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Theorem wilthlem3 23545
Description: Lemma for wilth 23546. Here we round out the argument of wilthlem2 23544 with the final step of the induction. The induction argument shows that every subset of  1 ... ( P  -  1 ) that is closed under inverse and contains  P  -  1 multiplies to  -u 1  mod  P, and clearly  1 ... ( P  -  1 ) itself is such a set. Thus, the product of all the elements is  -u 1, and all that is left is to translate the group sum notation (which we used for its unordered summing capabilities) into an ordered sequence to match the definition of the factorial. (Contributed by Mario Carneiro, 24-Jan-2015.) (Proof shortened by AV, 27-Jul-2019.)
Hypotheses
Ref Expression
wilthlem.t  |-  T  =  (mulGrp ` fld )
wilthlem.a  |-  A  =  { x  e.  ~P ( 1 ... ( P  -  1 ) )  |  ( ( P  -  1 )  e.  x  /\  A. y  e.  x  (
( y ^ ( P  -  2 ) )  mod  P )  e.  x ) }
Assertion
Ref Expression
wilthlem3  |-  ( P  e.  Prime  ->  P  ||  ( ( ! `  ( P  -  1
) )  +  1 ) )
Distinct variable groups:    x, y, A    x, P, y    x, T, y

Proof of Theorem wilthlem3
Dummy variables  t 
s  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prmuz2 14322 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
2 uz2m1nn 11157 . . . . . . . 8  |-  ( P  e.  ( ZZ>= `  2
)  ->  ( P  -  1 )  e.  NN )
31, 2syl 16 . . . . . . 7  |-  ( P  e.  Prime  ->  ( P  -  1 )  e.  NN )
4 nnuz 11117 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
53, 4syl6eleq 2552 . . . . . 6  |-  ( P  e.  Prime  ->  ( P  -  1 )  e.  ( ZZ>= `  1 )
)
6 eluzfz2 11697 . . . . . 6  |-  ( ( P  -  1 )  e.  ( ZZ>= `  1
)  ->  ( P  -  1 )  e.  ( 1 ... ( P  -  1 ) ) )
75, 6syl 16 . . . . 5  |-  ( P  e.  Prime  ->  ( P  -  1 )  e.  ( 1 ... ( P  -  1 ) ) )
8 simpl 455 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  y  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  Prime )
9 elfzelz 11691 . . . . . . . . 9  |-  ( y  e.  ( 1 ... ( P  -  1 ) )  ->  y  e.  ZZ )
109adantl 464 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  y  e.  ( 1 ... ( P  -  1 ) ) )  ->  y  e.  ZZ )
11 prmnn 14307 . . . . . . . . 9  |-  ( P  e.  Prime  ->  P  e.  NN )
12 fzm1ndvds 14125 . . . . . . . . 9  |-  ( ( P  e.  NN  /\  y  e.  ( 1 ... ( P  - 
1 ) ) )  ->  -.  P  ||  y
)
1311, 12sylan 469 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  y  e.  ( 1 ... ( P  -  1 ) ) )  ->  -.  P  ||  y )
14 eqid 2454 . . . . . . . . 9  |-  ( ( y ^ ( P  -  2 ) )  mod  P )  =  ( ( y ^
( P  -  2 ) )  mod  P
)
1514prmdiv 14402 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  y  e.  ZZ  /\  -.  P  ||  y )  ->  (
( ( y ^
( P  -  2 ) )  mod  P
)  e.  ( 1 ... ( P  - 
1 ) )  /\  P  ||  ( ( y  x.  ( ( y ^ ( P  - 
2 ) )  mod 
P ) )  - 
1 ) ) )
168, 10, 13, 15syl3anc 1226 . . . . . . 7  |-  ( ( P  e.  Prime  /\  y  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( ( y ^
( P  -  2 ) )  mod  P
)  e.  ( 1 ... ( P  - 
1 ) )  /\  P  ||  ( ( y  x.  ( ( y ^ ( P  - 
2 ) )  mod 
P ) )  - 
1 ) ) )
1716simpld 457 . . . . . 6  |-  ( ( P  e.  Prime  /\  y  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( y ^ ( P  -  2 ) )  mod  P )  e.  ( 1 ... ( P  -  1 ) ) )
1817ralrimiva 2868 . . . . 5  |-  ( P  e.  Prime  ->  A. y  e.  ( 1 ... ( P  -  1 ) ) ( ( y ^ ( P  - 
2 ) )  mod 
P )  e.  ( 1 ... ( P  -  1 ) ) )
19 ovex 6298 . . . . . . 7  |-  ( 1 ... ( P  - 
1 ) )  e. 
