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Theorem lgsqr 21091
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 21078) and the number is a quadratic residue  mod  P (it is  -u 1 for nonresidues by the process of elimination from lgsabs1 21079). 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 3437 . . . . . . . . . . 11  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  e.  Prime )
21adantl 453 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  P  e.  Prime )
3 prmz 13046 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  ZZ )
42, 3syl 16 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  P  e.  ZZ )
5 simpl 444 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  A  e.  ZZ )
6 gcdcom 12983 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  A  e.  ZZ )  ->  ( P  gcd  A
)  =  ( A  gcd  P ) )
74, 5, 6syl2anc 643 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( P  gcd  A )  =  ( A  gcd  P ) )
87eqeq1d 2420 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( P  gcd  A )  =  1  <->  ( A  gcd  P )  =  1 ) )
9 coprm 13063 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
102, 5, 9syl2anc 643 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
11 lgsne0 21078 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ZZ )  ->  ( ( A  / L P )  =/=  0  <->  ( A  gcd  P )  =  1 ) )
125, 4, 11syl2anc 643 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  / L P )  =/=  0  <->  ( A  gcd  P )  =  1 ) )
138, 10, 123bitr4d 277 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( -.  P  ||  A  <->  ( A  / L P )  =/=  0 ) )
1413necon4bbid 2640 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( P  ||  A  <->  ( A  / L P )  =  0 ) )
15 ax-1ne0 9023 . . . . . . 7  |-  1  =/=  0
1615necomi 2657 . . . . . 6  |-  0  =/=  1
17 neeq1 2583 . . . . . 6  |-  ( ( A  / L P
)  =  0  -> 
( ( A  / L P )  =/=  1  <->  0  =/=  1 ) )
1816, 17mpbiri 225 . . . . 5  |-  ( ( A  / L P
)  =  0  -> 
( A  / L P )  =/=  1
)
1914, 18syl6bi 220 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( P  ||  A  ->  ( A  / L P )  =/=  1 ) )
2019necon2bd 2624 . . 3  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  / L P )  =  1  ->  -.  P  ||  A ) )
21 lgsqrlem5 21090 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  ( A  / L P )  =  1 )  ->  E. x  e.  ZZ  P  ||  (
( x ^ 2 )  -  A ) )
22213expia 1155 . . 3  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  / L P )  =  1  ->  E. x  e.  ZZ  P  ||  (
( x ^ 2 )  -  A ) ) )
2320, 22jcad 520 . 2  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  / L P )  =  1  ->  ( -.  P  ||  A  /\  E. x  e.  ZZ  P  ||  ( ( x ^
2 )  -  A
) ) ) )
24 simprl 733 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  x  e.  ZZ )
2524zred 10339 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  x  e.  RR )
26 absresq 12070 . . . . . . 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 6063 . . . . 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 732 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  -.  P  ||  A )
301ad3antlr 712 . . . . . . . . . . . 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 11415 . . . . . . . . . . . 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 735 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  A  e.  ZZ )
35 simprr 734 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  P  ||  ( ( x ^ 2 )  -  A ) )
36 dvdssub2 12850 . . . . . . . . . . 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 1187 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( P  ||  (
x ^ 2 )  <-> 
P  ||  A )
)
3829, 37mtbird 293 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  -.  P  ||  ( x ^ 2 ) )
39 2nn 10097 . . . . . . . . . . 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 13077 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  x  e.  ZZ  /\  2  e.  NN )  ->  ( P  ||  ( x ^
2 )  <->  P  ||  x
) )
4230, 24, 40, 41syl3anc 1184 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( P  ||  (
x ^ 2 )  <-> 
P  ||  x )
)
4338, 42mtbid 292 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  -.  P  ||  x )
44 dvds0 12828 . . . . . . . . . . 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 4184 . . . . . . . . . 10  |-  ( x  =  0  ->  ( P  ||  x  <->  P  ||  0
) )
4745, 46syl5ibrcom 214 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( x  =  0  ->  P  ||  x
) )
4847necon3bd 2612 . . . . . . . 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 12092 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  x  =/=  0 )  -> 
( abs `  x
)  e.  NN )
5124, 49, 50syl2anc 643 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( abs `  x
)  e.  NN )
5251nnzd 10338 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( abs `  x
)  e.  ZZ )
53 gcdcom 12983 . . . . . . . 8  |-  ( ( ( abs `  x
)  e.  ZZ  /\  P  e.  ZZ )  ->  ( ( abs `  x
)  gcd  P )  =  ( P  gcd  ( abs `  x ) ) )
5452, 31, 53syl2anc 643 . . . . . . 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 12832 . . . . . . . . . 10  |-  ( ( P  e.  ZZ  /\  x  e.  ZZ )  ->  ( P  ||  x  <->  P 
||  ( abs `  x
) ) )
5631, 24, 55syl2anc 643 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( P  ||  x  <->  P 
||  ( abs `  x
) ) )
5743, 56mtbid 292 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  -.  P  ||  ( abs `  x ) )
