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Theorem gausslemma2dlem0i 15101
Description: Auxiliary lemma 9 for gausslemma2d 15113. (Contributed by AV, 14-Jul-2021.)
Hypotheses
Ref Expression
gausslemma2dlem0.p  |-  ( ph  ->  P  e.  ( Prime  \  { 2 } ) )
gausslemma2dlem0.m  |-  M  =  ( |_ `  ( P  /  4 ) )
gausslemma2dlem0.h  |-  H  =  ( ( P  - 
1 )  /  2
)
gausslemma2dlem0.n  |-  N  =  ( H  -  M
)
Assertion
Ref Expression
gausslemma2dlem0i  |-  ( ph  ->  ( ( ( 2  /L P )  mod  P )  =  ( ( -u 1 ^ N )  mod  P
)  ->  ( 2  /L P )  =  ( -u 1 ^ N ) ) )

Proof of Theorem gausslemma2dlem0i
StepHypRef Expression
1 2z 9331 . . . 4  |-  2  e.  ZZ
2 gausslemma2dlem0.p . . . . 5  |-  ( ph  ->  P  e.  ( Prime  \  { 2 } ) )
3 id 19 . . . . . . 7  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  e.  ( Prime  \  { 2 } ) )
43gausslemma2dlem0a 15093 . . . . . 6  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  e.  NN )
54nnzd 9424 . . . . 5  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  e.  ZZ )
62, 5syl 14 . . . 4  |-  ( ph  ->  P  e.  ZZ )
7 lgscl1 15067 . . . 4  |-  ( ( 2  e.  ZZ  /\  P  e.  ZZ )  ->  ( 2  /L
P )  e.  { -u 1 ,  0 ,  1 } )
81, 6, 7sylancr 414 . . 3  |-  ( ph  ->  ( 2  /L
P )  e.  { -u 1 ,  0 ,  1 } )
9 eltpg 3659 . . . 4  |-  ( ( 2  /L P )  e.  { -u
1 ,  0 ,  1 }  ->  (
( 2  /L
P )  e.  { -u 1 ,  0 ,  1 }  <->  ( (
2  /L P )  =  -u 1  \/  ( 2  /L
P )  =  0  \/  ( 2  /L P )  =  1 ) ) )
108, 9syl 14 . . 3  |-  ( ph  ->  ( ( 2  /L P )  e. 
{ -u 1 ,  0 ,  1 }  <->  ( (
2  /L P )  =  -u 1  \/  ( 2  /L
P )  =  0  \/  ( 2  /L P )  =  1 ) ) )
118, 10mpbid 147 . 2  |-  ( ph  ->  ( ( 2  /L P )  = 
-u 1  \/  (
2  /L P )  =  0  \/  ( 2  /L
P )  =  1 ) )
12 gausslemma2dlem0.m . . . . . . . . 9  |-  M  =  ( |_ `  ( P  /  4 ) )
13 gausslemma2dlem0.h . . . . . . . . 9  |-  H  =  ( ( P  - 
1 )  /  2
)
14 gausslemma2dlem0.n . . . . . . . . 9  |-  N  =  ( H  -  M
)
152, 12, 13, 14gausslemma2dlem0h 15100 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
1615nn0zd 9423 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
17 m1expcl2 10606 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( -u 1 ^ N )  e.  { -u 1 ,  1 } )
1816, 17syl 14 . . . . . 6  |-  ( ph  ->  ( -u 1 ^ N )  e.  { -u 1 ,  1 } )
19 elprg 3634 . . . . . . 7  |-  ( (
-u 1 ^ N
)  e.  { -u
1 ,  1 }  ->  ( ( -u
