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Theorem lgsqr 20585
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 20572) and the number is a quadratic residue  mod  P (it is  -u 1 for nonresidues by the process of elimination from lgsabs1 20573). 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 3298 . . . . . . . . . . 11  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  e.  Prime )
21adantl 452 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  P  e.  Prime )
3 prmz 12762 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  ZZ )
42, 3syl 15 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  P  e.  ZZ )
5 simpl 443 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  A  e.  ZZ )
6 gcdcom 12699 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  A  e.  ZZ )  ->  ( P  gcd  A
)  =  ( A  gcd  P ) )
74, 5, 6syl2anc 642 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( P  gcd  A )  =  ( A  gcd  P ) )
87eqeq1d 2291 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( P  gcd  A )  =  1  <->  ( A  gcd  P )  =  1 ) )
9 coprm 12779 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
102, 5, 9syl2anc 642 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( -.  P  ||  A  <->  ( P  gcd  A )  =  1 ) )
11 lgsne0 20572 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ZZ )  ->  ( ( A  / L P )  =/=  0  <->  ( A  gcd  P )  =  1 ) )
125, 4, 11syl2anc 642 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  / L P )  =/=  0  <->  ( A  gcd  P )  =  1 ) )
138, 10, 123bitr4d 276 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( -.  P  ||  A  <->  ( A  / L P )  =/=  0 ) )
1413necon4bbid 2511 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( P  ||  A  <->  ( A  / L P )  =  0 ) )
15 ax-1ne0 8806 . . . . . . 7  |-  1  =/=  0
1615necomi 2528 . . . . . 6  |-  0  =/=  1
17 neeq1 2454 . . . . . 6  |-  ( ( A  / L P
)  =  0  -> 
( ( A  / L P )  =/=  1  <->  0  =/=  1 ) )
1816, 17mpbiri 224 . . . . 5  |-  ( ( A  / L P
)  =  0  -> 
( A  / L P )  =/=  1
)
1914, 18syl6bi 219 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( P  ||  A  ->  ( A  / L P )  =/=  1 ) )
2019necon2bd 2495 . . 3  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  / L P )  =  1  ->  -.  P  ||  A ) )
21 lgsqrlem5 20584 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } )  /\  ( A  / L P )  =  1 )  ->  E. x  e.  ZZ  P  ||  (
( x ^ 2 )  -  A ) )
22213expia 1153 . . 3  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  / L P )  =  1  ->  E. x  e.  ZZ  P  ||  (
( x ^ 2 )  -  A ) ) )
2320, 22jcad 519 . 2  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  / L P )  =  1  ->  ( -.  P  ||  A  /\  E. x  e.  ZZ  P  ||  ( ( x ^
2 )  -  A
) ) ) )
24 simprl 732 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  x  e.  ZZ )
2524zred 10117 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  x  e.  RR )
26 absresq 11787 . . . . . . . 8  |-  ( x  e.  RR  ->  (
( abs `  x
) ^ 2 )  =  ( x ^
2 ) )
2725, 26syl 15 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( ( abs `  x
) ^ 2 )  =  ( x ^
2 ) )
2827oveq1d 5873 . . . . . 6  |-  ( ( ( ( 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 731 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  -.  P  ||  A )
302ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  P  e.  Prime )
3130, 3syl 15 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  P  e.  ZZ )
32 zsqcl 11174 . . . . . . . . . . . . 13  |-  ( x  e.  ZZ  ->  (
x ^ 2 )  e.  ZZ )
3324, 32syl 15 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( x ^ 2 )  e.  ZZ )
345ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  A  e.  ZZ )
35 simprr 733 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  P  ||  ( ( x ^ 2 )  -  A ) )
