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Theorem lgseisen 22635
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 3337 . . . 4  |-  ( ph  ->  Q  e.  Prime )
3 prmz 13763 . . . 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 22596 . . 3  |-  ( ( Q  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( Q  /L P )  =  ( ( ( ( Q ^ ( ( P  -  1 )  /  2 ) )  +  1 )  mod 
P )  -  1 ) )
74, 5, 6syl2anc 656 . 2  |-  ( ph  ->  ( Q  /L
P )  =  ( ( ( ( Q ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 ) )
8 prmnn 13762 . . . . . . . . 9  |-  ( Q  e.  Prime  ->  Q  e.  NN )
92, 8syl 16 . . . . . . . 8  |-  ( ph  ->  Q  e.  NN )
10 oddprm 13878 . . . . . . . . . 10  |-  ( P  e.  ( Prime  \  {
2 } )  -> 
( ( P  - 
1 )  /  2
)  e.  NN )
115, 10syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( P  - 
1 )  /  2
)  e.  NN )
1211nnnn0d 10632 . . . . . . . 8  |-  ( ph  ->  ( ( P  - 
1 )  /  2
)  e.  NN0 )
139, 12nnexpcld 12025 . . . . . . 7  |-  ( ph  ->  ( Q ^ (
( P  -  1 )  /  2 ) )  e.  NN )
1413nnred 10333 . . . . . 6  |-  ( ph  ->  ( Q ^ (
( P  -  1 )  /  2 ) )  e.  RR )
15 neg1rr 10422 . . . . . . . 8  |-  -u 1  e.  RR
1615a1i 11 . . . . . . 7  |-  ( ph  -> 
-u 1  e.  RR )
17 neg1ne0 10423 . . . . . . . 8  |-  -u 1  =/=  0
1817a1i 11 . . . . . . 7  |-  ( ph  -> 
-u 1  =/=  0
)
19 fzfid 11791 . . . . . . . 8  |-  ( ph  ->  ( 1 ... (
( P  -  1 )  /  2 ) )  e.  Fin )
209nnred 10333 . . . . . . . . . . . 12  |-  ( ph  ->  Q  e.  RR )
215eldifad 3337 . . . . . . . . . . . . 13  |-  ( ph  ->  P  e.  Prime )
22 prmnn 13762 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  P  e.  NN )
2321, 22syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  P  e.  NN )
2420, 23nndivred 10366 . . . . . . . . . . 11  |-  ( ph  ->  ( Q  /  P
)  e.  RR )
2524adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) )  ->  ( Q  /  P )  e.  RR )
26 2re 10387 . . . . . . . . . . 11  |-  2  e.  RR
27 elfznn 11474 . . . . . . . . . . . . 13  |-  ( x  e.  ( 1 ... ( ( P  - 
1 )  /  2
) )  ->  x  e.  NN )
2827adantl 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) )  ->  x  e.  NN )
2928nnred 10333 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) )  ->  x  e.  RR )
30 remulcl 9363 . . . . . . . . . . 11  |-  ( ( 2  e.  RR  /\  x  e.  RR )  ->  ( 2  x.  x
)  e.  RR )
3126, 29, 30sylancr 658 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) )  ->  (
2  x.  x )  e.  RR )
3225, 31remulcld 9410 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) )  ->  (
( Q  /  P
)  x.  ( 2  x.  x ) )  e.  RR )
3332flcld 11644 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) )  ->  ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x
) ) )  e.  ZZ )
3419, 33fsumzcl 13208 . . . . . . 7  |-  ( ph  -> 
sum_ x  e.  (
1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) )  e.  ZZ )
3516, 18, 34reexpclzd 12029 . . . . . 6  |-  ( ph  ->  ( -u 1 ^
sum_ x  e.  (
1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) ) )  e.  RR )
36 1re 9381 . . . . . . 7  |-  1  e.  RR
3736a1i 11 . . . . . 6  |-  ( ph  ->  1  e.  RR )
3823nnrpd 11022 . . . . . 6  |-  ( ph  ->  P  e.  RR+ )
39 lgseisen.3 . . . . . . 7  |-  ( ph  ->  P  =/=  Q )
40 eqid 2441 . . . . . . 7  |-  ( ( Q  x.  ( 2  x.  x ) )  mod  P )  =  ( ( Q  x.  ( 2  x.  x
) )  mod  P
)
41 eqid 2441 . . . . . . 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 ) )
42 eqid 2441 . . . . . . 7  |-  ( ( Q  x.  ( 2  x.  y ) )  mod  P )  =  ( ( Q  x.  ( 2  x.  y
) )  mod  P
)
43 eqid 2441 . . . . . . 7  |-  (ℤ/n `  P
)  =  (ℤ/n `  P
)
44 eqid 2441 . . . . . . 7  |-  (mulGrp `  (ℤ/n `  P ) )  =  (mulGrp `  (ℤ/n `  P ) )
