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Theorem lgsqr 21131
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 21118) and the number is a quadratic residue  mod  P (it is  -u 1 for nonresidues by the process of elimination from lgsabs1 21119). 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 3470 . . . . . . . . . . 11  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  e.  Prime )
21adantl 454 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  P  e.  Prime )
3 prmz 13084 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  ZZ )
42, 3syl 16 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  P  e.  ZZ )
5 simpl 445 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  A  e.  ZZ )
6 gcdcom 13021 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  A  e.  ZZ )  ->  ( P  gcd  A
)  =  ( A  gcd  P ) )
74, 5, 6syl2anc 644 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( P  gcd  A )  =  ( A  gcd  P ) )
87eqeq1d 2445 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( P  gcd  A )  =  1  <->  ( A  gcd  P )  =  1 ) )
9 coprm 13101 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
102, 5, 9syl2anc 644 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
11 lgsne0 21118 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ZZ )  ->  ( ( A  / L P )  =/=  0  <->  ( A  gcd  P )  =  1 ) )
125, 4, 11syl2anc 644 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  / L P )  =/=  0  <->  ( A  gcd  P )  =  1 ) )
138, 10, 123bitr4d 278 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( -.  P  ||  A  <->  ( A  / L P )  =/=  0 ) )
1413necon4bbid 2670 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( P  ||  A  <->  ( A  / L P )  =  0 ) )
15 ax-1ne0 9060 . . . . . . 7  |-  1  =/=  0
1615necomi 2687 . . . . . 6  |-  0  =/=  1
17 neeq1 2610 . . . . . 6  |-  ( ( A  / L P
)  =  0  -> 
( ( A  / L P )  =/=  1  <->  0  =/=  1 ) )
1816, 17mpbiri 226 . . . . 5  |-  ( ( A  / L P
)  =  0  -> 
( A  / L P )  =/=  1
)
1914, 18syl6bi 221 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( P  ||  A  ->  ( A  / L P )  =/=  1 ) )
2019necon2bd 2654 . . 3  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  / L P )  =  1  ->  -.  P  ||  A ) )
21 lgsqrlem5 21130 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  ( A  / L P )  =  1 )  ->  E. x  e.  ZZ  P  ||  (
( x ^ 2 )  -  A ) )
22213expia 1156 . . 3  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  / L P )  =  1  ->  E. x  e.  ZZ  P  ||  (
( x ^ 2 )  -  A ) ) )
2320, 22jcad 521 . 2  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  / L P )  =  1  ->  ( -.  P  ||  A  /\  E. x  e.  ZZ  P  ||  ( ( x ^
2 )  -  A
) ) ) )
24 simprl 734 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  x  e.  ZZ )
2524zred 10376 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  x  e.  RR )
26 absresq 12108 . . . . . . 7  |-  ( x  e.  RR  ->  (
( abs `  x
) ^ 2 )  =  ( x ^
2 ) )
2725, 26syl 16 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( ( abs `  x
) ^ 2 )  =  ( x ^
2 ) )
2827oveq1d 6097 . . . . 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 ) )
29 simplr 733 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  -.  P  ||  A )
301ad3antlr 713 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  P  e.  Prime )
3130, 3syl 16 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  P  e.  ZZ )
32 zsqcl 11453 . . . . . . . . . . . 12  |-  ( x  e.  ZZ  ->  (
x ^ 2 )  e.  ZZ )
3324, 32syl 16 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( x ^ 2 )  e.  ZZ )
34 simplll 736 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  A  e.  ZZ )
35 simprr 735 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  P  ||  ( ( x ^ 2 )  -  A ) )
36 dvdssub2 12888 . . . . . . . . . . 11  |-  ( ( ( P  e.  ZZ  /\  ( x ^ 2 )  e.  ZZ  /\  A  e.  ZZ )  /\  P  ||  ( ( x ^ 2 )  -  A ) )  ->  ( P  ||  ( x ^ 2 )  <->  P  ||  A ) )
3731, 33, 34, 35, 36syl31anc 1188 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( P  ||  (
x ^ 2 )  <-> 
P  ||  A )
)