1 ^ N )  e.  { -u 1 ,  1 }  <->  ( ( -u 1 ^ N )  =  -u 1  \/  ( -u 1 ^ N )  =  1 ) ) )
2018, 19syl 14 . . . . . 6  |-  ( ph  ->  ( ( -u 1 ^ N )  e.  { -u 1 ,  1 }  <-> 
( ( -u 1 ^ N )  =  -u
1  \/  ( -u
1 ^ N )  =  1 ) ) )
2118, 20mpbid 147 . . . . 5  |-  ( ph  ->  ( ( -u 1 ^ N )  =  -u
1  \/  ( -u
1 ^ N )  =  1 ) )
22 eqcom 2191 . . . . . . . 8  |-  ( (
-u 1 ^ N
)  =  -u 1  <->  -u 1  =  ( -u
1 ^ N ) )
2322biimpi 120 . . . . . . 7  |-  ( (
-u 1 ^ N
)  =  -u 1  -> 
-u 1  =  (
-u 1 ^ N
) )
24232a1d 23 . . . . . 6  |-  ( (
-u 1 ^ N
)  =  -u 1  ->  ( ph  ->  (
( -u 1  mod  P
)  =  ( (
-u 1 ^ N
)  mod  P )  -> 
-u 1  =  (
-u 1 ^ N
) ) ) )
252gausslemma2dlem0a 15093 . . . . . . . . . . 11  |-  ( ph  ->  P  e.  NN )
26 nnq 9684 . . . . . . . . . . 11  |-  ( P  e.  NN  ->  P  e.  QQ )
2725, 26syl 14 . . . . . . . . . 10  |-  ( ph  ->  P  e.  QQ )
282eldifad 3160 . . . . . . . . . . 11  |-  ( ph  ->  P  e.  Prime )
29 prmgt1 12244 . . . . . . . . . . 11  |-  ( P  e.  Prime  ->  1  < 
P )
3028, 29syl 14 . . . . . . . . . 10  |-  ( ph  ->  1  <  P )
31 q1mod 10413 . . . . . . . . . 10  |-  ( ( P  e.  QQ  /\  1  <  P )  -> 
( 1  mod  P
)  =  1 )
3227, 30, 31syl2anc 411 . . . . . . . . 9  |-  ( ph  ->  ( 1  mod  P
)  =  1 )
3332eqeq2d 2201 . . . . . . . 8  |-  ( ph  ->  ( ( -u 1  mod  P )  =  ( 1  mod  P )  <-> 
( -u 1  mod  P
)  =  1 ) )
34 oddprmge3 12247 . . . . . . . . 9  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  e.  ( ZZ>= ` 
3 ) )
35 m1modge3gt1 10428 . . . . . . . . . 10  |-  ( P  e.  ( ZZ>= `  3
)  ->  1  <  (
-u 1  mod  P
) )
36 breq2 4029 . . . . . . . . . . 11  |-  ( (
-u 1  mod  P
)  =  1  -> 
( 1  <  ( -u 1  mod  P )  <->  1  <  1 ) )
37 1re 8004 . . . . . . . . . . . . 13  |-  1  e.  RR
3837ltnri 8098 . . . . . . . . . . . 12  |-  -.  1  <  1
3938pm2.21i 647 . . . . . . . . . . 11  |-  ( 1  <  1  ->  -u 1  =  1 )
4036, 39biimtrdi 163 . . . . . . . . . 10  |-  ( (
-u 1  mod  P
)  =  1  -> 
( 1  <  ( -u 1  mod  P )  ->  -u 1  =  1 ) )
4135, 40syl5com 29 . . . . . . . . 9  |-  ( P  e.  ( ZZ>= `  3
)  ->  ( ( -u 1  mod  P )  =  1  ->  -u 1  =  1 ) )
422, 34, 413syl 17 . . . . . . . 8  |-  ( ph  ->  ( ( -u 1  mod  P )  =  1  ->  -u 1  =  1 ) )
4333, 42sylbid 150 . . . . . . 7  |-  ( ph  ->  ( ( -u 1  mod  P )  =  ( 1  mod  P )  ->  -u 1  =  1 ) )
44 oveq1 5913 . . . . . . . . 9  |-  ( (
-u 1 ^ N
)  =  1  -> 
( ( -u 1 ^ N )  mod  P