_V
2019pwid 4013 . . . . . 6  |-  ( 1 ... ( P  - 
1 ) )  e. 
~P ( 1 ... ( P  -  1 ) )
21 eleq2 2527 . . . . . . . 8  |-  ( x  =  ( 1 ... ( P  -  1 ) )  ->  (
( P  -  1 )  e.  x  <->  ( P  -  1 )  e.  ( 1 ... ( P  -  1 ) ) ) )
22 eleq2 2527 . . . . . . . . 9  |-  ( x  =  ( 1 ... ( P  -  1 ) )  ->  (
( ( y ^
( P  -  2 ) )  mod  P
)  e.  x  <->  ( (
y ^ ( P  -  2 ) )  mod  P )  e.  ( 1 ... ( P  -  1 ) ) ) )
2322raleqbi1dv 3059 . . . . . . . 8  |-  ( x  =  ( 1 ... ( P  -  1 ) )  ->  ( A. y  e.  x  ( ( y ^
( P  -  2 ) )  mod  P
)  e.  x  <->  A. y  e.  ( 1 ... ( P  -  1 ) ) ( ( y ^ ( P  - 
2 ) )  mod 
P )  e.  ( 1 ... ( P  -  1 ) ) ) )
2421, 23anbi12d 708 . . . . . . 7  |-  ( x  =  ( 1 ... ( P  -  1 ) )  ->  (
( ( P  - 
1 )  e.  x  /\  A. y  e.  x  ( ( y ^
( P  -  2 ) )  mod  P
)  e.  x )  <-> 
( ( P  - 
1 )  e.  ( 1 ... ( P  -  1 ) )  /\  A. y  e.  ( 1 ... ( P  -  1 ) ) ( ( y ^ ( P  - 
2 ) )  mod 
P )  e.  ( 1 ... ( P  -  1 ) ) ) ) )
25 wilthlem.a . . . . . . 7  |-  A  =  { x  e.  ~P ( 1 ... ( P  -  1 ) )  |  ( ( P  -  1 )  e.  x  /\  A. y  e.  x  (
( y ^ ( P  -  2 ) )  mod  P )  e.  x ) }
2624, 25elrab2 3256 . . . . . 6  |-  ( ( 1 ... ( P  -  1 ) )  e.  A  <->  ( (
1 ... ( P  - 
1 ) )  e. 
~P ( 1 ... ( P  -  1 ) )  /\  (
( P  -  1 )  e.  ( 1 ... ( P  - 
1 ) )  /\  A. y  e.  ( 1 ... ( P  - 
1 ) ) ( ( y ^ ( P  -  2 ) )  mod  P )  e.  ( 1 ... ( P  -  1 ) ) ) ) )
2720, 26mpbiran 916 . . . . 5  |-  ( ( 1 ... ( P  -  1 ) )  e.  A  <->  ( ( P  -  1 )  e.  ( 1 ... ( P  -  1 ) )  /\  A. y  e.  ( 1 ... ( P  - 
1 ) ) ( ( y ^ ( P  -  2 ) )  mod  P )  e.  ( 1 ... ( P  -  1 ) ) ) )
287, 18, 27sylanbrc 662 . . . 4  |-  ( P  e.  Prime  ->  ( 1 ... ( P  - 
1 ) )  e.  A )
29 fzfi 12067 . . . . 5  |-  ( 1 ... ( P  - 
1 ) )  e. 
