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Theorem gausslemma2dlem0i 15173
Description: Auxiliary lemma 9 for gausslemma2d 15185. (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 9345 . . . 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 15165 . . . . . 6  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  e.  NN )
54nnzd 9438 . . . . 5  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  e.  ZZ )
62, 5syl 14 . . . 4  |-  ( ph  ->  P  e.  ZZ )
7 lgscl1 15139 . . . 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 3663 . . . 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 15172 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
1615nn0zd 9437 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
17 m1expcl2 10632 . . . . . . 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 3638 . . . . . . 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 2195 . . . . . . . 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 15165 . . . . . . . . . . 11  |-  ( ph  ->  P  e.  NN )
26 nnq 9698 . . . . . . . . . . 11  |-  ( P  e.  NN  ->  P  e.  QQ )
2725, 26syl 14 . . . . . . . . . 10  |-  ( ph  ->  P  e.  QQ )
282eldifad 3164 . . . . . . . . . . 11  |-  ( ph  ->  P  e.  Prime )
29 prmgt1 12270 . . . . . . . . . . 11  |-  ( P  e.  Prime  ->  1  < 
P )
3028, 29syl 14 . . . . . . . . . 10  |-  ( ph  ->  1  <  P )
31 q1mod 10427 . . . . . . . . . 10  |-  ( ( P  e.  QQ  /\  1  <  P )  -> 
( 1  mod  P
)  =  1 )
3227, 30, 31syl2anc 411 . . . . . . . . 9  |-  ( ph  ->  ( 1  mod  P
)  =  1 )
3332eqeq2d 2205 . . . . . . . 8  |-  ( ph  ->  ( ( -u 1  mod  P )  =  ( 1  mod  P )  <-> 
( -u 1  mod  P
)  =  1 ) )
34 oddprmge3 12273 . . . . . . . . 9  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  e.  ( ZZ>= ` 
3 ) )
35 m1modge3gt1 10442 . . . . . . . . . 10  |-  ( P  e.  ( ZZ>= `  3
)  ->  1  <  (
-u 1  mod  P
) )
36 breq2 4033 . . . . . . . . . . 11  |-  ( (
-u 1  mod  P
)  =  1  -> 
( 1  <  ( -u 1  mod  P )  <->  1  <  1 ) )
37 1re 8018 . . . . . . . . . . . . 13  |-  1  e.  RR
3837ltnri 8112 . . . . . . . . . . . 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 5925 . . . . . . . . 9  |-  ( (
-u 1 ^ N
)  =  1  -> 
( ( -u 1 ^ N )  mod  P
)  =  ( 1  mod  P ) )
4544eqeq2d 2205 . . . . . . . 8  |-  ( (
-u 1 ^ N
)  =  1  -> 
( ( -u 1  mod  P )  =  ( ( -u 1 ^ N )  mod  P
)  <->  ( -u 1  mod  P )  =  ( 1  mod  P ) ) )
46 eqeq2 2203 . . . . . . . 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 5925 . . . . . 6  |-  ( ( 2  /L P )  =  -u 1  ->  ( ( 2  /L P )  mod 
P )  =  (
-u 1  mod  P
) )
5251eqeq1d 2202 . . . . 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 2200 . . . . 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 9026 . . . . . . 7  |-  ( ph  ->  0  <  P )
57 q0mod 10426 . . . . . . 7  |-  ( ( P  e.  QQ  /\  0  <  P )  -> 
( 0  mod  P
)  =  0 )
5827, 56, 57syl2anc 411 . . . . . 6  |-  ( ph  ->  ( 0  mod  P
)  =  0 )
5958eqeq1d 2202 . . . . 5  |-  ( ph  ->  ( ( 0  mod 
P )  =  ( ( -u 1 ^ N )  mod  P
)  <->  0  =  ( ( -u 1 ^ N )  mod  P
) ) )
60 oveq1 5925 . . . . . . . . . . 11  |-  ( (
-u 1 ^ N
)  =  -u 1  ->  ( ( -u 1 ^ N )  mod  P
)  =  ( -u
1  mod  P )
)
6160eqeq2d 2205 . . . . . . . . . 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 9343 . . . . . . . . . . . . . 14  |-  1  e.  ZZ
64 zq 9691 . . . . . . . . . . . . . 14  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
6563, 64ax-mp 5 . . . . . . . . . . . . 13  |-  1  e.  QQ
66 negqmod0 10402 . . . . . . . . . . . . 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 2195 . . . . . . . . . . . 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 2202 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1  mod 
P )  =  0  <->  1  =  0 ) )
71 1ne0 9050 . . . . . . . . . . . . 13  |-  1  =/=  0
72 eqneqall 2374 . . . . . . . . . . . . 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 2205 . . . . . . . . . 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 2195 . . . . . . . . . . . 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 5925 . . . . . 6  |-  ( ( 2  /L P )  =  0  -> 
( ( 2  /L P )  mod 
P )  =  ( 0  mod  P ) )
9190eqeq1d 2202 . . . . 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 2200 . . . . 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 2202 . . . . 5  |-  ( ph  ->  ( ( 1  mod 
P )  =  ( ( -u 1 ^ N )  mod  P
)  <->  1  =  ( ( -u 1 ^ N )  mod  P
) ) )
96 eqcom 2195 . . . . . . . . 9  |-  ( 1  =  ( -u 1  mod  P )  <->  ( -u 1  mod  P )  =  1 )
97 eqcom 2195 . . . . . . . . 9  |-  ( 1  =  -u 1  <->  -u 1  =  1 )
9842, 96, 973imtr4g 205 . . . . . . . 8  |-  ( ph  ->  ( 1  =  (
-u 1  mod  P
)  ->  1  =  -u 1 ) )
9960eqeq2d 2205 . . . . . . . . 9  |-  ( (
-u 1 ^ N
)  =  -u 1  ->  ( 1  =  ( ( -u 1 ^ N )  mod  P
)  <->  1  =  (
-u 1  mod  P
) ) )
100 eqeq2 2203 . . . . . . . . 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 2195 . . . . . . . . 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 5925 . . . . . 6  |-  ( ( 2  /L P )  =  1  -> 
( ( 2  /L P )  mod 
P )  =  ( 1  mod  P ) )
110109eqeq1d 2202 . . . . 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 2200 . . . . 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 2164    =/= wne 2364    \ cdif 3150   {csn 3618   {cpr 3619   {ctp 3620   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   0cc0 7872   1c1 7873    < clt 8054    - cmin 8190   -ucneg 8191    / cdiv 8691   NNcn 8982   2c2 9033   3c3 9034   4c4 9035   ZZcz 9317   ZZ>=cuz 9592   QQcq 9684   |_cfl 10337    mod cmo 10393   ^cexp 10609   Primecprime 12245    /Lclgs 15113
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 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992
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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-tp 3626  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-frec 6444  df-1o 6469  df-2o 6470  df-oadd 6473  df-er 6587  df-en 6795  df-dom 6796  df-fin 6797  df-sup 7043  df-inf 7044  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045  df-7 9046  df-8 9047  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-ihash 10847  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-clim 11422  df-proddc 11694  df-dvds 11931  df-gcd 12080  df-prm 12246  df-phi 12349  df-pc 12423  df-lgs 15114
This theorem is referenced by:  gausslemma2d  15185
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