MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lgseisen Unicode version

Theorem lgseisen 21090
Description: Eisenstein's lemma, an expression for  ( P  / L Q ) when  P ,  Q are distinct odd primes. (Contributed by Mario Carneiro, 18-Jun-2015.)
Hypotheses
Ref Expression
lgseisen.1  |-  ( ph  ->  P  e.  ( Prime  \  { 2 } ) )
lgseisen.2  |-  ( ph  ->  Q  e.  ( Prime  \  { 2 } ) )
lgseisen.3  |-  ( ph  ->  P  =/=  Q )
Assertion
Ref Expression
lgseisen  |-  ( ph  ->  ( Q  / L P )  =  (
-u 1 ^ sum_ x  e.  ( 1 ... ( ( P  - 
1 )  /  2
) ) ( |_
`  ( ( Q  /  P )  x.  ( 2  x.  x
) ) ) ) )
Distinct variable groups:    x, P    ph, x    x, Q

Proof of Theorem lgseisen
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 lgseisen.2 . . . . 5  |-  ( ph  ->  Q  e.  ( Prime  \  { 2 } ) )
21eldifad 3292 . . . 4  |-  ( ph  ->  Q  e.  Prime )
3 prmz 13038 . . . 4  |-  ( Q  e.  Prime  ->  Q  e.  ZZ )
42, 3syl 16 . . 3  |-  ( ph  ->  Q  e.  ZZ )
5 lgseisen.1 . . 3  |-  ( ph  ->  P  e.  ( Prime  \  { 2 } ) )
6 lgsval3 21051 . . 3  |-  ( ( Q  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( Q  / L P )  =  ( ( ( ( Q ^ ( ( P  -  1 )  /  2 ) )  +  1 )  mod 
P )  -  1 ) )
74, 5, 6syl2anc 643 . 2  |-  ( ph  ->  ( Q  / L P )  =  ( ( ( ( Q ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 ) )
8 prmnn 13037 . . . . . . . . 9  |-  ( Q  e.  Prime  ->  Q  e.  NN )
92, 8syl 16 . . . . . . . 8  |-  ( ph  ->  Q  e.  NN )
10 oddprm 13144 . . . . . . . . . 10  |-  ( P  e.  ( Prime  \  {
2 } )  -> 
( ( P  - 
1 )  /  2
)  e.  NN )
115, 10syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( P  - 
1 )  /  2
)  e.  NN )
1211nnnn0d 10230 . . . . . . . 8  |-  ( ph  ->  ( ( P  - 
1 )  /  2
)  e.  NN0 )
139, 12nnexpcld 11499 . . . . . . 7  |-  ( ph  ->  ( Q ^ (
( P  -  1 )  /  2 ) )  e.  NN )
1413nnred 9971 . . . . . 6  |-  ( ph  ->  ( Q ^ (
( P  -  1 )  /  2 ) )  e.  RR )
15 1re 9046 . . . . . . . . 9  |-  1  e.  RR
1615renegcli 9318 . . . . . . . 8  |-  -u 1  e.  RR
1716a1i 11 . . . . . . 7  |-  ( ph  -> 
-u 1  e.  RR )
18 ax-1cn 9004 . . . . . . . . 9  |-  1  e.  CC
19 ax-1ne0 9015 . . . . . . . . 9  |-  1  =/=  0
2018, 19negne0i 9331 . . . . . . . 8  |-  -u 1  =/=  0
2120a1i 11 . . . . . . 7  |-  ( ph  -> 
-u 1  =/=  0
)
22 fzfid 11267 . . . . . . . 8  |-  ( ph  ->  ( 1 ... (
( P  -  1 )  /  2 ) )  e.  Fin )
239nnred 9971 . . . . . . . . . . . 12  |-  ( ph  ->  Q  e.  RR )
245eldifad 3292 . . . . . . . . . . . . 13  |-  ( ph  ->  P  e.  Prime )
25 prmnn 13037 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  P  e.  NN )
2624, 25syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  P  e.  NN )
2723, 26nndivred 10004 . . . . . . . . . . 11  |-  ( ph  ->  ( Q  /  P
)  e.  RR )
2827adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) )  ->  ( Q  /  P )  e.  RR )
29 2re 10025 . . . . . . . . . . 11  |-  2  e.  RR
30 elfznn 11036 . . . . . . . . . . . . 13  |-  ( x  e.  ( 1 ... ( ( P  - 
1 )  /  2
) )  ->  x  e.  NN )
3130adantl 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) )  ->  x  e.  NN )
3231nnred 9971 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) )  ->  x  e.  RR )