Fin
30 eleq1 2526 . . . . . . . 8  |-  ( s  =  t  ->  (
s  e.  A  <->  t  e.  A ) )
31 reseq2 5257 . . . . . . . . . . 11  |-  ( s  =  t  ->  (  _I  |`  s )  =  (  _I  |`  t
) )
3231oveq2d 6286 . . . . . . . . . 10  |-  ( s  =  t  ->  ( T  gsumg  (  _I  |`  s
) )  =  ( T  gsumg  (  _I  |`  t
) ) )
3332oveq1d 6285 . . . . . . . . 9  |-  ( s  =  t  ->  (
( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( ( T  gsumg  (  _I  |`  t
) )  mod  P
) )
3433eqeq1d 2456 . . . . . . . 8  |-  ( s  =  t  ->  (
( ( T  gsumg  (  _I  |`  s ) )  mod 
P )  =  (
-u 1  mod  P
)  <->  ( ( T 
gsumg  (  _I  |`  t ) )  mod  P )  =  ( -u 1  mod  P ) ) )
3530, 34imbi12d 318 . . . . . . 7  |-  ( s  =  t  ->  (
( s  e.  A  ->  ( ( T  gsumg  (  _I  |`  s ) )  mod 
P )  =  (
-u 1  mod  P
) )  <->  ( t  e.  A  ->  ( ( T  gsumg  (  _I  |`  t
) )  mod  P
)  =  ( -u
1  mod  P )
) ) )
3635imbi2d 314 . . . . . 6  |-  ( s  =  t  ->  (
( P  e.  Prime  -> 
( s  e.  A  ->  ( ( T  gsumg  (  _I  |`  s ) )  mod 
P )  =  (
-u 1  mod  P
) ) )  <->  ( P  e.  Prime  ->  ( t  e.  A  ->  ( ( T  gsumg  (  _I  |`  t
) )  mod  P
)  =  ( -u
1  mod  P )
) ) ) )
37 eleq1 2526 . . . . . . . 8  |-  ( s  =  ( 1 ... ( P  -  1 ) )  ->  (
s  e.  A  <->  ( 1 ... ( P  - 
1 ) )  e.  A ) )
38 reseq2 5257 . . . . . . . . . . 11  |-  ( s  =  ( 1 ... ( P  -  1 ) )  ->  (  _I  |`  s )  =  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )
3938oveq2d 6286 . . . . . . . . . 10  |-  ( s  =  ( 1 ... ( P  -  1 ) )  ->  ( T  gsumg  (  _I  |`  s
) )  =  ( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) )
4039oveq1d 6285 . . . . . . . . 9  |-  ( s  =  ( 1 ... ( P  -  1 ) )  ->  (
( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( ( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  mod  P ) )
4140eqeq1d 2456 . . . . . . . 8  |-  ( s  =  ( 1 ... ( P  -  1 ) )  ->  (
( ( T  gsumg  (  _I  |`  s ) )  mod 
P )  =  (
-u 1  mod  P
)  <->  ( ( T 
gsumg  (  _I  |`  ( 1 ... ( P  - 
1 ) ) ) )  mod  P )  =  ( -u 1  mod  P ) ) )
4237, 41imbi12d 318 . . . . . . 7  |-  ( s  =  ( 1 ... ( P  -  1 ) )  ->  (
( s  e.  A  ->  ( ( T  gsumg  (  _I  |`  s ) )  mod 
P )  =  (
-u 1  mod  P
) )  <->  ( (
1 ... ( P  - 
1 ) )  e.  A  ->  ( ( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  mod  P )  =  ( -u 1  mod  P ) ) ) )
4342imbi2d 314 . . . . . 6  |-  ( s  =  ( 1 ... ( P  -  1 ) )  ->  (
( P  e.  Prime  -> 
( s  e.  A  ->  ( ( T  gsumg  (  _I  |`  s ) )  mod 
P )  =  (
-u 1  mod  P
) ) )  <->  ( P  e.  Prime  ->  ( (
1 ... ( P  - 
1 ) )  e.  A  ->  ( ( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  mod  P )  =  ( -u 1  mod  P ) ) ) ) )
44 bi2.04 359 . . . . . . . . . . . 12  |-  ( ( s  C.  t  ->  ( P  e.  Prime  ->  ( s  e.  A  -> 
( ( T  gsumg  (  _I  |`  s ) )  mod 
P )  =  (
-u 1  mod  P
) ) ) )  <-> 
( P  e.  Prime  -> 
( s  C.  t  ->  ( s  e.  A  ->  ( ( T  gsumg  (  _I  |`  s ) )  mod 
P )  =  (
-u 1  mod  P
) ) ) ) )
45 pm2.27 39 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  ( ( P  e.  Prime  ->  ( s  C.  t  ->  ( s  e.  A  -> 
( ( T  gsumg  (  _I  |`  s ) )  mod 
P )  =  (
-u 1  mod  P
) ) ) )  ->  ( s  C.  t  ->  ( s  e.  A  ->  ( ( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( -u
