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Theorem lgsqr 23365
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 23352) and the number is a quadratic residue  mod  P (it is  -u 1 for nonresidues by the process of elimination from lgsabs1 23353). 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 3626 . . . . . . . . . . 11  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  e.  Prime )
21adantl 466 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  P  e.  Prime )
3 prmz 14079 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  ZZ )
42, 3syl 16 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  P  e.  ZZ )
5 simpl 457 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  A  e.  ZZ )
6 gcdcom 14016 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  A  e.  ZZ )  ->  ( P  gcd  A
)  =  ( A  gcd  P ) )
74, 5, 6syl2anc 661 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( P  gcd  A )  =  ( A  gcd  P ) )
87eqeq1d 2469 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( P  gcd  A )  =  1  <->  ( A  gcd  P )  =  1 ) )
9 coprm 14099 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
102, 5, 9syl2anc 661 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
11 lgsne0 23352 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ZZ )  ->  ( ( A  /L P )  =/=  0  <->  ( A  gcd  P )  =  1 ) )
125, 4, 11syl2anc 661 . . . . . . 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 2720 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( P  ||  A  <->  ( A  /L P )  =  0 ) )
15 0ne1 10602 . . . . . 6  |-  0  =/=  1
16 neeq1 2748 . . . . . 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 2682 . . 3  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  /L P )  =  1  ->  -.  P  ||  A ) )
20 lgsqrlem5 23364 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  ( A  /L P )  =  1 )  ->  E. x  e.  ZZ  P  ||  (
( x ^ 2 )  -  A ) )
21203expia 1198 . . 3  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  /L P )  =  1  ->  E. x  e.  ZZ  P  ||  (
( x ^ 2 )  -  A ) ) )
2219, 21jcad 533 . 2  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  /L P )  =  1  ->  ( -.  P  ||  A  /\  E. x  e.  ZZ  P  ||  ( ( x ^
2 )  -  A
) ) ) )
23 simprl 755 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  x  e.  ZZ )
2423zred 10965 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  x  e.  RR )
25 absresq 13097 . . . . . . 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 6298 . . . . 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 754 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  -.  P  ||  A )
291ad3antlr 730 . . . . . . . . . . . 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 12205 . . . . . . . . . . . 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 757 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  A  e.  ZZ )
34 simprr 756 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  P  ||  ( ( x ^ 2 )  -  A ) )
35 dvdssub2 13881 . . . . . . . . . . 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 1231 . . . . . . . . . 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 10692 . . . . . . . . . . 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 14113 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  x  e.  ZZ  /\  2  e.  NN )  ->  ( P  ||  ( x ^
2 )  <->  P  ||  x
) )
4129, 23, 39, 40syl3anc 1228 . . . . . . . . 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 13859 . . . . . . . . . . 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 4451 . . . . . . . . . 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 2679 . . . . . . . 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 13120 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  x  =/=  0 )  -> 
( abs `  x
)  e.  NN )
5023, 48, 49syl2anc 661 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( abs `  x
)  e.  NN )
5150nnzd 10964 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( abs `  x
)  e.  ZZ )
52 gcdcom 14016 . . . . . . . 8  |-  ( ( ( abs `  x
)  e.  ZZ  /\  P  e.  ZZ )  ->  ( ( abs `  x
)  gcd  P )  =  ( P  gcd  ( abs `  x ) ) )
5351, 30, 52syl2anc 661 . . . . . . 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 13863 . . . . . . . . . 10  |-  ( ( P  e.  ZZ  /\  x  e.  ZZ )  ->  ( P  ||  x  <->  P 
||  ( abs `  x
) ) )
5530, 23, 54syl2anc 661 . . . . . . . . 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 14099 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( abs `  x )  e.  ZZ )  ->  ( -.  P  ||  ( abs `  x )  <->  ( P  gcd  ( abs `  x
) )  =  1 ) )
5829, 51, 57syl2anc 661 . . . . . . . 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 2508 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( ( abs `  x
)  gcd  P )  =  1 )