3829, 37mtbird 294 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  -.  P  ||  ( x ^ 2 ) )
39 2nn 10134 . . . . . . . . . . 11  |-  2  e.  NN
4039a1i 11 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
2  e.  NN )
41 prmdvdsexp 13115 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  x  e.  ZZ  /\  2  e.  NN )  ->  ( P  ||  ( x ^
2 )  <->  P  ||  x
) )
4230, 24, 40, 41syl3anc 1185 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( P  ||  (
x ^ 2 )  <-> 
P  ||  x )
)
4338, 42mtbid 293 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  -.  P  ||  x )
44 dvds0 12866 . . . . . . . . . . 11  |-  ( P  e.  ZZ  ->  P  ||  0 )
4531, 44syl 16 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  P  ||  0 )
46 breq2 4217 . . . . . . . . . 10  |-  ( x  =  0  ->  ( P  ||  x  <->  P  ||  0
) )
4745, 46syl5ibrcom 215 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( x  =  0  ->  P  ||  x
) )
4847necon3bd 2639 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( -.  P  ||  x  ->  x  =/=  0
) )
4943, 48mpd 15 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  x  =/=  0 )
50 nnabscl 12130 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  x  =/=  0 )  -> 
( abs `  x
)  e.  NN )
5124, 49, 50syl2anc 644 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( abs `  x
)  e.  NN )
5251nnzd 10375 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( abs `  x
)  e.  ZZ )
53 gcdcom 13021 . . . . . . . 8  |-  ( ( ( abs `  x
)  e.  ZZ  /\  P  e.  ZZ )  ->  ( ( abs `  x
)  gcd  P )  =  ( P  gcd  ( abs `  x ) ) )
5452, 31, 53syl2anc 644 . . . . . . 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 ) ) )
55 dvdsabsb 12870 . . . . . . . . . 10  |-  ( ( P  e.  ZZ  /\  x  e.  ZZ )  ->  ( P  ||  x  <->  P 
||  ( abs `  x
) ) )
5631, 24, 55syl2anc 644 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( P  ||  x  <->  P 
||  ( abs `  x
) ) )
5743, 56mtbid 293 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  -.  P  ||  ( abs `  x ) )
58 coprm 13101 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( abs `  x )  e.  ZZ )  ->  ( -.  P  ||  ( abs `  x )  <->  ( P  gcd  ( abs `  x
) )  =  1 ) )
5930, 52, 58syl2anc 644 . . . . . . . 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 ) )
6057, 59mpbid 203 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( P  gcd  ( abs `  x ) )  =  1 )
6154, 60eqtrd 2469 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( ( abs `  x
)  gcd  P )  =  1 )
62 lgssq 21120 . . . . . 6  |-  ( ( ( abs `  x
)  e.  NN  /\  P  e.  ZZ  /\  (
( abs `  x
)  gcd  P )  =  1 )  -> 
( ( ( abs `  x ) ^ 2 )  / L P
)  =  1 )
6351, 31, 61, 62syl3anc 1185 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( ( ( abs `  x ) ^ 2 )  / L P
)  =  1 )
64 prmnn 13083 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
6530, 64syl 16 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  P  e.  NN )
66 moddvds 12860 . . . . . . . . 9  |-  ( ( P  e.  NN  /\  ( x ^ 2 )  e.  ZZ  /\  A  e.  ZZ )  ->  ( ( ( x ^ 2 )  mod 
P )  =  ( A  mod  P )  <-> 
P  ||  ( (
x ^ 2 )  -  A ) ) )
6765, 33, 34, 66syl3anc 1185 . . . . . . . 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 ) ) )
6835, 67mpbird 225 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( ( x ^
2 )  mod  P
)  =  ( A  mod  P ) )
6968oveq1d 6097 . . . . . 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
) )
70 eldifsni 3929 . . . . . . . . . 10  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  =/=  2 )
7170ad3antlr 713 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  P  =/=  2 )
7271necomd 2688 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
2  =/=  P )
73 2z 10313 . . . . . . . . . 10  |-  2  e.  ZZ
74 uzid 10501 . . . . . . . . . 10  |-  ( 2  e.  ZZ  ->  2  e.  ( ZZ>= `  2 )
)
7573, 74ax-mp 8 . . . . . . . . 9  |-  2  e.  ( ZZ>= `  2 )
76 dvdsprm 13100 . . . . . . . . . 10  |-  ( ( 2  e.  ( ZZ>= ` 
2 )  /\  P  e.  Prime )  ->  (
2  ||  P  <->  2  =  P ) )
7776necon3bbid 2636 . . . . . . . . 9  |-  ( ( 2  e.  ( ZZ>= ` 