1  mod  P )
) ) ) )
4645com34 83 . . . . . . . . . . . 12  |-  ( P  e.  Prime  ->  ( ( P  e.  Prime  ->  ( s  C.  t  ->  ( s  e.  A  -> 
( ( T  gsumg  (  _I  |`  s ) )  mod 
P )  =  (
-u 1  mod  P
) ) ) )  ->  ( s  e.  A  ->  ( s  C.  t  ->  ( ( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( -u
1  mod  P )
) ) ) )
4744, 46syl5bi 217 . . . . . . . . . . 11  |-  ( P  e.  Prime  ->  ( ( s  C.  t  ->  ( P  e.  Prime  ->  ( s  e.  A  -> 
( ( T  gsumg  (  _I  |`  s ) )  mod 
P )  =  (
-u 1  mod  P
) ) ) )  ->  ( s  e.  A  ->  ( s  C.  t  ->  ( ( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( -u
1  mod  P )
) ) ) )
4847alimdv 1714 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  ( A. s ( s  C.  t  ->  ( P  e. 
Prime  ->  ( s  e.  A  ->  ( ( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( -u
1  mod  P )
) ) )  ->  A. s ( s  e.  A  ->  ( s  C.  t  ->  ( ( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( -u
1  mod  P )
) ) ) )
49 df-ral 2809 . . . . . . . . . 10  |-  ( A. s  e.  A  (
s  C.  t  ->  ( ( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( -u
1  mod  P )
)  <->  A. s ( s  e.  A  ->  (
s  C.  t  ->  ( ( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( -u
1  mod  P )
) ) )
5048, 49syl6ibr 227 . . . . . . . . 9  |-  ( P  e.  Prime  ->  ( A. s ( s  C.  t  ->  ( P  e. 
Prime  ->  ( s  e.  A  ->  ( ( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( -u
1  mod  P )
) ) )  ->  A. s  e.  A  ( s  C.  t  ->  ( ( T  gsumg  (  _I  |`  s ) )  mod 
P )  =  (
-u 1  mod  P
) ) ) )
5150com12 31 . . . . . . . 8  |-  ( A. s ( s  C.  t  ->  ( P  e. 
Prime  ->  ( s  e.  A  ->  ( ( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( -u
1  mod  P )
) ) )  -> 
( P  e.  Prime  ->  A. s  e.  A  ( s  C.  t  ->  ( ( T  gsumg  (  _I  |`  s ) )  mod 
P )  =  (
-u 1  mod  P
) ) ) )
52 wilthlem.t . . . . . . . . . 10  |-  T  =  (mulGrp ` fld )
53 simp1 994 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A. s  e.  A  (
s  C.  t  ->  ( ( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( -u
1  mod  P )
)  /\  t  e.  A )  ->  P  e.  Prime )
54 simp3 996 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A. s  e.  A  (
s  C.  t  ->  ( ( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( -u
1  mod  P )
)  /\  t  e.  A )  ->  t  e.  A )
55 simp2 995 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A. s  e.  A  (
s  C.  t  ->  ( ( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( -u
1  mod  P )
)  /\  t  e.  A )  ->  A. s  e.  A  ( s  C.  t  ->  ( ( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( -u
1  mod  P )
) )
5652, 25, 53, 54, 55wilthlem2 23544 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A. s  e.  A  (
s  C.  t  ->  ( ( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( -u
1  mod  P )
)  /\  t  e.  A )  ->  (
( T  gsumg  (  _I  |`  t
) )  mod  P
)  =  ( -u
1  mod  P )
)
57563exp 1193 . . . . . . . 8  |-  ( P  e.  Prime  ->  ( A. s  e.  A  (
s  C.  t  ->  ( ( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( -u
1  mod  P )
)  ->  ( t  e.  A  ->  ( ( T  gsumg  (  _I  |`  t
) )  mod  P
)  =  ( -u
1  mod  P )
) ) )
5851, 57sylcom 29 . . . . . . 7  |-  ( A. s ( s  C.  t  ->  ( P  e. 
