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Theorem wilthlem3 24074
Description: Lemma for wilth 24075. Here we round out the argument of wilthlem2 24073 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 14721 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
2 uz2m1nn 11256 . . . . . . . 8  |-  ( P  e.  ( ZZ>= `  2
)  ->  ( P  -  1 )  e.  NN )
31, 2syl 17 . . . . . . 7  |-  ( P  e.  Prime  ->  ( P  -  1 )  e.  NN )
4 nnuz 11218 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
53, 4syl6eleq 2559 . . . . . 6  |-  ( P  e.  Prime  ->  ( P  -  1 )  e.  ( ZZ>= `  1 )
)
6 eluzfz2 11833 . . . . . 6  |-  ( ( P  -  1 )  e.  ( ZZ>= `  1
)  ->  ( P  -  1 )  e.  ( 1 ... ( P  -  1 ) ) )
75, 6syl 17 . . . . 5  |-  ( P  e.  Prime  ->  ( P  -  1 )  e.  ( 1 ... ( P  -  1 ) ) )
8 simpl 464 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  y  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  Prime )
9 elfzelz 11826 . . . . . . . . 9  |-  ( y  e.  ( 1 ... ( P  -  1 ) )  ->  y  e.  ZZ )
109adantl 473 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  y  e.  ( 1 ... ( P  -  1 ) ) )  ->  y  e.  ZZ )
11 prmnn 14704 . . . . . . . . 9  |-  ( P  e.  Prime  ->  P  e.  NN )
12 fzm1ndvds 14434 . . . . . . . . 9  |-  ( ( P  e.  NN  /\  y  e.  ( 1 ... ( P  - 
1 ) ) )  ->  -.  P  ||  y
)
1311, 12sylan 479 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  y  e.  ( 1 ... ( P  -  1 ) ) )  ->  -.  P  ||  y )
14 eqid 2471 . . . . . . . . 9  |-  ( ( y ^ ( P  -  2 ) )  mod  P )  =  ( ( y ^
( P  -  2 ) )  mod  P
)
1514prmdiv 14812 . . . . . . . 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 1292 . . . . . . 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 466 . . . . . 6  |-  ( ( P  e.  Prime  /\  y  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( y ^ ( P  -  2 ) )  mod  P )  e.  ( 1 ... ( P  -  1 ) ) )
1817ralrimiva 2809 . . . . 5  |-  ( P  e.  Prime  ->  A. y  e.  ( 1 ... ( P  -  1 ) ) ( ( y ^ ( P  - 
2 ) )  mod 
P )  e.  ( 1 ... ( P  -  1 ) ) )
19 ovex 6336 . . . . . . 7  |-  ( 1 ... ( P  - 
1 ) )  e. 
_V
2019pwid 3956 . . . . . 6  |-  ( 1 ... ( P  - 
1 ) )  e. 
~P ( 1 ... ( P  -  1 ) )
21 eleq2 2538 . . . . . . . 8  |-  ( x  =  ( 1 ... ( P  -  1 ) )  ->  (
( P  -  1 )  e.  x  <->  ( P  -  1 )  e.  ( 1 ... ( P  -  1 ) ) ) )
22 eleq2 2538 . . . . . . . . 9  |-  ( x  =  ( 1 ... ( P  -  1 ) )  ->  (
( ( y ^
( P  -  2 ) )  mod  P
)  e.  x  <->  ( (
y ^ ( P  -  2 ) )  mod  P )  e.  ( 1 ... ( P  -  1 ) ) ) )
2322raleqbi1dv 2981 . . . . . . . 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 725 . . . . . . 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 3186 . . . . . 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 932 . . . . 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 677 . . . 4  |-  ( P  e.  Prime  ->  ( 1 ... ( P  - 
1 ) )  e.  A )
29 fzfi 12223 . . . . 5  |-  ( 1 ... ( P  - 
1 ) )  e. 