)  =  ( 1  mod  P ) )
4544eqeq2d 2201 . . . . . . . 8  |-  ( (
-u 1 ^ N
)  =  1  -> 
( ( -u 1  mod  P )  =  ( ( -u 1 ^ N )  mod  P
)  <->  ( -u 1  mod  P )  =  ( 1  mod  P ) ) )
46 eqeq2 2199 . . . . . . . 8  |-  ( (
-u 1 ^ N
)  =  1  -> 
( -u 1  =  (
-u 1 ^ N
)  <->  -u 1  =  1 ) )
4745, 46imbi12d 234 . . . . . . 7  |-  ( (
-u 1 ^ N
)  =  1  -> 
( ( ( -u
1  mod  P )  =  ( ( -u
1 ^ N )  mod  P )  ->  -u 1  =  ( -u
1 ^ N ) )  <->  ( ( -u
1  mod  P )  =  ( 1  mod 
P )  ->  -u 1  =  1 ) ) )
4843, 47imbitrrid 156 . . . . . 6  |-  ( (
-u 1 ^ N
)  =  1  -> 
( ph  ->  ( (
-u 1  mod  P
)  =  ( (
-u 1 ^ N
)  mod  P )  -> 
-u 1  =  (
-u 1 ^ N
) ) ) )
4924, 48jaoi 717 . . . . 5  |-  ( ( ( -u 1 ^ N )  =  -u
1  \/  ( -u
1 ^ N )  =  1 )  -> 
( ph  ->  ( (
-u 1  mod  P
)  =  ( (
-u 1 ^ N
)  mod  P )  -> 
-u 1  =  (
-u 1 ^ N
) ) ) )
5021, 49mpcom 36 . . . 4  |-  ( ph  ->  ( ( -u 1  mod  P )  =  ( ( -u 1 ^ N )  mod  P
)  ->  -u 1  =  ( -u 1 ^ N ) ) )
51 oveq1 5913 . . . . . 6  |-  ( ( 2  /L P )  =  -u 1  ->  ( ( 2  /L P )  mod 
P )  =  (
-u 1  mod  P
) )
5251eqeq1d 2198 . . . . 5  |-  ( ( 2  /L P )  =  -u 1  ->  ( ( ( 2  /L P )  mod  P )  =  ( ( -u 1 ^ N )  mod  P
)  <->  ( -u 1  mod  P )  =  ( ( -u 1 ^ N )  mod  P
) ) )
53 eqeq1 2196 . . . . 5  |-  ( ( 2  /L P )  =  -u 1  ->  ( ( 2  /L P )  =  ( -u 1 ^ N )  <->  -u 1  =  ( -u 1 ^ N ) ) )
5452, 53imbi12d 234 . . . 4  |-  ( ( 2  /L P )  =  -u 1  ->  ( ( ( ( 2  /L P )  mod  P )  =  ( ( -u
1 ^ N )  mod  P )  -> 
( 2  /L
P )  =  (
-u 1 ^ N
) )  <->  ( ( -u 1  mod  P )  =  ( ( -u
1 ^ N )  mod  P )  ->  -u 1  =  ( -u
1 ^ N ) ) ) )
5550, 54imbitrrid 156 . . 3  |-  ( ( 2  /L P )  =  -u 1  ->  ( ph  ->  (
( ( 2  /L P )  mod 
P )  =  ( ( -u 1 ^ N )  mod  P
)  ->  ( 2  /L P )  =  ( -u 1 ^ N ) ) ) )
5625nngt0d 9012 . . . . . . 7  |-  ( ph  ->  0  <  P )
57 q0mod 10412 . . . . . . 7  |-  ( ( P  e.  QQ  /\  0  <  P )  -> 
( 0  mod  P
)  =  0 )
5827, 56, 57syl2anc 411 . . . . . 6  |-  ( ph  ->  ( 0  mod  P
)  =  0 )
5958eqeq1d 2198 . . . . 5  |-  ( ph  ->  ( ( 0  mod 
P )  =  ( ( -u 1 ^ N )  mod  P
)  <->  0  =  ( ( -u 1 ^ N )  mod  P
) ) )
60 oveq1 5913 . . . . . . . . . . 11  |-  ( (
-u 1 ^ N
)  =  -u 1  ->  ( ( -u 1 ^ N )  mod  P
)  =  ( -u
1  mod  P )
)
6160eqeq2d 2201 . . . . . . . . . 10  |-  ( (
-u 1 ^ N
)  =  -u 1  ->  ( 0  =  ( ( -u 1 ^ N )  mod  P
)  <->  0  =  (
-u 1  mod  P
) ) )