33 remulcl 9031 . . . . . . . . . . 11  |-  ( ( 2  e.  RR  /\  x  e.  RR )  ->  ( 2  x.  x
)  e.  RR )
3429, 32, 33sylancr 645 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) )  ->  (
2  x.  x )  e.  RR )
3528, 34remulcld 9072 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) )  ->  (
( Q  /  P
)  x.  ( 2  x.  x ) )  e.  RR )
3635flcld 11162 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) )  ->  ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x
) ) )  e.  ZZ )
3722, 36fsumzcl 12484 . . . . . . 7  |-  ( ph  -> 
sum_ x  e.  (
1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) )  e.  ZZ )
3817, 21, 37reexpclzd 11503 . . . . . 6  |-  ( ph  ->  ( -u 1 ^
sum_ x  e.  (
1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) ) )  e.  RR )
3915a1i 11 . . . . . 6  |-  ( ph  ->  1  e.  RR )
4026nnrpd 10603 . . . . . 6  |-  ( ph  ->  P  e.  RR+ )
41 lgseisen.3 . . . . . . 7  |-  ( ph  ->  P  =/=  Q )
42 eqid 2404 . . . . . . 7  |-  ( ( Q  x.  ( 2  x.  x ) )  mod  P )  =  ( ( Q  x.  ( 2  x.  x
) )  mod  P
)
43 eqid 2404 . . . . . . 7  |-  ( x  e.  ( 1 ... ( ( P  - 
1 )  /  2
) )  |->  ( ( ( ( -u 1 ^ ( ( Q  x.  ( 2  x.  x ) )  mod 
P ) )  x.  ( ( Q  x.  ( 2  x.  x
) )  mod  P
) )  mod  P
)  /  2 ) )  =  ( x  e.  ( 1 ... ( ( P  - 
1 )  /  2
) )  |->  ( ( ( ( -u 1 ^ ( ( Q  x.  ( 2  x.  x ) )  mod 
P ) )  x.  ( ( Q  x.  ( 2  x.  x
) )  mod  P
) )  mod  P
)  /  2 ) )
44 eqid 2404 . . . . . . 7  |-  ( ( Q  x.  ( 2  x.  y ) )  mod  P )  =  ( ( Q  x.  ( 2  x.  y
) )  mod  P
)
45 eqid 2404 . . . . . . 7  |-  (ℤ/n `  P
)  =  (ℤ/n `  P
)
46 eqid 2404 . . . . . . 7  |-  (mulGrp `  (ℤ/n `  P ) )  =  (mulGrp `  (ℤ/n `  P ) )
47 eqid 2404 . . . . . . 7  |-  ( ZRHom `  (ℤ/n `  P ) )  =  ( ZRHom `  (ℤ/n `  P
) )
485, 1, 41, 42, 43, 44, 45, 46, 47lgseisenlem4 21089 . . . . . 6  |-  ( ph  ->  ( ( Q ^
( ( P  - 
1 )  /  2
) )  mod  P
)  =  ( (
-u 1 ^ sum_ x  e.  ( 1 ... ( ( P  - 
1 )  /  2
) ) ( |_
`  ( ( Q  /  P )  x.  ( 2  x.  x
) ) ) )  mod  P ) )
49 modadd1 11233 . . . . . 6  |-  ( ( ( ( Q ^
( ( P  - 
1 )  /  2
) )  e.  RR  /\  ( -u 1 ^
sum_ x  e.  (
1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) ) )  e.  RR )  /\  ( 1  e.  RR  /\  P  e.  RR+ )  /\  (
( Q ^ (
( P  -  1 )  /  2 ) )  mod  P )  =  ( ( -u
1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) )  mod 
P ) )  -> 
( ( ( Q ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  =  ( ( ( -u 1 ^
sum_ x  e.  (
1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) ) )  +  1 )  mod  P ) )
5014, 38, 39, 40, 48, 49syl221anc 1195 . . . . 5  |-  ( ph  ->  ( ( ( Q ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  =  ( ( ( -u 1 ^
sum_ x  e.  (
1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) ) )  +  1 )  mod  P ) )
51 peano2re 9195 . . . . . . 7  |-  ( (
-u 1 ^ sum_ x  e.  ( 1 ... ( ( P  - 
1 )  /  2
) ) ( |_
`  ( ( Q  /  P )  x.  ( 2  x.  x
) ) ) )  e.  RR  ->  (
( -u 1 ^ sum_ x  e.  ( 1 ... ( ( P  - 
1 )  /  2
) ) ( |_
`  ( ( Q  /  P )  x.  ( 2  x.  x
) ) ) )  +  1 )  e.  RR )
5238, 51syl 16 . . . . . 6  |-  ( ph  ->  ( ( -u 1 ^ sum_ x  e.  ( 1 ... ( ( P  -  1 )  /  2 ) ) ( |_ `  (
( Q  /  P
)  x.  ( 2  x.  x ) ) ) )  +  1 )  e.  RR )