Prime  ->  ( s  e.  A  ->  ( ( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( -u
1  mod  P )
) ) )  -> 
( P  e.  Prime  -> 
( t  e.  A  ->  ( ( T  gsumg  (  _I  |`  t ) )  mod 
P )  =  (
-u 1  mod  P
) ) ) )
5958a1i 11 . . . . . 6  |-  ( t  e.  Fin  ->  ( A. s ( s  C.  t  ->  ( P  e. 
Prime  ->  ( s  e.  A  ->  ( ( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( -u
1  mod  P )
) ) )  -> 
( P  e.  Prime  -> 
( t  e.  A  ->  ( ( T  gsumg  (  _I  |`  t ) )  mod 
P )  =  (
-u 1  mod  P
) ) ) ) )
6036, 43, 59findcard3 7755 . . . . 5  |-  ( ( 1 ... ( P  -  1 ) )  e.  Fin  ->  ( P  e.  Prime  ->  (
( 1 ... ( P  -  1 ) )  e.  A  -> 
( ( T  gsumg  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) )  mod 
P )  =  (
-u 1  mod  P
) ) ) )
6129, 60ax-mp 5 . . . 4  |-  ( P  e.  Prime  ->  ( ( 1 ... ( P  -  1 ) )  e.  A  ->  (
( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  mod  P )  =  ( -u 1  mod  P ) ) )
6228, 61mpd 15 . . 3  |-  ( P  e.  Prime  ->  ( ( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  mod  P )  =  ( -u 1  mod  P ) )
63 cnfld1 18641 . . . . . 6  |-  1  =  ( 1r ` fld )
6452, 63ringidval 17353 . . . . 5  |-  1  =  ( 0g `  T )
65 cncrng 18637 . . . . . 6  |-fld  e.  CRing
6652crngmgp 17404 . . . . . 6  |-  (fld  e.  CRing  ->  T  e. CMnd )
6765, 66mp1i 12 . . . . 5  |-  ( P  e.  Prime  ->  T  e. CMnd
)
6829a1i 11 . . . . 5  |-  ( P  e.  Prime  ->  ( 1 ... ( P  - 
1 ) )  e. 
Fin )
69 zsubrg 18669 . . . . . 6  |-  ZZ  e.  (SubRing ` fld )
7052subrgsubm 17640 . . . . . 6  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubMnd `  T ) )
7169, 70mp1i 12 . . . . 5  |-  ( P  e.  Prime  ->  ZZ  e.  (SubMnd `  T ) )
72 f1oi 5833 . . . . . . . 8  |-  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) : ( 1 ... ( P  -  1 ) ) -1-1-onto-> ( 1 ... ( P  -  1 ) )
73 f1of 5798 . . . . . . . 8  |-  ( (  _I  |`  ( 1 ... ( P  - 
1 ) ) ) : ( 1 ... ( P  -  1 ) ) -1-1-onto-> ( 1 ... ( P  -  1 ) )  ->  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) : ( 1 ... ( P  -  1 ) ) --> ( 1 ... ( P  -  1 ) ) )
7472, 73ax-mp 5 . . . . . . 7  |-  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) : ( 1 ... ( P  -  1 ) ) --> ( 1 ... ( P  -  1 ) )
759ssriv 3493 . . . . . . 7  |-  ( 1 ... ( P  - 
1 ) )  C_  ZZ
76 fss 5721 . . . . . . 7  |-  ( ( (  _I  |`  (
1 ... ( P  - 
1 ) ) ) : ( 1 ... ( P  -  1 ) ) --> ( 1 ... ( P  - 
1 ) )  /\  ( 1 ... ( P  -  1 ) )  C_  ZZ )  ->  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) : ( 1 ... ( P  -  1 ) ) --> ZZ )
7774, 75, 76mp2an 670 . . . . . 6  |-  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) : ( 1 ... ( P  -  1 ) ) --> ZZ
7877a1i 11 . . . . 5  |-  ( P  e.  Prime  ->  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) : ( 1 ... ( P  -  1 ) ) --> ZZ )
79 1ex 9580 . . . . . . 7  |-  1  e.  _V
8079a1i 11 . . . . . 6  |-  ( P  e.  Prime  ->  1  e. 