2 )  /\  P  e.  Prime )  ->  ( -.  2  ||  P  <->  2  =/=  P ) )
7875, 30, 77sylancr 646 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( -.  2  ||  P 
<->  2  =/=  P ) )
7972, 78mpbird 225 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  -.  2  ||  P )
80 lgsmod 21106 . . . . . . 7  |-  ( ( ( x ^ 2 )  e.  ZZ  /\  P  e.  NN  /\  -.  2  ||  P )  -> 
( ( ( x ^ 2 )  mod 
P )  / L P )  =  ( ( x ^ 2 )  / L P
) )
8133, 65, 79, 80syl3anc 1185 . . . . . 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
) )
82 lgsmod 21106 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  NN  /\  -.  2  ||  P )  -> 
( ( A  mod  P )  / L P
)  =  ( A  / L P ) )
8334, 65, 79, 82syl3anc 1185 . . . . . 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 ) )
8469, 81, 833eqtr3d 2477 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( ( x ^
2 )  / L P )  =  ( A  / L P
) )
8528, 63, 843eqtr3rd 2478 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( A  / L P )  =  1 )
8685rexlimdvaa 2832 . . 3  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  ->  ( E. x  e.  ZZ  P  ||  ( ( x ^ 2 )  -  A )  ->  ( A  / L P )  =  1 ) )
8786expimpd 588 . 2  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( -.  P  ||  A  /\  E. x  e.  ZZ  P  ||  ( ( x ^
2 )  -  A
) )  ->  ( A  / L P )  =  1 ) )
8823, 87impbid 185 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2600   E.wrex 2707    \ cdif 3318   {csn 3815   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   RRcr 8990   0cc0 8991   1c1 8992    - cmin 9292   NNcn 10001   2c2 10050   ZZcz 10283   ZZ>=cuz 10489    mod cmo 11251   ^cexp 11383   abscabs 12040    || cdivides 12853    gcd cgcd 13007   Primecprime 13080    / Lclgs 21079
This theorem is referenced by:  2sqlem11  21160  2sqblem  21162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069  ax-addf 9070  ax-mulf 9071
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-iin 4097  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6306  df-ofr 6307  df-1st 6350  df-2nd 6351  df-tpos 6480  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-2o 6726  df-oadd 6729  df-er 6906  df-ec 6908  df-qs 6912  df-map 7021  df-pm 7022  df-ixp 7065  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-sup 7447  df-oi 7480  df-card 7827  df-cda 8049  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-5 10062  df-6 10063  df-7 10064  df-8 10065  df-9 10066  df-10 10067  df-n0 10223  df-z 10284  df-dec 10384  df-uz 10490  df-q 10576  df-rp 10614  df-fz 11045  df-fzo 11137  df-fl 11203  df-mod 11252  df-seq 11325  df-exp 11384  df-hash 11620  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-dvds 12854  df-gcd 13008  df-prm 13081  df-phi 13156  df-pc 13212  df-struct 13472  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-ress 13477  df-plusg 13543  df-mulr 13544  df-starv 13545  df-sca 13546  df-vsca 13547  df-tset 13549  df-ple 13550  df-ds 13552  df-unif 13553  df-hom 13554  df-cco 13555  df-prds 13672  df-pws 13674  df-0g 13728  df-gsum 13729  df-imas 13735  df-divs 13736  df-mre 13812  df-mrc 13813  df-acs 13815  df-mnd 14691  df-mhm 14739  df-submnd 14740  df-grp 14813  df-minusg 14814  df-sbg 14815  df-mulg 14816  df-subg 14942  df-nsg 14943  df-eqg 14944  df-ghm 15005  df-cntz 15117  df-cmn 15415  df-abl 15416  df-mgp 15650  df-rng 15664  df-cring 15665  df-ur 15666  df-oppr 15729  df-dvdsr 15747  df-unit 15748  df-invr 15778  df-rnghom 15820  df-drng 15838  df-field 15839  df-subrg 15867  df-lmod 15953  df-lss 16010  df-lsp 16049  df-sra 16245  df-rgmod 16246  df-lidl 16247  df-rsp 16248  df-2idl 16304  df-nzr 16330  df-rlreg 16344  df-domn 16345  df-idom 16346  df-assa 16373  df-asp 16374  df-ascl 16375  df-psr 16418  df-mvr 16419  df-mpl 16420  df-evls 16421  df-evl 16422  df-opsr 16426  df-psr1 16577  df-vr1 16578  df-ply1 16579  df-evl1 16581  df-coe1 16582  df-cnfld 16705  df-zrh 16783  df-zn 16786  df-mdeg 19979  df-deg1 19980  df-mon1 20054  df-uc1p 20055  df-q1p 20056  df-r1p 20057  df-lgs 21080
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