58 coprm 13063 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( abs `  x )  e.  ZZ )  ->  ( -.  P  ||  ( abs `  x )  <->  ( P  gcd  ( abs `  x
) )  =  1 ) )
5930, 52, 58syl2anc 643 . . . . . . . 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 202 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( P  gcd  ( abs `  x ) )  =  1 )
6154, 60eqtrd 2444 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( ( abs `  x
)  gcd  P )  =  1 )
62 lgssq 21080 . . . . . 6  |-  ( ( ( abs `  x
)  e.  NN  /\  P  e.  ZZ  /\  (
( abs `  x
)  gcd  P )  =  1 )  -> 
( ( ( abs `  x ) ^ 2 )  / L P
)  =  1 )
6351, 31, 61, 62syl3anc 1184 . . . . 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 13045 . . . . . . . . . 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 12822 . . . . . . . . 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 1184 . . . . . . . 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 224 . . . . . . 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 6063 . . . . . 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 3896 . . . . . . . . . 10  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  =/=  2 )
7170ad3antlr 712 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  P  =/=  2 )
7271necomd 2658 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
2  =/=  P )
73 2z 10276 . . . . . . . . . 10  |-  2  e.  ZZ
74 uzid 10464 . . . . . . . . . 10  |-  ( 2  e.  ZZ  ->  2  e.  ( ZZ>= `  2 )
)
7573, 74ax-mp 8 . . . . . . . . 9  |-  2  e.  ( ZZ>= `  2 )
76 dvdsprm 13062 . . . . . . . . . 10  |-  ( ( 2  e.  ( ZZ>= ` 
2 )  /\  P  e.  Prime )  ->  (
2  ||  P  <->  2  =  P ) )
7776necon3bbid 2609 . . . . . . . . 9  |-  ( ( 2  e.  ( ZZ>= ` 
2 )  /\  P  e.  Prime )  ->  ( -.  2  ||  P  <->  2  =/=  P ) )
7875, 30, 77sylancr 645 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( -.  2  ||  P 
<->  2  =/=  P ) )
7972, 78mpbird 224 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  -.  2  ||  P )
80 lgsmod 21066 . . . . . . 7  |-  ( ( ( x ^ 2 )  e.  ZZ  /\  P  e.  NN  /\  -.  2  ||  P )  -> 
( ( ( x ^ 2 )  mod 
P )  / L P )  =  ( ( x ^ 2 )  / L P
) )
8133, 65, 79, 80syl3anc 1184 . . . . . 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 21066 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  NN  /\  -.  2  ||  P )  -> 
( ( A  mod  P )  / L P
)  =  ( A  / L P ) )
8334, 65, 79, 82syl3anc 1184 . . . . . 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 2452 . . . . 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 2453 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( A  / L P )  =  1 )
8685rexlimdvaa 2799 . . 3  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  ->  ( E. x  e.  ZZ  P  ||  ( ( x ^ 2 )  -  A )  ->  ( A  / L P )  =  1 ) )
8786expimpd 587 . 2  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( -.  P  ||  A  /\  E. x  e.  ZZ  P  ||  ( ( x ^
2 )  -  A
) )  ->  ( A  / L P )  =  1 ) )
8823, 87impbid 184 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 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2575   E.wrex 2675    \ cdif 3285   {csn 3782   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   RRcr 8953   0cc0 8954   1c1 8955    - cmin 9255   NNcn 9964   2c2 10013   ZZcz 10246   ZZ>=cuz 10452    mod cmo 11213   ^cexp 11345   abscabs 12002    || cdivides 12815    gcd cgcd 12969   Primecprime 13042    / Lclgs 21039
This theorem is referenced by:  2sqlem11  21120  2sqblem  21122
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033  ax-mulf 9034
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-ofr 6273  df-1st 6316  df-2nd 6317  df-tpos 6446  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-ec 6874  df-qs 6878  df-map 6987  df-pm 6988  df-ixp 7031  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-oi 7443  df-card 7790  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-q 10539  df-rp 10577  df-fz 11008  df-fzo 11099  df-fl 11165  df-mod 11214  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-dvds 12816  df-gcd 12970  df-prm 13043  df-phi 13118  df-pc 13174  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-starv 13507  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-unif 13515  df-hom 13516  df-cco 13517  df-prds 13634  df-pws 13636  df-0g 13690  df-gsum 13691  df-imas 13697  df-divs 13698  df-mre 13774  df-mrc 13775  df-acs 13777  df-mnd 14653  df-mhm 14701  df-submnd 14702  df-grp 14775  df-minusg 14776  df-sbg 14777  df-mulg 14778  df-subg 14904  df-nsg 14905  df-eqg 14906  df-ghm 14967  df-cntz 15079  df-cmn 15377  df-abl 15378  df-mgp 15612  df-rng 15626  df-cring 15627  df-ur 15628  df-oppr 15691  df-dvdsr 15709  df-unit 15710  df-invr 15740  df-rnghom 15782  df-drng 15800  df-field 15801  df-subrg 15829  df-lmod 15915  df-lss 15972  df-lsp 16011  df-sra 16207  df-rgmod 16208  df-lidl 16209  df-rsp 16210  df-2idl 16266  df-nzr 16292  df-rlreg 16306  df-domn 16307  df-idom 16308  df-assa 16335  df-asp 16336  df-ascl 16337  df-psr 16380  df-mvr 16381  df-mpl 16382  df-evls 16383  df-evl 16384  df-opsr 16388  df-psr1 16539  df-vr1 16540  df-ply1 16541  df-evl1 16543  df-coe1 16544  df-cnfld 16667  df-zrh 16745  df-zn 16748  df-mdeg 19939  df-deg1 19940  df-mon1 20014  df-uc1p 20015  df-q1p 20016  df-r1p 20017  df-lgs 21040
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