53 df-neg 9250 . . . . . . . 8  |-  -u 1  =  ( 0  -  1 )
54 neg1cn 10023 . . . . . . . . . . . . . 14  |-  -u 1  e.  CC
5554a1i 11 . . . . . . . . . . . . 13  |-  ( ph  -> 
-u 1  e.  CC )
56 absexpz 12065 . . . . . . . . . . . . 13  |-  ( (
-u 1  e.  CC  /\  -u 1  =/=  0  /\  sum_ x  e.  ( 1 ... ( ( P  -  1 )  /  2 ) ) ( |_ `  (
( Q  /  P
)  x.  ( 2  x.  x ) ) )  e.  ZZ )  ->  ( abs `  ( -u 1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) ) )  =  ( ( abs `  -u 1 ) ^ sum_ x  e.  ( 1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) ) ) )
5755, 21, 37, 56syl3anc 1184 . . . . . . . . . . . 12  |-  ( ph  ->  ( abs `  ( -u 1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) ) )  =  ( ( abs `  -u 1 ) ^ sum_ x  e.  ( 1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) ) ) )
5818absnegi 12158 . . . . . . . . . . . . . . 15  |-  ( abs `  -u 1 )  =  ( abs `  1
)
59 abs1 12057 . . . . . . . . . . . . . . 15  |-  ( abs `  1 )  =  1
6058, 59eqtri 2424 . . . . . . . . . . . . . 14  |-  ( abs `  -u 1 )  =  1
6160oveq1i 6050 . . . . . . . . . . . . 13  |-  ( ( abs `  -u 1
) ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) )  =  ( 1 ^ sum_ x  e.  ( 1 ... ( ( P  - 
1 )  /  2
) ) ( |_
`  ( ( Q  /  P )  x.  ( 2  x.  x
) ) ) )
62 1exp 11364 . . . . . . . . . . . . . 14  |-  ( sum_ x  e.  ( 1 ... ( ( P  - 
1 )  /  2
) ) ( |_
`  ( ( Q  /  P )  x.  ( 2  x.  x
) ) )  e.  ZZ  ->  ( 1 ^ sum_ x  e.  ( 1 ... ( ( P  -  1 )  /  2 ) ) ( |_ `  (
( Q  /  P
)  x.  ( 2  x.  x ) ) ) )  =  1 )
6337, 62syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 1 ^ sum_ x  e.  ( 1 ... ( ( P  - 
1 )  /  2
) ) ( |_
`  ( ( Q  /  P )  x.  ( 2  x.  x
) ) ) )  =  1 )
6461, 63syl5eq 2448 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( abs `  -u 1
) ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) )  =  1 )
6557, 64eqtrd 2436 . . . . . . . . . . 11  |-  ( ph  ->  ( abs `  ( -u 1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) ) )  =  1 )
66 1le1 9606 . . . . . . . . . . 11  |-  1  <_  1
6765, 66syl6eqbr 4209 . . . . . . . . . 10  |-  ( ph  ->  ( abs `  ( -u 1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) ) )  <_  1 )
68 absle 12074 . . . . . . . . . . 11  |-  ( ( ( -u 1 ^
sum_ x  e.  (
1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) ) )  e.  RR  /\  1  e.  RR )  ->  ( ( abs `  ( -u 1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) ) )  <_  1  <->  ( -u 1  <_  ( -u 1 ^
sum_ x  e.  (
1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) ) )  /\  ( -u
1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) )  <_ 
1 ) ) )
6938, 15, 68sylancl 644 . . . . . . . . . 10  |-  ( ph  ->  ( ( abs `  ( -u 1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) ) )  <_  1  <->  ( -u 1  <_  ( -u 1 ^
sum_ x  e.  (
1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) ) )  /\  ( -u
1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) )  <_ 
1 ) ) )
7067, 69mpbid 202 . . . . . . . . 9  |-  ( ph  ->  ( -u 1  <_ 
( -u 1 ^ sum_ x  e.  ( 1 ... ( ( P  - 
1 )  /  2
) ) ( |_
`  ( ( Q  /  P )  x.  ( 2  x.  x
) ) ) )  /\  ( -u 1 ^ sum_ x  e.  ( 1 ... ( ( P  -  1 )  /  2 ) ) ( |_ `  (
( Q  /  P