Fin
30 eleq1 2537 . . . . . . . 8  |-  ( s  =  t  ->  (
s  e.  A  <->  t  e.  A ) )
31 reseq2 5106 . . . . . . . . . . 11  |-  ( s  =  t  ->  (  _I  |`  s )  =  (  _I  |`  t
) )
3231oveq2d 6324 . . . . . . . . . 10  |-  ( s  =  t  ->  ( T  gsumg  (  _I  |`  s
) )  =  ( T  gsumg  (  _I  |`  t
) ) )
3332oveq1d 6323 . . . . . . . . 9  |-  ( s  =  t  ->  (
( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( ( T  gsumg  (  _I  |`  t
) )  mod  P
) )
3433eqeq1d 2473 . . . . . . . 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 327 . . . . . . 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 323 . . . . . 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 2537 . . . . . . . 8  |-  ( s  =  ( 1 ... ( P  -  1 ) )  ->  (
s  e.  A  <->  ( 1 ... ( P  - 
1 ) )  e.  A ) )
38 reseq2 5106 . . . . . . . . . . 11  |-  ( s  =  ( 1 ... ( P  -  1 ) )  ->  (  _I  |`  s )  =  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )
3938oveq2d 6324 . . . . . . . . . 10  |-  ( s  =  ( 1 ... ( P  -  1 ) )  ->  ( T  gsumg  (  _I  |`  s
) )  =  ( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) )
4039oveq1d 6323 . . . . . . . . 9  |-  ( s  =  ( 1 ... ( P  -  1 ) )  ->  (
( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( ( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  mod  P ) )
4140eqeq1d 2473 . . . . . . . 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 327 . . . . . . 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 323 . . . . . 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 bi2.04 368 . . . . . . . . . . . 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 85 . . . . . . . . . . . 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 225 . . . . . . . . . . 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 1771 . . . . . . . . . 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 2761 . . . . . . . . . 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 235 . . . . . . . . 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 1030 . . . . . . . . . 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 1032 . . . . . . . . . 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 1031 . . . . . . . . . 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 24073 . . . . . . . . 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 1230 . . . . . . . 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 7832 . . . . 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 19070 . . . . . 6  |-  1  =  ( 1r ` fld )
6452, 63ringidval 17815 . . . . 5  |-  1  =  ( 0g `  T )
65 cncrng 19066 . . . . . 6  |-fld  e.  CRing
6652crngmgp 17866 . . . . . 6  |-  (fld  e.  CRing  ->  T  e. CMnd )
6765, 66mp1i 13 . . . . 5  |-  ( P  e.  Prime  ->  T  e. CMnd
)
6829a1i 11 . . . . 5  |-  ( P  e.  Prime  ->  ( 1 ... ( P  - 
1 ) )  e. 
Fin )
69 zsubrg 19098 . . . . . 6  |-  ZZ  e.  (SubRing ` fld )
7052subrgsubm 18099 . . . . . 6  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubMnd `  T ) )
7169, 70mp1i 13 . . . . 5  |-  ( P  e.  Prime  ->  ZZ  e.  (SubMnd `  T ) )
72 f1oi 5864 . . . . . . . 8  |-  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) : ( 1 ... ( P  -  1 ) ) -1-1-onto-> ( 1 ... ( P  -  1 ) )
73 f1of 5828 . . . . . . . 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 3422 . . . . . . 7  |-  ( 1 ... ( P  - 
1 ) )  C_  ZZ
76 fss 5749 . . . . . . 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 686 . . . . . 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 9656 . . . . . . 7  |-  1  e.  _V
8079a1i 11 . . . . . 6  |-  ( P  e.  Prime  ->  1  e. 
_V )
8178, 68, 80fdmfifsupp 7911 . . . . 5  |-  ( P  e.  Prime  ->  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) finSupp  1 )
8264, 67, 68, 71, 78, 81gsumsubmcl 17630 . . . 4  |-  ( P  e.  Prime  ->  ( T 
gsumg  (  _I  |`  ( 1 ... ( P  - 
1 ) ) ) )  e.  ZZ )
83 1z 10991 . . . . 5  |-  1  e.  ZZ
84 znegcl 10996 . . . . 5  |-  ( 1  e.  ZZ  ->  -u 1  e.  ZZ )
8583, 84mp1i 13 . . . 4  |-  ( P  e.  Prime  ->  -u 1  e.  ZZ )
86 moddvds 14389 . . . 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 1292 . . 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 215 . 2  |-  ( P  e.  Prime  ->  P  ||  ( ( T  gsumg  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) )  -  -u 1 ) )
89 fcoi1 5769 . . . . . . . . . 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 5880 . . . . . . . 8  |-  ( ( (  _I  |`  (
1 ... ( P  - 
1 ) ) )  o.  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) `  k )  =  ( (  _I  |`  ( 1 ... ( P  -  1 ) ) ) `  k
)
92 fvres 5893 . . . . . . . 8  |-  ( k  e.  ( 1 ... ( P  -  1 ) )  ->  (
(  _I  |`  (
1 ... ( P  - 
1 ) ) ) `
 k )  =  (  _I  `  k
) )
9391, 92syl5eq 2517 . . . . . . 7  |-  ( k  e.  ( 1 ... ( P  -  1 ) )  ->  (
( (  _I  |`  (
1 ... ( P  - 
1 ) ) )  o.  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) `  k )  =  (  _I  `  k ) )
9493adantl 473 . . . . . 6  |-  ( ( P  e.  Prime  /\  k  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( (  _I  |`  (
1 ... ( P  - 
1 ) ) )  o.  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) `  k )  =  (  _I  `  k ) )
955, 94seqfveq 12275 . . . . 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 19051 . . . . . . 7  |-  CC  =  ( Base ` fld )
9752, 96mgpbas 17807 . . . . . 6  |-  CC  =  ( Base `  T )
98 cnfldmul 19053 . . . . . . 7  |-  x.  =  ( .r ` fld )
9952, 98mgpplusg 17805 . . . . . 6  |-  x.  =  ( +g  `  T )
100 eqid 2471 . . . . . 6  |-  (Cntz `  T )  =  (Cntz `  T )
101 cnring 19067 . . . . . . 7  |-fld  e.  Ring
10252ringmgp 17864 . . . . . . 7  |-  (fld  e.  Ring  ->  T  e.  Mnd )
103101, 102mp1i 13 . . . . . 6  |-  ( P  e.  Prime  ->  T  e. 