6261adantr 276 . . . . . . . . 9  |-  ( ( ( -u 1 ^ N )  =  -u
1  /\  ph )  -> 
( 0  =  ( ( -u 1 ^ N )  mod  P
)  <->  0  =  (
-u 1  mod  P
) ) )
63 1z 9329 . . . . . . . . . . . . . 14  |-  1  e.  ZZ
64 zq 9677 . . . . . . . . . . . . . 14  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
6563, 64ax-mp 5 . . . . . . . . . . . . 13  |-  1  e.  QQ
66 negqmod0 10388 . . . . . . . . . . . . 13  |-  ( ( 1  e.  QQ  /\  P  e.  QQ  /\  0  <  P )  ->  (
( 1  mod  P
)  =  0  <->  ( -u 1  mod  P )  =  0 ) )
6765, 27, 56, 66mp3an2i 1353 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1  mod 
P )  =  0  <-> 
( -u 1  mod  P
)  =  0 ) )
68 eqcom 2191 . . . . . . . . . . . 12  |-  ( (
-u 1  mod  P
)  =  0  <->  0  =  ( -u 1  mod  P ) )
6967, 68bitrdi 196 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 1  mod 
P )  =  0  <->  0  =  ( -u
1  mod  P )
) )
7032eqeq1d 2198 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1  mod 
P )  =  0  <->  1  =  0 ) )
71 1ne0 9036 . . . . . . . . . . . . 13  |-  1  =/=  0
72 eqneqall 2370 . . . . . . . . . . . . 13  |-  ( 1  =  0  ->  (
1  =/=  0  -> 
0  =  ( -u
1 ^ N ) ) )
7371, 72mpi 15 . . . . . . . . . . . 12  |-  ( 1  =  0  ->  0  =  ( -u 1 ^ N ) )
7470, 73biimtrdi 163 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 1  mod 
P )  =  0  ->  0  =  (
-u 1 ^ N
) ) )
7569, 74sylbird 170 . . . . . . . . . 10  |-  ( ph  ->  ( 0  =  (
-u 1  mod  P
)  ->  0  =  ( -u 1 ^ N
) ) )
7675adantl 277 . . . . . . . . 9  |-  ( ( ( -u 1 ^ N )  =  -u
1  /\  ph )  -> 
( 0  =  (
-u 1  mod  P
)  ->  0  =  ( -u 1 ^ N
) ) )
7762, 76sylbid 150 . . . . . . . 8  |-  ( ( ( -u 1 ^ N )  =  -u
1  /\  ph )  -> 
( 0  =  ( ( -u 1 ^ N )  mod  P
)  ->  0  =  ( -u 1 ^ N
) ) )
7877ex 115 . . . . . . 7  |-  ( (
-u 1 ^ N
)  =  -u 1  ->  ( ph  ->  (
0  =  ( (
-u 1 ^ N
)  mod  P )  ->  0  =  ( -u
1 ^ N ) ) ) )
7944eqeq2d 2201 . . . . . . . . . 10  |-  ( (
-u 1 ^ N
)  =  1  -> 
( 0  =  ( ( -u 1 ^ N )  mod  P
)  <->  0  =  ( 1  mod  P ) ) )
8079adantr 276 . . . . . . . . 9  |-  ( ( ( -u 1 ^ N )  =  1  /\  ph )  -> 
( 0  =  ( ( -u 1 ^ N )  mod  P
)  <->  0  =  ( 1  mod  P ) ) )
81 eqcom 2191 . . . . . . . . . . . 12  |-  ( 0  =  ( 1  mod 
P )  <->  ( 1  mod  P )  =  0 )
8281, 70bitrid 192 . . . . . . . . . . 11  |-  ( ph  ->  ( 0  =  ( 1  mod  P )  <->  1  =  0 ) )
8382, 73biimtrdi 163 . . . . . . . . . 10  |-  ( ph  ->  ( 0  =  ( 1  mod  P )  ->  0  =  (