_V )
8178, 68, 80fdmfifsupp 7831 . . . . 5  |-  ( P  e.  Prime  ->  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) finSupp  1 )
8264, 67, 68, 71, 78, 81gsumsubmcl 17132 . . . 4  |-  ( P  e.  Prime  ->  ( T 
gsumg  (  _I  |`  ( 1 ... ( P  - 
1 ) ) ) )  e.  ZZ )
83 1z 10890 . . . . 5  |-  1  e.  ZZ
84 znegcl 10895 . . . . 5  |-  ( 1  e.  ZZ  ->  -u 1  e.  ZZ )
8583, 84mp1i 12 . . . 4  |-  ( P  e.  Prime  ->  -u 1  e.  ZZ )
86 moddvds 14080 . . . 4  |-  ( ( P  e.  NN  /\  ( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  e.  ZZ  /\  -u 1  e.  ZZ )  ->  ( ( ( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  mod  P )  =  ( -u 1  mod  P )  <->  P  ||  (
( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  -  -u 1
) ) )
8711, 82, 85, 86syl3anc 1226 . . 3  |-  ( P  e.  Prime  ->  ( ( ( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  mod  P )  =  ( -u 1  mod  P )  <->  P  ||  (
( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  -  -u 1
) ) )
8862, 87mpbid 210 . 2  |-  ( P  e.  Prime  ->  P  ||  ( ( T  gsumg  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) )  -  -u 1 ) )
89 fcoi1 5741 . . . . . . . . . 10  |-  ( (  _I  |`  ( 1 ... ( P  - 
1 ) ) ) : ( 1 ... ( P  -  1 ) ) --> ( 1 ... ( P  - 
1 ) )  -> 
( (  _I  |`  (
1 ... ( P  - 
1 ) ) )  o.  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  =  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) )
9074, 89ax-mp 5 . . . . . . . . 9  |-  ( (  _I  |`  ( 1 ... ( P  - 
1 ) ) )  o.  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  =  (  _I  |`  ( 1 ... ( P  -  1 ) ) )
9190fveq1i 5849 . . . . . . . 8  |-  ( ( (  _I  |`  (
1 ... ( P  - 
1 ) ) )  o.  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) `  k )  =  ( (  _I  |`  ( 1 ... ( P  -  1 ) ) ) `  k
)
92 fvres 5862 . . . . . . . 8  |-  ( k  e.  ( 1 ... ( P  -  1 ) )  ->  (
(  _I  |`  (
1 ... ( P  - 
1 ) ) ) `
 k )  =  (  _I  `  k
) )
9391, 92syl5eq 2507 . . . . . . 7  |-  ( k  e.  ( 1 ... ( P  -  1 ) )  ->  (
( (  _I  |`  (
1 ... ( P  - 
1 ) ) )  o.  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) `  k )  =  (  _I  `  k ) )
9493adantl 464 . . . . . 6  |-  ( ( P  e.  Prime  /\  k  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( (  _I  |`  (
1 ... ( P  - 
1 ) ) )  o.  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) `  k )  =  (  _I  `  k ) )
955, 94seqfveq 12116 . . . . 5  |-  ( P  e.  Prime  ->  (  seq 1 (  x.  , 
( (  _I  |`  (
1 ... ( P  - 
1 ) ) )  o.  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) ) `  ( P  -  1 ) )  =  (  seq 1 (  x.  ,  _I  ) `  ( P  -  1 ) ) )
96 cnfldbas 18622 . . . . . . 7  |-  CC  =  ( Base ` fld )
9752, 96mgpbas 17345 . . . . . 6  |-  CC  =  ( Base `  T )
98 cnfldmul 18624 . . . . . . 7  |-  x.  =  ( .r ` fld )
9952, 98mgpplusg 17343 . . . . . 6  |-  x.  =  ( +g  `  T )
100 eqid 2454 . . . . . 6  |-  (Cntz `  T )  =  (Cntz `  T )
101 cnring 18638 . . . . . . 7  |-fld  e.  Ring
10252ringmgp 17402 . . . . . . 7  |-  (fld  e.  Ring  ->  T  e.  Mnd )
103101, 102mp1i 12 . . . . . 6  |-  ( P  e.  Prime  ->  T  e. 