)  x.  ( 2  x.  x ) ) ) )  <_  1
) )
7170simpld 446 . . . . . . . 8  |-  ( ph  -> 
-u 1  <_  ( -u 1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) ) )
7253, 71syl5eqbrr 4206 . . . . . . 7  |-  ( ph  ->  ( 0  -  1 )  <_  ( -u 1 ^ sum_ x  e.  ( 1 ... ( ( P  -  1 )  /  2 ) ) ( |_ `  (
( Q  /  P
)  x.  ( 2  x.  x ) ) ) ) )
73 0re 9047 . . . . . . . . 9  |-  0  e.  RR
7473a1i 11 . . . . . . . 8  |-  ( ph  ->  0  e.  RR )
7574, 39, 38lesubaddd 9579 . . . . . . 7  |-  ( ph  ->  ( ( 0  -  1 )  <_  ( -u 1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) )  <->  0  <_  ( ( -u 1 ^
sum_ x  e.  (
1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) ) )  +  1 ) ) )
7672, 75mpbid 202 . . . . . 6  |-  ( ph  ->  0  <_  ( ( -u 1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) )  +  1 ) )
7726nnred 9971 . . . . . . . . 9  |-  ( ph  ->  P  e.  RR )
78 peano2rem 9323 . . . . . . . . 9  |-  ( P  e.  RR  ->  ( P  -  1 )  e.  RR )
7977, 78syl 16 . . . . . . . 8  |-  ( ph  ->  ( P  -  1 )  e.  RR )
8070simprd 450 . . . . . . . 8  |-  ( ph  ->  ( -u 1 ^
sum_ x  e.  (
1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) ) )  <_  1 )
81 df-2 10014 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
82 eldifsni 3888 . . . . . . . . . . . 12  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  =/=  2 )
835, 82syl 16 . . . . . . . . . . 11  |-  ( ph  ->  P  =/=  2 )
8429a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  2  e.  RR )
85 prmuz2 13052 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
86 eluzle 10454 . . . . . . . . . . . . 13  |-  ( P  e.  ( ZZ>= `  2
)  ->  2  <_  P )
8724, 85, 863syl 19 . . . . . . . . . . . 12  |-  ( ph  ->  2  <_  P )
8884, 77, 87leltned 9180 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  <  P  <->  P  =/=  2 ) )
8983, 88mpbird 224 . . . . . . . . . 10  |-  ( ph  ->  2  <  P )
9081, 89syl5eqbrr 4206 . . . . . . . . 9  |-  ( ph  ->  ( 1  +  1 )  <  P )
9139, 39, 77ltaddsubd 9582 . . . . . . . . 9  |-  ( ph  ->  ( ( 1  +  1 )  <  P  <->  1  <  ( P  - 
1 ) ) )
9290, 91mpbid 202 . . . . . . . 8  |-  ( ph  ->  1  <  ( P  -  1 ) )
9338, 39, 79, 80, 92lelttrd 9184 . . . . . . 7  |-  ( ph  ->  ( -u 1 ^
sum_ x  e.  (
1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) ) )  <  ( P  -  1 ) )
9438, 39, 77ltaddsubd 9582 . . . . . . 7  |-  ( ph  ->  ( ( ( -u
1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) )  +  1 )  <  P  <->  (
-u 1 ^ sum_ x  e.  ( 1 ... ( ( P  - 
1 )  /  2
) ) ( |_
`  ( ( Q  /  P )  x.  ( 2  x.  x
) ) ) )  <  ( P  - 
1 ) ) )
9593, 94mpbird 224 . . . . . 6  |-  ( ph  ->  ( ( -u 1 ^ sum_ x  e.  ( 1 ... ( ( P  -  1 )  /  2 ) ) ( |_ `  (
( Q  /  P
)  x.  ( 2  x.  x ) ) ) )  +  1 )  <  P )
96 modid 11225 . . . . . 6  |-  ( ( ( ( ( -u
1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) )  +  1 )  e.  RR  /\  P  e.  RR+ )  /\  ( 0  <_  (
( -u 1 ^ sum_ x  e.  ( 1 ... ( ( P  - 
1 )  /  2
) ) ( |_
`  ( ( Q  /  P )  x.  ( 2  x.  x
) ) ) )  +  1 )  /\  ( ( -u 1 ^ sum_ x  e.  ( 1 ... ( ( P  -  1 )  /  2 ) ) ( |_ `  (
( Q  /  P
)  x.  ( 2  x.  x ) ) ) )  +  1 )  <  P ) )  ->  ( (
( -u 1 ^ sum_ x  e.  ( 1 ... ( ( P  - 
1 )  /  2