-u 1 ^ N
) ) )
8483adantl 277 . . . . . . . . 9  |-  ( ( ( -u 1 ^ N )  =  1  /\  ph )  -> 
( 0  =  ( 1  mod  P )  ->  0  =  (
-u 1 ^ N
) ) )
8580, 84sylbid 150 . . . . . . . 8  |-  ( ( ( -u 1 ^ N )  =  1  /\  ph )  -> 
( 0  =  ( ( -u 1 ^ N )  mod  P
)  ->  0  =  ( -u 1 ^ N
) ) )
8685ex 115 . . . . . . 7  |-  ( (
-u 1 ^ N
)  =  1  -> 
( ph  ->  ( 0  =  ( ( -u
1 ^ N )  mod  P )  -> 
0  =  ( -u
1 ^ N ) ) ) )
8778, 86jaoi 717 . . . . . 6  |-  ( ( ( -u 1 ^ N )  =  -u
1  \/  ( -u
1 ^ N )  =  1 )  -> 
( ph  ->  ( 0  =  ( ( -u
1 ^ N )  mod  P )  -> 
0  =  ( -u
1 ^ N ) ) ) )
8821, 87mpcom 36 . . . . 5  |-  ( ph  ->  ( 0  =  ( ( -u 1 ^ N )  mod  P
)  ->  0  =  ( -u 1 ^ N
) ) )
8959, 88sylbid 150 . . . 4  |-  ( ph  ->  ( ( 0  mod 
P )  =  ( ( -u 1 ^ N )  mod  P
)  ->  0  =  ( -u 1 ^ N
) ) )
90 oveq1 5913 . . . . . 6  |-  ( ( 2  /L P )  =  0  -> 
( ( 2  /L P )  mod 
P )  =  ( 0  mod  P ) )
9190eqeq1d 2198 . . . . 5  |-  ( ( 2  /L P )  =  0  -> 
( ( ( 2  /L P )  mod  P )  =  ( ( -u 1 ^ N )  mod  P
)  <->  ( 0  mod 
P )  =  ( ( -u 1 ^ N )  mod  P
) ) )
92 eqeq1 2196 . . . . 5  |-  ( ( 2  /L P )  =  0  -> 
( ( 2  /L P )  =  ( -u 1 ^ N )  <->  0  =  ( -u 1 ^ N
) ) )
9391, 92imbi12d 234 . . . 4  |-  ( ( 2  /L P )  =  0  -> 
( ( ( ( 2  /L P )  mod  P )  =  ( ( -u
1 ^ N )  mod  P )  -> 
( 2  /L
P )  =  (
-u 1 ^ N
) )  <->  ( (
0  mod  P )  =  ( ( -u
1 ^ N )  mod  P )  -> 
0  =  ( -u
1 ^ N ) ) ) )
9489, 93imbitrrid 156 . . 3  |-  ( ( 2  /L P )  =  0  -> 
( ph  ->  ( ( ( 2  /L
P )  mod  P
)  =  ( (
-u 1 ^ N
)  mod  P )  ->  ( 2  /L
P )  =  (
-u 1 ^ N
) ) ) )
9532eqeq1d 2198 . . . . 5  |-  ( ph  ->  ( ( 1  mod 
P )  =  ( ( -u 1 ^ N )  mod  P
)  <->  1  =  ( ( -u 1 ^ N )  mod  P
) ) )
96 eqcom 2191 . . . . . . . . 9  |-  ( 1  =  ( -u 1  mod  P )  <->  ( -u 1  mod  P )  =  1 )
97 eqcom 2191 . . . . . . . . 9  |-  ( 1  =  -u 1  <->  -u 1  =  1 )
9842, 96, 973imtr4g 205 . . . . . . . 8  |-  ( ph  ->  ( 1  =  (
-u 1  mod  P
)  ->  1  =  -u 1 ) )
9960eqeq2d 2201 . . . . . . . . 9  |-  ( (
-u 1 ^ N
)  =  -u 1  ->  ( 1  =  ( ( -u 1 ^ N )  mod  P
)  <->  1  =  (
-u 1  mod  P
) ) )
100 eqeq2 2199 . . . . . . . . 9  |-  ( (
-u 1 ^ N
)  =  -u 1  ->  ( 1  =  (
-u 1 ^ N
)  <->  1  =  -u
1 ) )
10199, 100imbi12d 234 . . . . . . . 8  |-  ( (
-u 1 ^ N
)  =  -u 1  ->  ( ( 1  =  ( ( -u 1 ^ N )  mod  P
)  ->  1  =  ( -u 1 ^ N