61 lgssq 23354 . . . . . 6  |-  ( ( ( abs `  x
)  e.  NN  /\  P  e.  ZZ  /\  (
( abs `  x
)  gcd  P )  =  1 )  -> 
( ( ( abs `  x ) ^ 2 )  /L P )  =  1 )
6250, 30, 60, 61syl3anc 1228 . . . . 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 14078 . . . . . . . . . 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 13853 . . . . . . . . 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 1228 . . . . . . . 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 6298 . . . . . 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 4153 . . . . . . . . . 10  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  =/=  2 )
7069ad3antlr 730 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  P  =/=  2 )
7170necomd 2738 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
2  =/=  P )
72 2z 10895 . . . . . . . . . 10  |-  2  e.  ZZ
73 uzid 11095 . . . . . . . . . 10  |-  ( 2  e.  ZZ  ->  2  e.  ( ZZ>= `  2 )
)
7472, 73ax-mp 5 . . . . . . . . 9  |-  2  e.  ( ZZ>= `  2 )
75 dvdsprm 14098 . . . . . . . . . 10  |-  ( ( 2  e.  ( ZZ>= ` 
2 )  /\  P  e.  Prime )  ->  (
2  ||  P  <->  2  =  P ) )
7675necon3bbid 2714 . . . . . . . . 9  |-  ( ( 2  e.  ( ZZ>= ` 
2 )  /\  P  e.  Prime )  ->  ( -.  2  ||  P  <->  2  =/=  P ) )
7774, 29, 76sylancr 663 . . . . . . . 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 23340 . . . . . . 7  |-  ( ( ( x ^ 2 )  e.  ZZ  /\  P  e.  NN  /\  -.  2  ||  P )  -> 
( ( ( x ^ 2 )  mod 
P )  /L
P )  =  ( ( x ^ 2 )  /L P ) )
8032, 64, 78, 79syl3anc 1228 . . . . . 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 23340 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  NN  /\  -.  2  ||  P )  -> 
( ( A  mod  P )  /L P )  =  ( A  /L P ) )
8233, 64, 78, 81syl3anc 1228 . . . . . 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 2516 . . . . 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 2517 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( A  /L
P )  =  1 )
8584rexlimdvaa 2956 . . 3  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  ->  ( E. x  e.  ZZ  P  ||  ( ( x ^ 2 )  -  A )  ->  ( A  /L P )  =  1 ) )
8685expimpd 603 . 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 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815    \ cdif 3473   {csn 4027   class class class wbr 4447   ` cfv 5587  (class class class)co 6283   RRcr 9490   0cc0 9491   1c1 9492    - cmin 9804   NNcn 10535   2c2 10584   ZZcz 10863   ZZ>=cuz 11081    mod cmo 11963   ^cexp 12133   abscabs 13029    || cdivides 13846    gcd cgcd 14002   Primecprime 14075    /Lclgs 23313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569  ax-addf 9570  ax-mulf 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-ofr 6524  df-om 6680  df-1st 6784  df-2nd 6785  df-supp 6902  df-tpos 6955  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-ec 7313  df-qs 7317  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7829  df-sup 7900  df-oi 7934  df-card 8319  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-7 10598  df-8 10599  df-9 10600  df-10 10601  df-n0 10795  df-z 10864  df-dec 10976  df-uz 11082  df-q 11182  df-rp 11220  df-fz 11672  df-fzo 11792  df-fl 11896  df-mod 11964  df-seq 12075  df-exp 12134  df-hash 12373  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-dvds 13847  df-gcd 14003  df-prm 14076  df-phi 14154  df-pc 14219  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-mulr 14568  df-starv 14569  df-sca 14570  df-vsca 14571  df-ip 14572  df-tset 14573  df-ple 14574  df-ds 14576  df-unif 14577  df-hom 14578  df-cco 14579  df-0g 14696  df-gsum 14697  df-prds 14702  df-pws 14704  df-imas 14762  df-divs 14763  df-mre 14840  df-mrc 14841  df-acs 14843  df-mnd 15731  df-mhm 15783  df-submnd 15784  df-grp 15864  df-minusg 15865  df-sbg 15866  df-mulg 15867  df-subg 16000  df-nsg 16001  df-eqg 16002  df-ghm 16067  df-cntz 16157  df-cmn 16603  df-abl 16604  df-mgp 16941  df-ur 16953  df-srg 16957  df-rng 16997  df-cring 16998  df-oppr 17068  df-dvdsr 17086  df-unit 17087  df-invr 17117  df-dvr 17128  df-rnghom 17160  df-drng 17193  df-field 17194  df-subrg 17222  df-lmod 17309  df-lss 17374  df-lsp 17413  df-sra 17613  df-rgmod 17614  df-lidl 17615  df-rsp 17616  df-2idl 17674  df-nzr 17700  df-rlreg 17718  df-domn 17719  df-idom 17720  df-assa 17748  df-asp 17749  df-ascl 17750  df-psr 17792  df-mvr 17793  df-mpl 17794  df-opsr 17796  df-evls 17958  df-evl 17959  df-psr1 18006  df-vr1 18007  df-ply1 18008  df-coe1 18009  df-evl1 18140  df-cnfld 18208  df-zring 18273  df-zrh 18324  df-zn 18327  df-mdeg 22204  df-deg1 22205  df-mon1 22282  df-uc1p 22283  df-q1p 22284  df-r1p 22285  df-lgs 23314
This theorem is referenced by:  2sqlem11  23394  2sqblem  23396
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