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Theorem lgseisen 21137
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 3332 . . . 4  |-  ( ph  ->  Q  e.  Prime )
3 prmz 13083 . . . 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 21098 . . 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 13082 . . . . . . . . 9  |-  ( Q  e.  Prime  ->  Q  e.  NN )
92, 8syl 16 . . . . . . . 8  |-  ( ph  ->  Q  e.  NN )
10 oddprm 13189 . . . . . . . . . 10  |-  ( P  e.  ( Prime  \  {
2 } )  -> 
( ( P  - 
1 )  /  2
)  e.  NN )
115, 10syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( P  - 
1 )  /  2
)  e.  NN )
1211nnnn0d 10274 . . . . . . . 8  |-  ( ph  ->  ( ( P  - 
1 )  /  2
)  e.  NN0 )
139, 12nnexpcld 11544 . . . . . . 7  |-  ( ph  ->  ( Q ^ (
( P  -  1 )  /  2 ) )  e.  NN )
1413nnred 10015 . . . . . 6  |-  ( ph  ->  ( Q ^ (
( P  -  1 )  /  2 ) )  e.  RR )
15 1re 9090 . . . . . . . . 9  |-  1  e.  RR
1615renegcli 9362 . . . . . . . 8  |-  -u 1  e.  RR
1716a1i 11 . . . . . . 7  |-  ( ph  -> 
-u 1  e.  RR )
18 ax-1cn 9048 . . . . . . . . 9  |-  1  e.  CC
19 ax-1ne0 9059 . . . . . . . . 9  |-  1  =/=  0
2018, 19negne0i 9375 . . . . . . . 8  |-  -u 1  =/=  0
2120a1i 11 . . . . . . 7  |-  ( ph  -> 
-u 1  =/=  0
)
22 fzfid 11312 . . . . . . . 8  |-  ( ph  ->  ( 1 ... (
( P  -  1 )  /  2 ) )  e.  Fin )
239nnred 10015 . . . . . . . . . . . 12  |-  ( ph  ->  Q  e.  RR )
245eldifad 3332 . . . . . . . . . . . . 13  |-  ( ph  ->  P  e.  Prime )
25 prmnn 13082 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  P  e.  NN )
2624, 25syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  P  e.  NN )
2723, 26nndivred 10048 . . . . . . . . . . 11  |-  ( ph  ->  ( Q  /  P
)  e.  RR )
2827adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) )  ->  ( Q  /  P )  e.  RR )
29 2re 10069 . . . . . . . . . . 11  |-  2  e.  RR
30 elfznn 11080 . . . . . . . . . . . . 13  |-  ( x  e.  ( 1 ... ( ( P  - 
1 )  /  2
) )  ->  x  e.  NN )
3130adantl 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) )  ->  x  e.  NN )
3231nnred 10015 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) )  ->  x  e.  RR )
33 remulcl 9075 . . . . . . . . . . 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 9116 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) )  ->  (
( Q  /  P
)  x.  ( 2  x.  x ) )  e.  RR )
3635flcld 11207 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) )  ->  ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x
) ) )  e.  ZZ )
3722, 36fsumzcl 12529 . . . . . . 7  |-  ( ph  -> 
sum_ x  e.  (
1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) )  e.  ZZ )
3817, 21, 37reexpclzd 11548 . . . . . 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 10647 . . . . . 6  |-  ( ph  ->  P  e.  RR+ )
41 lgseisen.3 . . . . . . 7  |-  ( ph  ->  P  =/=  Q )
42 eqid 2436 . . . . . . 7  |-  ( ( Q  x.  ( 2  x.  x ) )  mod  P )  =  ( ( Q  x.  ( 2  x.  x
) )  mod  P
)
43 eqid 2436 . . . . . . 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 2436 . . . . . . 7  |-  ( ( Q  x.  ( 2  x.  y ) )  mod  P )  =  ( ( Q  x.  ( 2  x.  y
) )  mod  P
)
45 eqid 2436 . . . . . . 7  |-  (ℤ/n `  P
)  =  (ℤ/n `  P
)
46 eqid 2436 . . . . . . 7  |-  (mulGrp `  (ℤ/n `  P ) )  =  (mulGrp `  (ℤ/n `  P ) )
47 eqid 2436 . . . . . . 7  |-  ( ZRHom `  (ℤ/n `  P ) )  =  ( ZRHom `  (ℤ/n `  P
) )
485, 1, 41, 42, 43, 44, 45, 46, 47lgseisenlem4 21136 . . . . . 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 11278 . . . . . 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 9239 . . . . . . 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 9294 . . . . . . . 8  |-  -u 1  =  ( 0  -  1 )
54 neg1cn 10067 . . . . . . . . . . . . . 14  |-  -u 1  e.  CC
5554a1i 11 . . . . . . . . . . . . 13  |-  ( ph  -> 
-u 1  e.  CC )
56 absexpz 12110 . . . . . . . . . . . . 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 12203 . . . . . . . . . . . . . . 15  |-  ( abs `  -u 1 )  =  ( abs `  1
)
59 abs1 12102 . . . . . . . . . . . . . . 15  |-  ( abs `  1 )  =  1
6058, 59eqtri 2456 . . . . . . . . . . . . . 14  |-  ( abs `  -u 1 )  =  1
6160oveq1i 6091 . . . . . . . . . . . . 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 11409 . . . . . . . . . . . . . 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 2480 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( abs `  -u 1
) ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) )  =  1 )
6557, 64eqtrd 2468 . . . . . . . . . . 11  |-  ( ph  ->  ( abs `  ( -u 1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) ) )  =  1 )
66 1le1 9650 . . . . . . . . . . 11  |-  1  <_  1
6765, 66syl6eqbr 4249 . . . . . . . . . 10  |-  ( ph  ->  ( abs `  ( -u 1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) ) )  <_  1 )
68 absle 12119 . . . . . . . . . . 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 4246 . . . . . . 7  |-  ( ph  ->  ( 0  -  1 )  <_  ( -u 1 ^ sum_ x  e.  ( 1 ... ( ( P  -  1 )  /  2 ) ) ( |_ `  (
( Q  /  P
)  x.  ( 2  x.  x ) ) ) ) )
73 0re 9091 . . . . . . . . 9  |-  0  e.  RR
7473a1i 11 . . . . . . . 8  |-  ( ph  ->  0  e.  RR )
7574, 39, 38lesubaddd 9623 . . . . . . 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 10015 . . . . . . . . 9  |-  ( ph  ->  P  e.  RR )
78 peano2rem 9367 . . . . . . . . 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 10058 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
82 eldifsni 3928 . . . . . . . . . . . 12  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  =/=  2 )
835, 82syl 16 . . . . . . . . . . 11  |-  ( ph  ->  P  =/=  2 )
8429a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  2  e.  RR )
85 prmuz2 13097 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
86 eluzle 10498 . . . . . . . . . . . . 13  |-  ( P  e.  ( ZZ>= `  2
)  ->  2  <_  P )
8724, 85, 863syl 19 . . . . . . . . . . . 12  |-  ( ph  ->  2  <_  P )
8884, 77, 87leltned 9224 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  <  P  <->  P  =/=  2 ) )
8983, 88mpbird 224 . . . . . . . . . 10  |-  ( ph  ->  2  <  P )
9081, 89syl5eqbrr 4246 . . . . . . . . 9  |-  ( ph  ->  ( 1  +  1 )  <  P )
9139, 39, 77ltaddsubd 9626 . . . . . . . . 9  |-  ( ph  ->  ( ( 1  +  1 )  <  P  <->  1  <  ( P  - 
1 ) ) )
9290, 91mpbid 202 . . . . . . . 8  |-  ( ph  ->  1  <  ( P  -  1 ) )
9338, 39, 79, 80, 92lelttrd 9228 . . . . . . 7  |-  ( ph  ->  ( -u 1 ^
sum_ x  e.  (
1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) ) )  <  ( P  -  1 ) )
9438, 39, 77ltaddsubd 9626 . . . . . . 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 11270 . . . . . 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 2468 . . . 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 6096 . . 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 9114 . . . 4  |-  ( ph  ->  ( -u 1 ^
sum_ x  e.  (
1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) ) )  e.  CC )
101 pncan 9311 . . . 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 2468 . 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 2468 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 1652    e. wcel 1725    =/= wne 2599    \ cdif 3317   {csn 3814   class class class wbr 4212    e. cmpt 4266   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    < clt 9120    <_ cle 9121    - cmin 9291   -ucneg 9292    / cdiv 9677   NNcn 10000   2c2 10049   ZZcz 10282   ZZ>=cuz 10488   RR+crp 10612   ...cfz 11043   |_cfl 11201    mod cmo 11250   ^cexp 11382   abscabs 12039   sum_csu 12479   Primecprime 13079  mulGrpcmgp 15648   ZRHomczrh 16778  ℤ/nczn 16781    / Lclgs 21078
This theorem is referenced by:  lgsquadlem2  21139
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-ec 6907  df-qs 6911  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480  df-dvds 12853  df-gcd 13007  df-prm 13080  df-phi 13155  df-pc 13211  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-0g 13727  df-gsum 13728  df-imas 13734  df-divs 13735  df-mnd 14690  df-mhm 14738  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-mulg 14815  df-subg 14941  df-nsg 14942  df-eqg 14943  df-ghm 15004  df-cntz 15116  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-cring 15664  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-dvr 15788  df-rnghom 15819  df-drng 15837  df-field 15838  df-subrg 15866  df-lmod 15952  df-lss 16009  df-lsp 16048  df-sra 16244  df-rgmod 16245  df-lidl 16246  df-rsp 16247  df-2idl 16303  df-nzr 16329  df-rlreg 16343  df-domn 16344  df-idom 16345  df-cnfld 16704  df-zrh 16782  df-zn 16785  df-lgs 21079
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