Mnd )
104 zsscn 10868 . . . . . . . 8  |-  ZZ  C_  CC
105 fss 5721 . . . . . . . 8  |-  ( ( (  _I  |`  (
1 ... ( P  - 
1 ) ) ) : ( 1 ... ( P  -  1 ) ) --> ZZ  /\  ZZ  C_  CC )  -> 
(  _I  |`  (
1 ... ( P  - 
1 ) ) ) : ( 1 ... ( P  -  1 ) ) --> CC )
10677, 104, 105mp2an 670 . . . . . . 7  |-  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) : ( 1 ... ( P  -  1 ) ) --> CC
107106a1i 11 . . . . . 6  |-  ( P  e.  Prime  ->  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) : ( 1 ... ( P  -  1 ) ) --> CC )
10897, 100, 67, 107cntzcmnf 17053 . . . . . 6  |-  ( P  e.  Prime  ->  ran  (  _I  |`  ( 1 ... ( P  -  1 ) ) )  C_  ( (Cntz `  T ) `  ran  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) )
109 f1of1 5797 . . . . . . 7  |-  ( (  _I  |`  ( 1 ... ( P  - 
1 ) ) ) : ( 1 ... ( P  -  1 ) ) -1-1-onto-> ( 1 ... ( P  -  1 ) )  ->  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) : ( 1 ... ( P  -  1 ) )
-1-1-> ( 1 ... ( P  -  1 ) ) )
11072, 109mp1i 12 . . . . . 6  |-  ( P  e.  Prime  ->  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) : ( 1 ... ( P  -  1 ) )
-1-1-> ( 1 ... ( P  -  1 ) ) )
111 suppssdm 6904 . . . . . . . . 9  |-  ( (  _I  |`  ( 1 ... ( P  - 
1 ) ) ) supp  1 )  C_  dom  (  _I  |`  ( 1 ... ( P  - 
1 ) ) )
112 dmresi 5317 . . . . . . . . 9  |-  dom  (  _I  |`  ( 1 ... ( P  -  1 ) ) )  =  ( 1 ... ( P  -  1 ) )
113111, 112sseqtri 3521 . . . . . . . 8  |-  ( (  _I  |`  ( 1 ... ( P  - 
1 ) ) ) supp  1 )  C_  (
1 ... ( P  - 
1 ) )
114 rnresi 5338 . . . . . . . 8  |-  ran  (  _I  |`  ( 1 ... ( P  -  1 ) ) )  =  ( 1 ... ( P  -  1 ) )
115113, 114sseqtr4i 3522 . . . . . . 7  |-  ( (  _I  |`  ( 1 ... ( P  - 
1 ) ) ) supp  1 )  C_  ran  (  _I  |`  ( 1 ... ( P  - 
1 ) ) )
116115a1i 11 . . . . . 6  |-  ( P  e.  Prime  ->  ( (  _I  |`  ( 1 ... ( P  - 
1 ) ) ) supp  1 )  C_  ran  (  _I  |`  ( 1 ... ( P  - 
1 ) ) ) )
117 eqid 2454 . . . . . 6  |-  ( ( (  _I  |`  (
1 ... ( P  - 
1 ) ) )  o.  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) supp  1 )  =  ( ( (  _I  |`  ( 1 ... ( P  -  1 ) ) )  o.  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) ) supp  1 )
11897, 64, 99, 100, 103, 68, 107, 108, 3, 110, 116, 117gsumval3 17113 . . . . 5  |-  ( P  e.  Prime  ->  ( T 
gsumg  (  _I  |`  ( 1 ... ( P  - 
1 ) ) ) )  =  (  seq 1 (  x.  , 
( (  _I  |`  (
1 ... ( P  - 
1 ) ) )  o.  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) ) `  ( P  -  1 ) ) )
119 facnn 12340 . . . . . 6  |-  ( ( P  -  1 )  e.  NN  ->  ( ! `  ( P  -  1 ) )  =  (  seq 1
(  x.  ,  _I  ) `  ( P  -  1 ) ) )
1203, 119syl 16 . . . . 5  |-  ( P  e.  Prime  ->  ( ! `
 ( P  - 
1 ) )  =  (  seq 1 (  x.  ,  _I  ) `  ( P  -  1 ) ) )
12195, 118, 1203eqtr4d 2505 . . . 4  |-  ( P  e.  Prime  ->  ( T 
gsumg  (  _I  |`  ( 1 ... ( P  - 
1 ) ) ) )  =  ( ! `
 ( P  - 
1 ) ) )
122121oveq1d 6285 . . 3  |-  ( P  e.  Prime  ->  ( ( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  -  -u 1
)  =  ( ( ! `  ( P  -  1 ) )  -  -u 1 ) )
123 nnm1nn0 10833 . . . . . . 7  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
12411, 123syl 16 . . . . . 6  |-  ( P  e.  Prime  ->  ( P  -  1 )  e. 