36 dvdssub2 12566 . . . . . . . . . . . 12  |-  ( ( ( P  e.  ZZ  /\  ( x ^ 2 )  e.  ZZ  /\  A  e.  ZZ )  /\  P  ||  ( ( x ^ 2 )  -  A ) )  ->  ( P  ||  ( x ^ 2 )  <->  P  ||  A ) )
3731, 33, 34, 35, 36syl31anc 1185 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( P  ||  (
x ^ 2 )  <-> 
P  ||  A )
)
3829, 37mtbird 292 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  -.  P  ||  ( x ^ 2 ) )
39 2nn 9877 . . . . . . . . . . . 12  |-  2  e.  NN
4039a1i 10 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
2  e.  NN )
41 prmdvdsexp 12793 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  x  e.  ZZ  /\  2  e.  NN )  ->  ( P  ||  ( x ^
2 )  <->  P  ||  x
) )
4230, 24, 40, 41syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( P  ||  (
x ^ 2 )  <-> 
P  ||  x )
)
4338, 42mtbid 291 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  -.  P  ||  x )
44 dvds0 12544 . . . . . . . . . . . 12  |-  ( P  e.  ZZ  ->  P  ||  0 )
4531, 44syl 15 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  P  ||  0 )
46 breq2 4027 . . . . . . . . . . 11  |-  ( x  =  0  ->  ( P  ||  x  <->  P  ||  0
) )
4745, 46syl5ibrcom 213 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( x  =  0  ->  P  ||  x
) )
4847necon3bd 2483 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( -.  P  ||  x  ->  x  =/=  0
) )
4943, 48mpd 14 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  x  =/=  0 )
50 nnabscl 11809 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  x  =/=  0 )  -> 
( abs `  x
)  e.  NN )
5124, 49, 50syl2anc 642 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( abs `  x
)  e.  NN )
5251nnzd 10116 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( abs `  x
)  e.  ZZ )
53 gcdcom 12699 . . . . . . . . 9  |-  ( ( ( abs `  x
)  e.  ZZ  /\  P  e.  ZZ )  ->  ( ( abs `  x
)  gcd  P )  =  ( P  gcd  ( abs `  x ) ) )
5452, 31, 53syl2anc 642 . . . . . . . 8  |-  ( ( ( ( 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 12548 . . . . . . . . . . 11  |-  ( ( P  e.  ZZ  /\  x  e.  ZZ )  ->  ( P  ||  x  <->  P 
||  ( abs `  x
) ) )
5631, 24, 55syl2anc 642 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( P  ||  x  <->  P 
||  ( abs `  x
) ) )
5743, 56mtbid 291 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  -.  P  ||  ( abs `  x ) )
58 coprm 12779 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  ( abs `  x )  e.  ZZ )  ->  ( -.  P  ||  ( abs `  x )  <->  ( P  gcd  ( abs `  x
) )  =  1 ) )
5930, 52, 58syl2anc 642 . . . . . . . . 9  |-  ( ( ( ( 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 201 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( P  gcd  ( abs `  x ) )  =  1 )
6154, 60eqtrd 2315 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( ( abs `  x
)  gcd  P )  =  1 )
62 lgssq 20574 . . . . . . 7  |-  ( ( ( abs `  x
)  e.  NN  /\  P  e.  ZZ  /\  (
( abs `  x
)  gcd  P )  =  1 )  -> 
( ( ( abs `  x ) ^ 2 )  / L P
)  =  1 )
6351, 31, 61, 62syl3anc 1182 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( ( ( abs `  x ) ^ 2 )  / L P
)  =  1 )
64 prmnn 12761 . . . . . . . . . . 11  |-  ( P  e.  Prime  ->  P  e.  NN )
6530, 64syl 15 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  P  e.  NN )
66 moddvds 12538 . . . . . . . . . 10  |-  ( ( 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 1182 . . . . . . . . 9  |-  ( ( ( ( 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 223 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( ( x ^
2 )  mod  P
)  =  ( A  mod  P ) )
6968oveq1d 5873 . . . . . . 7  |-  ( ( ( ( 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 simpllr 735 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  P  e.  ( Prime  \  { 2 } ) )