) )  <->  ( 1  =  ( -u 1  mod  P )  ->  1  =  -u 1 ) ) )
10298, 101imbitrrid 156 . . . . . . 7  |-  ( (
-u 1 ^ N
)  =  -u 1  ->  ( ph  ->  (
1  =  ( (
-u 1 ^ N
)  mod  P )  ->  1  =  ( -u
1 ^ N ) ) ) )
103 eqcom 2191 . . . . . . . . 9  |-  ( (
-u 1 ^ N
)  =  1  <->  1  =  ( -u 1 ^ N ) )
104103biimpi 120 . . . . . . . 8  |-  ( (
-u 1 ^ N
)  =  1  -> 
1  =  ( -u
1 ^ N ) )
1051042a1d 23 . . . . . . 7  |-  ( (
-u 1 ^ N
)  =  1  -> 
( ph  ->  ( 1  =  ( ( -u
1 ^ N )  mod  P )  -> 
1  =  ( -u
1 ^ N ) ) ) )
106102, 105jaoi 717 . . . . . 6  |-  ( ( ( -u 1 ^ N )  =  -u
1  \/  ( -u
1 ^ N )  =  1 )  -> 
( ph  ->  ( 1  =  ( ( -u
1 ^ N )  mod  P )  -> 
1  =  ( -u
1 ^ N ) ) ) )
10721, 106mpcom 36 . . . . 5  |-  ( ph  ->  ( 1  =  ( ( -u 1 ^ N )  mod  P
)  ->  1  =  ( -u 1 ^ N
) ) )
10895, 107sylbid 150 . . . 4  |-  ( ph  ->  ( ( 1  mod 
P )  =  ( ( -u 1 ^ N )  mod  P
)  ->  1  =  ( -u 1 ^ N
) ) )
109 oveq1 5913 . . . . . 6  |-  ( ( 2  /L P )  =  1  -> 
( ( 2  /L P )  mod 
P )  =  ( 1  mod  P ) )
110109eqeq1d 2198 . . . . 5  |-  ( ( 2  /L P )  =  1  -> 
( ( ( 2  /L P )  mod  P )  =  ( ( -u 1 ^ N )  mod  P
)  <->  ( 1  mod 
P )  =  ( ( -u 1 ^ N )  mod  P
) ) )
111 eqeq1 2196 . . . . 5  |-  ( ( 2  /L P )  =  1  -> 
( ( 2  /L P )  =  ( -u 1 ^ N )  <->  1  =  ( -u 1 ^ N
) ) )
112110, 111imbi12d 234 . . . 4  |-  ( ( 2  /L P )  =  1  -> 
( ( ( ( 2  /L P )  mod  P )  =  ( ( -u
1 ^ N )  mod  P )  -> 
( 2  /L
P )  =  (
-u 1 ^ N
) )  <->  ( (
1  mod  P )  =  ( ( -u
1 ^ N )  mod  P )  -> 
1  =  ( -u
1 ^ N ) ) ) )
113108, 112imbitrrid 156 . . 3  |-  ( ( 2  /L P )  =  1  -> 
( ph  ->  ( ( ( 2  /L
P )  mod  P
)  =  ( (
-u 1 ^ N
)  mod  P )  ->  ( 2  /L
P )  =  (
-u 1 ^ N
) ) ) )
11455, 94, 1133jaoi 1314 . 2  |-  ( ( ( 2  /L
P )  =  -u
1  \/  ( 2  /L P )  =  0  \/  (
2  /L P )  =  1 )  ->  ( ph  ->  ( ( ( 2  /L P )  mod 
P )  =  ( ( -u 1 ^ N )  mod  P
)  ->  ( 2  /L P )  =  ( -u 1 ^ N ) ) ) )
11511, 114mpcom 36 1  |-  ( ph  ->  ( ( ( 2  /L P )  mod  P )  =  ( ( -u 1 ^ N )  mod  P
)  ->  ( 2  /L P )  =  ( -u 1 ^ N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709    \/ w3o 979    = wceq 1364    e. wcel 2160    =/= wne 2360    \ cdif 3146   {csn 3614   {cpr 3615   {ctp 3616   class class class wbr 4025   ` cfv 5242  (class class class)co 5906   0cc0 7858   1c1 7859    < clt 8040    - cmin 8176   -ucneg 8177    / cdiv 8677   NNcn 8968   