45 eqid 2441 . . . . . . 7  |-  ( ZRHom `  (ℤ/n `  P ) )  =  ( ZRHom `  (ℤ/n `  P
) )
465, 1, 39, 40, 41, 42, 43, 44, 45lgseisenlem4 22634 . . . . . 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 ) )
47 modadd1 11741 . . . . . 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 ) )
4814, 35, 37, 38, 46, 47syl221anc 1224 . . . . 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 ) )
49 peano2re 9538 . . . . . . 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 )
5035, 49syl 16 . . . . . 6  |-  ( ph  ->  ( ( -u 1 ^ sum_ x  e.  ( 1 ... ( ( P  -  1 )  /  2 ) ) ( |_ `  (
( Q  /  P
)  x.  ( 2  x.  x ) ) ) )  +  1 )  e.  RR )
51 df-neg 9594 . . . . . . . 8  |-  -u 1  =  ( 0  -  1 )
52 neg1cn 10421 . . . . . . . . . . . . . 14  |-  -u 1  e.  CC
5352a1i 11 . . . . . . . . . . . . 13  |-  ( ph  -> 
-u 1  e.  CC )
54 absexpz 12790 . . . . . . . . . . . . 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 ) ) ) ) )
5553, 18, 34, 54syl3anc 1213 . . . . . . . . . . . 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 ) ) ) ) )
56 ax-1cn 9336 . . . . . . . . . . . . . . . 16  |-  1  e.  CC
5756absnegi 12883 . . . . . . . . . . . . . . 15  |-  ( abs `  -u 1 )  =  ( abs `  1
)
58 abs1 12782 . . . . . . . . . . . . . . 15  |-  ( abs `  1 )  =  1
5957, 58eqtri 2461 . . . . . . . . . . . . . 14  |-  ( abs `  -u 1 )  =  1
6059oveq1i 6100 . . . . . . . . . . . . 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
) ) ) )
61 1exp 11889 . . . . . . . . . . . . . 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 )
6234, 61syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 1 ^ sum_ x  e.  ( 1 ... ( ( P  - 
1 )  /  2
) ) ( |_
`  ( ( Q  /  P )  x.  ( 2  x.  x
) ) ) )  =  1 )
6360, 62syl5eq 2485 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( abs `  -u 1
) ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) )  =  1 )
6455, 63eqtrd 2473 . . . . . . . . . . 11  |-  ( ph  ->  ( abs `  ( -u 1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) ) )  =  1 )
65 1le1 9960 . . . . . . . . . . 11  |-  1  <_  1
6664, 65syl6eqbr 4326 . . . . . . . . . 10  |-  ( ph  ->  ( abs `  ( -u 1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) ) )  <_  1 )
67 absle 12799 . . . . . . . . . . 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 ) ) )
6835, 36, 67sylancl 657 . . . . . . . . . 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 ) ) )
6966, 68mpbid 210 . . . . . . . . 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
) )
7069simpld 456 . . . . . . . 8  |-  ( ph  -> 
-u 1  <_  ( -u 1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) ) )
7151, 70syl5eqbrr 4323 . . . . . . 7  |-  ( ph  ->  ( 0  -  1 )  <_  ( -u 1 ^ sum_ x  e.  ( 1 ... ( ( P  -  1 )  /  2 ) ) ( |_ `  (
( Q  /  P
)  x.  ( 2  x.  x ) ) ) ) )
72 0red 9383 . . . . . . . 8  |-  ( ph  ->  0  e.  RR )
7372, 37, 35lesubaddd 9932 . . . . . . 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 ) ) )
7471, 73mpbid 210 . . . . . 6  |-  ( ph  ->  0  <_  ( ( -u 1 ^ sum_ x  e.  ( 1 ... (
( P  -  1 )  /  2 ) ) ( |_ `  ( ( Q  /  P )  x.  (
2  x.  x ) ) ) )  +  1 ) )
7523nnred 10333 . . . . . . . . 9  |-  ( ph  ->  P  e.  RR )
76 peano2rem 9671 . . . . . . . . 9  |-  ( P  e.  RR  ->  ( P  -  1 )  e.  RR )
7775, 76syl 16 . . . . . . . 8  |-  ( ph  ->  ( P  -  1 )  e.  RR )
7869simprd 460 . . . . . . . 8  |-  ( ph  ->  ( -u 1 ^
sum_ x  e.  (
1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) ) )  <_  1 )
79 df-2 10376 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
80 eldifsni 3998 . . . . . . . . . . . 12  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  =/=  2 )
815, 80syl 16 . . . . . . . . . . 11  |-  ( ph  ->  P  =/=  2 )