71 eldifsni 3750 . . . . . . . . . . 11  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  =/=  2 )
7270, 71syl 15 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  P  =/=  2 )
7372necomd 2529 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
2  =/=  P )
74 2z 10054 . . . . . . . . . . 11  |-  2  e.  ZZ
75 uzid 10242 . . . . . . . . . . 11  |-  ( 2  e.  ZZ  ->  2  e.  ( ZZ>= `  2 )
)
7674, 75ax-mp 8 . . . . . . . . . 10  |-  2  e.  ( ZZ>= `  2 )
77 dvdsprm 12778 . . . . . . . . . . 11  |-  ( ( 2  e.  ( ZZ>= ` 
2 )  /\  P  e.  Prime )  ->  (
2  ||  P  <->  2  =  P ) )
7877necon3bbid 2480 . . . . . . . . . 10  |-  ( ( 2  e.  ( ZZ>= ` 
2 )  /\  P  e.  Prime )  ->  ( -.  2  ||  P  <->  2  =/=  P ) )
7976, 30, 78sylancr 644 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( -.  2  ||  P 
<->  2  =/=  P ) )
8073, 79mpbird 223 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  ->  -.  2  ||  P )
81 lgsmod 20560 . . . . . . . 8  |-  ( ( ( x ^ 2 )  e.  ZZ  /\  P  e.  NN  /\  -.  2  ||  P )  -> 
( ( ( x ^ 2 )  mod 
P )  / L P )  =  ( ( x ^ 2 )  / L P
) )
8233, 65, 80, 81syl3anc 1182 . . . . . . 7  |-  ( ( ( ( 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
) )
83 lgsmod 20560 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  NN  /\  -.  2  ||  P )  -> 
( ( A  mod  P )  / L P
)  =  ( A  / L P ) )
8434, 65, 80, 83syl3anc 1182 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( ( A  mod  P )  / L P
)  =  ( A  / L P ) )
8569, 82, 843eqtr3d 2323 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( ( x ^
2 )  / L P )  =  ( A  / L P
) )
8628, 63, 853eqtr3rd 2324 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  ( x  e.  ZZ  /\  P  ||  ( ( x ^
2 )  -  A
) ) )  -> 
( A  / L P )  =  1 )
8786expr 598 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  /\  x  e.  ZZ )  ->  ( P  ||  ( ( x ^
2 )  -  A
)  ->  ( A  / L P )  =  1 ) )
8887rexlimdva 2667 . . 3  |-  ( ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  /\  -.  P  ||  A )  ->  ( E. x  e.  ZZ  P  ||  ( ( x ^ 2 )  -  A )  ->  ( A  / L P )  =  1 ) )
8988expimpd 586 . 2  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( -.  P  ||  A  /\  E. x  e.  ZZ  P  ||  ( ( x ^
2 )  -  A
) )  ->  ( A  / L P )  =  1 ) )
9023, 89impbid 183 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 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    \ cdif 3149   {csn 3640   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    - cmin 9037   NNcn 9746   2c2 9795   ZZcz 10024   ZZ>=cuz 10230    mod cmo 10973   ^cexp 11104   abscabs 11719    || cdivides 12531    gcd cgcd 12685   Primecprime 12758    / Lclgs 20533
This theorem is referenced by:  2sqlem11  20614  2sqblem  20616
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-prm 12759  df-phi 12834  df-pc 12890  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-pws 13350  df-0g 13404  df-gsum 13405  df-imas 13411  df-divs 13412  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-nsg 14619  df-eqg 14620  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-rnghom 15496  df-drng 15514  df-field 15515  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-sra 15925  df-rgmod 15926  df-lidl 15927  df-rsp 15928  df-2idl 15984  df-nzr 16010  df-rlreg 16024  df-domn 16025  df-idom 16026  df-assa 16053  df-asp 16054  df-ascl 16055  df-psr 16098  df-mvr 16099  df-mpl 16100  df-evls 16101  df-evl 16102  df-opsr 16106  df-psr1 16257  df-vr1 16258  df-ply1 16259  df-evl1 16261  df-coe1 16262  df-cnfld 16378  df-zrh 16455  df-zn 16458  df-mdeg 19441  df-deg1 19442  df-mon1 19516  df-uc1p 19517  df-q1p 19518  df-r1p 19519  df-lgs 20534
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