2c2 9019   3c3 9020   4c4 9021   ZZcz 9303   ZZ>=cuz 9578   QQcq 9670   |_cfl 10323    mod cmo 10379   ^cexp 10583   Primecprime 12219    /Lclgs 15041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4140  ax-sep 4143  ax-nul 4151  ax-pow 4199  ax-pr 4234  ax-un 4458  ax-setind 4561  ax-iinf 4612  ax-cnex 7949  ax-resscn 7950  ax-1cn 7951  ax-1re 7952  ax-icn 7953  ax-addcl 7954  ax-addrcl 7955  ax-mulcl 7956  ax-mulrcl 7957  ax-addcom 7958  ax-mulcom 7959  ax-addass 7960  ax-mulass 7961  ax-distr 7962  ax-i2m1 7963  ax-0lt1 7964  ax-1rid 7965  ax-0id 7966  ax-rnegex 7967  ax-precex 7968  ax-cnre 7969  ax-pre-ltirr 7970  ax-pre-ltwlin 7971  ax-pre-lttrn 7972  ax-pre-apti 7973  ax-pre-ltadd 7974  ax-pre-mulgt0 7975  ax-pre-mulext 7976  ax-arch 7977  ax-caucvg 7978
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-xor 1387  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2758  df-sbc 2982  df-csb 3077  df-dif 3151  df-un 3153  df-in 3155  df-ss 3162  df-nul 3443  df-if 3554  df-pw 3599  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3832  df-int 3867  df-iun 3910  df-br 4026  df-opab 4087  df-mpt 4088  df-tr 4124  df-id 4318  df-po 4321  df-iso 4322  df-iord 4391  df-on 4393  df-ilim 4394  df-suc 4396  df-iom 4615  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-rn 4662  df-res 4663  df-ima 4664  df-iota 5203  df-fun 5244  df-fn 5245  df-f 5246  df-f1 5247  df-fo 5248  df-f1o 5249  df-fv 5250  df-isom 5251  df-riota 5861  df-ov 5909  df-oprab 5910  df-mpo 5911  df-1st 6180  df-2nd 6181  df-recs 6345  df-irdg 6410  df-frec 6431  df-1o 6456  df-2o 6457  df-oadd 6460  df-er 6574  df-en 6782  df-dom 6783  df-fin 6784  df-sup 7029  df-inf 7030  df-pnf 8042  df-mnf 8043  df-xr 8044  df-ltxr 8045  df-le 8046  df-sub 8178  df-neg 8179  df-reap 8580  df-ap 8587  df-div 8678  df-inn 8969  df-2 9027  df-3 9028  df-4 9029  df-5 9030  df-6 9031  df-7 9032  df-8 9033  df-n0 9227  df-z 9304  df-uz 9579  df-q 9671  df-rp 9706  df-fz 10061  df-fzo 10195  df-fl 10325  df-mod 10380  df-seqfrec 10505  df-exp 10584  df-ihash 10821  df-cj 10960  df-re 10961  df-im 10962  df-rsqrt 11116  df-abs 11117  df-clim 11396  df-proddc 11668  df-dvds 11905  df-gcd 12054  df-prm 12220  df-phi 12323  df-pc 12397  df-lgs 15042
This theorem is referenced by:  gausslemma2d  15113
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