8226a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  2  e.  RR )
83 prmuz2 13777 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
84 eluzle 10869 . . . . . . . . . . . . 13  |-  ( P  e.  ( ZZ>= `  2
)  ->  2  <_  P )
8521, 83, 843syl 20 . . . . . . . . . . . 12  |-  ( ph  ->  2  <_  P )
8682, 75, 85leltned 9521 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  <  P  <->  P  =/=  2 ) )
8781, 86mpbird 232 . . . . . . . . . 10  |-  ( ph  ->  2  <  P )
8879, 87syl5eqbrr 4323 . . . . . . . . 9  |-  ( ph  ->  ( 1  +  1 )  <  P )
8937, 37, 75ltaddsubd 9935 . . . . . . . . 9  |-  ( ph  ->  ( ( 1  +  1 )  <  P  <->  1  <  ( P  - 
1 ) ) )
9088, 89mpbid 210 . . . . . . . 8  |-  ( ph  ->  1  <  ( P  -  1 ) )
9135, 37, 77, 78, 90lelttrd 9525 . . . . . . 7  |-  ( ph  ->  ( -u 1 ^
sum_ x  e.  (
1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) ) )  <  ( P  -  1 ) )
9235, 37, 75ltaddsubd 9935 . . . . . . 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 ) ) )
9391, 92mpbird 232 . . . . . 6  |-  ( ph  ->  ( ( -u 1 ^ sum_ x  e.  ( 1 ... ( ( P  -  1 )  /  2 ) ) ( |_ `  (
( Q  /  P
)  x.  ( 2  x.  x ) ) ) )  +  1 )  <  P )
94 modid 11728 . . . . . 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 ) )
9550, 38, 74, 93, 94syl22anc 1214 . . . . 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 ) )
9648, 95eqtrd 2473 . . . 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 ) )
9796oveq1d 6105 . . 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 ) )
9835recnd 9408 . . . 4  |-  ( ph  ->  ( -u 1 ^
sum_ x  e.  (
1 ... ( ( P  -  1 )  / 
2 ) ) ( |_ `  ( ( Q  /  P )  x.  ( 2  x.  x ) ) ) )  e.  CC )
99 pncan 9612 . . . 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 ) ) ) ) )
10098, 56, 99sylancl 657 . . 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 ) ) ) ) )
10197, 100eqtrd 2473 . 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 ) ) ) ) )
1027, 101eqtrd 2473 1  |-  ( ph  ->  ( Q  /L
P )  =  (
-u 1 ^ sum_ x  e.  ( 1 ... ( ( P  - 
1 )  /  2
) ) ( |_
`  ( ( Q  /  P )  x.  ( 2  x.  x
) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604    \ cdif 3322   {csn 3874   class class class wbr 4289    e. cmpt 4347   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    + caddc 9281    x. cmul 9283    < clt 9414    <_ cle 9415    - cmin 9591   -ucneg 9592    / cdiv 9989   NNcn 10318   2c2 10367   ZZcz 10642   ZZ>=cuz 10857   RR+crp 10987   ...cfz 11433   |_cfl 11636    mod cmo 11704   ^cexp 11861   abscabs 12719   sum_csu 13159   Primecprime 13759  mulGrpcmgp 16581   ZRHomczrh 17831  ℤ/nczn 17834    /Lclgs 22576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-ec 7099  df-qs 7103  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-sum 13160  df-dvds 13532  df-gcd 13687  df-prm 13760  df-phi 13837  df-pc 13900  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-0g 14376  df-gsum 14377  df-imas 14442  df-divs 14443  df-mnd 15411  df-mhm 15460  df-submnd 15461  df-grp 15538  df-minusg 15539  df-sbg 15540  df-mulg 15541  df-subg 15671  df-nsg 15672  df-eqg 15673  df-ghm 15738  df-cntz 15828  df-cmn 16272  df-abl 16273  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-invr 16754  df-dvr 16765  df-rnghom 16796  df-drng 16814  df-field 16815  df-subrg 16843  df-lmod 16930  df-lss 16992  df-lsp 17031  df-sra 17231  df-rgmod 17232  df-lidl 17233  df-rsp 17234  df-2idl 17292  df-nzr 17318  df-rlreg 17332  df-domn 17333  df-idom 17334  df-cnfld 17719  df-zring 17784  df-zrh 17835  df-zn 17838  df-lgs 22577
This theorem is referenced by:  lgsquadlem2  22637
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