NN0 )
125 faccl 12348 . . . . . 6  |-  ( ( P  -  1 )  e.  NN0  ->  ( ! `
 ( P  - 
1 ) )  e.  NN )
126124, 125syl 16 . . . . 5  |-  ( P  e.  Prime  ->  ( ! `
 ( P  - 
1 ) )  e.  NN )
127126nncnd 10547 . . . 4  |-  ( P  e.  Prime  ->  ( ! `
 ( P  - 
1 ) )  e.  CC )
128 ax-1cn 9539 . . . 4  |-  1  e.  CC
129 subneg 9859 . . . 4  |-  ( ( ( ! `  ( P  -  1 ) )  e.  CC  /\  1  e.  CC )  ->  ( ( ! `  ( P  -  1
) )  -  -u 1
)  =  ( ( ! `  ( P  -  1 ) )  +  1 ) )
130127, 128, 129sylancl 660 . . 3  |-  ( P  e.  Prime  ->  ( ( ! `  ( P  -  1 ) )  -  -u 1 )  =  ( ( ! `  ( P  -  1
) )  +  1 ) )
131122, 130eqtrd 2495 . 2  |-  ( P  e.  Prime  ->  ( ( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  -  -u 1
)  =  ( ( ! `  ( P  -  1 ) )  +  1 ) )
13288, 131breqtrd 4463 1  |-  ( P  e.  Prime  ->  P  ||  ( ( ! `  ( P  -  1
) )  +  1 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971   A.wal 1396    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808   _Vcvv 3106    C_ wss 3461    C. wpss 3462   ~Pcpw 3999   class class class wbr 4439    _I cid 4779   dom cdm 4988   ran crn 4989    |` cres 4990    o. ccom 4992   -->wf 5566   -1-1->wf1 5567   -1-1-onto->wf1o 5569   ` cfv 5570  (class class class)co 6270   supp csupp 6891   Fincfn 7509   CCcc 9479   1c1 9482    + caddc 9484    x. cmul 9486    - cmin 9796   -ucneg 9797   NNcn 10531   2c2 10581   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11082   ...cfz 11675    mod cmo 11978    seqcseq 12092   ^cexp 12151   !cfa 12338    || cdvds 14073   Primecprime 14304    gsumg cgsu 14933   Mndcmnd 16121  SubMndcsubmnd 16167  Cntzccntz 16555  CMndccmn 17000  mulGrpcmgp 17339   Ringcrg 17396   CRingccrg 17397  SubRingcsubrg 17623  ℂfldccnfld 18618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-rp 11222  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12093  df-exp 12152  df-fac 12339  df-hash 12391  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-dvds 14074  df-gcd 14232  df-prm 14305  df-phi 14383  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-starv 14802  df-tset 14806  df-ple 14807  df-ds 14809  df-unif 14810  df-0g 14934  df-gsum 14935  df-mre 15078  df-mrc 15079  df-acs 15081  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-submnd 16169  df-grp 16259  df-minusg 16260  df-mulg 16262  df-subg 16400  df-cntz 16557  df-cmn 17002  df-mgp 17340  df-ur 17352  df-ring 17398  df-cring 17399  df-subrg 17625  df-cnfld 18619
This theorem is referenced by:  wilth  23546
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