) ) ( |_
`  ( ( Q  /  P )  x.  ( 2  x.  x
) ) ) )  +  1 )  mod 
P )  =  ( ( -u 1 ^
sum_ x  e.  (
1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) ) )  +  1 ) )
9752, 40, 76, 95, 96syl22anc 1185 . . . . 5  |-  ( ph  ->  ( ( ( -u
1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) )  +  1 )  mod  P
)  =  ( (
-u 1 ^ sum_ x  e.  ( 1 ... ( ( P  - 
1 )  /  2
) ) ( |_
`  ( ( Q  /  P )  x.  ( 2  x.  x
) ) ) )  +  1 ) )
9850, 97eqtrd 2436 . . . 4  |-  ( ph  ->  ( ( ( Q ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  =  ( (
-u 1 ^ sum_ x  e.  ( 1 ... ( ( P  - 
1 )  /  2
) ) ( |_
`  ( ( Q  /  P )  x.  ( 2  x.  x
) ) ) )  +  1 ) )
9998oveq1d 6055 . . 3  |-  ( ph  ->  ( ( ( ( Q ^ ( ( P  -  1 )  /  2 ) )  +  1 )  mod 
P )  -  1 )  =  ( ( ( -u 1 ^
sum_ x  e.  (
1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) ) )  +  1 )  -  1 ) )
10038recnd 9070 . . . 4  |-  ( ph  ->  ( -u 1 ^
sum_ x  e.  (
1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) ) )  e.  CC )
101 pncan 9267 . . . 4  |-  ( ( ( -u 1 ^
sum_ x  e.  (
1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) ) )  e.  CC  /\  1  e.  CC )  ->  ( ( ( -u
1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) )  +  1 )  -  1 )  =  ( -u
1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) ) )
102100, 18, 101sylancl 644 . . 3  |-  ( ph  ->  ( ( ( -u
1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) )  +  1 )  -  1 )  =  ( -u
1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) ) )
10399, 102eqtrd 2436 . 2  |-  ( ph  ->  ( ( ( ( Q ^ ( ( P  -  1 )  /  2 ) )  +  1 )  mod 
P )  -  1 )  =  ( -u
1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) ) )
1047, 103eqtrd 2436 1  |-  ( ph  ->  ( Q  / L P )  =  (
-u 1 ^ sum_ x  e.  ( 1 ... ( ( P  - 
1 )  /  2
) ) ( |_
`  ( ( Q  /  P )  x.  ( 2  x.  x
) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567    \ cdif 3277   {csn 3774   class class class wbr 4172    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    < clt 9076    <_ cle 9077    - cmin 9247   -ucneg 9248    / cdiv 9633   NNcn 9956   2c2 10005   ZZcz 10238   ZZ>=cuz 10444   RR+crp 10568   ...cfz 10999   |_cfl 11156    mod cmo 11205   ^cexp 11337   abscabs 11994   sum_csu 12434   Primecprime 13034  mulGrpcmgp 15603   ZRHomczrh 16733  ℤ/nczn 16736    / Lclgs 21031
This theorem is referenced by:  lgsquadlem2  21092
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-ec 6866  df-qs 6870  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-dvds 12808  df-gcd 12962  df-prm 13035  df-phi 13110  df-pc 13166  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-0g 13682  df-gsum 13683  df-imas 13689  df-divs 13690  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-nsg 14897  df-eqg 14898  df-ghm 14959  df-cntz 15071  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-rnghom 15774  df-drng 15792  df-field 15793  df-subrg 15821  df-lmod 15907  df-lss 15964  df-lsp 16003  df-sra 16199  df-rgmod 16200  df-lidl 16201  df-rsp 16202  df-2idl 16258  df-nzr 16284  df-rlreg 16298  df-domn 16299  df-idom 16300  df-cnfld 16659  df-zrh 16737  df-zn 16740  df-lgs 21032
  Copyright terms: Public domain W3C validator