ( ZZ>= `  2 )
)
7472, 73ax-mp 5 . . . . . . . . 9  |-  2  e.  ( ZZ>= `  2 )
75 dvdsprm 13781 . . . . . . . . . 10  |-  ( ( 2  e.  ( ZZ>= ` 
2 )  /\  P  e.  Prime )  ->  (
2  ||  P  <->  2  =  P ) )
7675necon3bbid 2640 . . . . . . . . 9  |-  ( ( 2  e.  ( ZZ>= ` 
2 )  /\  P  e.  Prime )  ->  ( -.  2  ||  P  <->  2  =/=  P ) )
7774, 29, 76sylancr 658 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( -.  2  ||  P 
<->  2  =/=  P ) )
7871, 77mpbird 232 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  -.  2  ||  P )
79 lgsmod 22603 . . . . . . 7  |-  ( ( ( x ^ 2 )  e.  ZZ  /\  P  e.  NN  /\  -.  2  ||  P )  -> 
( ( ( x ^ 2 )  mod 
P )  /L
P )  =  ( ( x ^ 2 )  /L P ) )
8032, 64, 78, 79syl3anc 1213 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( ( ( x ^ 2 )  mod 
P )  /L
P )  =  ( ( x ^ 2 )  /L P ) )
81 lgsmod 22603 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  NN  /\  -.  2  ||  P )  -> 
( ( A  mod  P )  /L P )  =  ( A  /L P ) )
8233, 64, 78, 81syl3anc 1213 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( ( A  mod  P )  /L P )  =  ( A  /L P ) )
8368, 80, 823eqtr3d 2481 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( ( x ^
2 )  /L
P )  =  ( A  /L P ) )
8427, 62, 833eqtr3rd 2482 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( A  /L
P )  =  1 )
8584rexlimdvaa 2840 . . 3  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  ->  ( E. x  e.  ZZ  P  ||  ( ( x ^ 2 )  -  A )  ->  ( A  /L P )  =  1 ) )
8685expimpd 600 . 2  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( -.  P  ||  A  /\  E. x  e.  ZZ  P  ||  ( ( x ^
2 )  -  A
) )  ->  ( A  /L P )  =  1 ) )
8722, 86impbid 191 1  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  /L P )  =  1  <->  ( -.  P  ||  A  /\  E. x  e.  ZZ  P  ||  ( ( x ^
2 )  -  A
) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604   E.wrex 2714    \ cdif 3322   {csn 3874   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   RRcr 9277   0cc0 9278   1c1 9279    - cmin 9591   NNcn 10318   2c2 10367   ZZcz 10642   ZZ>=cuz 10857    mod cmo 11704   ^cexp 11861   abscabs 12719    || cdivides 13531    gcd cgcd 13686   Primecprime 13759    /Lclgs 22576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-ofr 6320  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-ec 7099  df-qs 7103  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-dvds 13532  df-gcd 13687  df-prm 13760  df-phi 13837  df-pc 13900  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-0g 14376  df-gsum 14377  df-prds 14382  df-pws 14384  df-imas 14442  df-divs 14443  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-mhm 15460  df-submnd 15461  df-grp 15538  df-minusg 15539  df-sbg 15540  df-mulg 15541  df-subg 15671  df-nsg 15672  df-eqg 15673  df-ghm 15738  df-cntz 15828  df-cmn 16272  df-abl 16273  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-invr 16754  df-dvr 16765  df-rnghom 16796  df-drng 16814  df-field 16815  df-subrg 16843  df-lmod 16930  df-lss 16992  df-lsp 17031  df-sra 17231  df-rgmod 17232  df-lidl 17233  df-rsp 17234  df-2idl 17292  df-nzr 17318  df-rlreg 17332  df-domn 17333  df-idom 17334  df-assa 17362  df-asp 17363  df-ascl 17364  df-psr 17407  df-mvr 17408  df-mpl 17409  df-evls 17410  df-evl 17411  df-opsr 17415  df-psr1 17589  df-vr1 17590  df-ply1 17591  df-evl1 17593  df-coe1 17594  df-cnfld 17719  df-zring 17784  df-zrh 17835  df-zn 17838  df-mdeg 21467  df-deg1 21468  df-mon1 21545  df-uc1p 21546  df-q1p 21547  df-r1p 21548  df-lgs 22577
This theorem is referenced by:  2sqlem11  22657  2sqblem  22659
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