Mnd )
104 zsscn 10969 . . . . . . . 8  |-  ZZ  C_  CC
105 fss 5749 . . . . . . . 8  |-  ( ( (  _I  |`  (
1 ... ( P  - 
1 ) ) ) : ( 1 ... ( P  -  1 ) ) --> ZZ  /\  ZZ  C_  CC )  -> 
(  _I  |`  (
1 ... ( P  - 
1 ) ) ) : ( 1 ... ( P  -  1 ) ) --> CC )
10677, 104, 105mp2an 686 . . . . . . 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 17561 . . . . . 6  |-  ( P  e.  Prime  ->  ran  (  _I  |`  ( 1 ... ( P  -  1 ) ) )  C_  ( (Cntz `  T ) `  ran  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) )
109 f1of1 5827 . . . . . . 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 13 . . . . . 6  |-  ( P  e.  Prime  ->  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) : ( 1 ... ( P  -  1 ) )
-1-1-> ( 1 ... ( P  -  1 ) ) )
111 suppssdm 6946 . . . . . . . . 9  |-  ( (  _I  |`  ( 1 ... ( P  - 
1 ) ) ) supp  1 )  C_  dom  (  _I  |`  ( 1 ... ( P  - 
1 ) ) )
112 dmresi 5166 . . . . . . . . 9  |-  dom  (  _I  |`  ( 1 ... ( P  -  1 ) ) )  =  ( 1 ... ( P  -  1 ) )
113111, 112sseqtri 3450 . . . . . . . 8  |-  ( (  _I  |`  ( 1 ... ( P  - 
1 ) ) ) supp  1 )  C_  (
1 ... ( P  - 
1 ) )
114 rnresi 5187 . . . . . . . 8  |-  ran  (  _I  |`  ( 1 ... ( P  -  1 ) ) )  =  ( 1 ... ( P  -  1 ) )
115113, 114sseqtr4i 3451 . . . . . . 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 2471 . . . . . 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 17619 . . . . 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 12499 . . . . . 6  |-  ( ( P  -  1 )  e.  NN  ->  ( ! `  ( P  -  1 ) )  =  (  seq 1
(  x.  ,  _I  ) `  ( P  -  1 ) ) )
1203, 119syl 17 . . . . 5  |-  ( P  e.  Prime  ->  ( ! `
 ( P  - 
1 ) )  =  (  seq 1 (  x.  ,  _I  ) `  ( P  -  1 ) ) )
12195, 118, 1203eqtr4d 2515 . . . 4  |-  ( P  e.  Prime  ->  ( T 
gsumg  (  _I  |`  ( 1 ... ( P  - 
1 ) ) ) )  =  ( ! `
 ( P  - 
1 ) ) )
122121oveq1d 6323 . . 3  |-  ( P  e.  Prime  ->  ( ( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  -  -u 1
)  =  ( ( ! `  ( P  -  1 ) )  -  -u 1 ) )
123 nnm1nn0 10935 . . . . . . 7  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
12411, 123syl 17 . . . . . 6  |-  ( P  e.  Prime  ->  ( P  -  1 )  e. 
NN0 )
125 faccl 12507 . . . . . 6  |-  ( ( P  -  1 )  e.  NN0  ->  ( ! `
 ( P  - 
1 ) )  e.  NN )
126124, 125syl 17 . . . . 5  |-  ( P  e.  Prime  ->  ( ! `
 ( P  - 
1 ) )  e.  NN )
127126nncnd 10647 . . . 4  |-  ( P  e.  Prime  ->  ( ! `
 ( P  - 
1 ) )  e.  CC )
128 ax-1cn 9615 . . . 4  |-  1  e.  CC
129 subneg 9943 . . . 4  |-  ( ( ( ! `  ( P  -  1 ) )  e.  CC  /\  1  e.  CC )  ->  ( ( ! `  ( P  -  1
) )  -  -u 1
)  =  ( ( ! `  ( P  -  1 ) )  +  1 ) )
130127, 128, 129sylancl 675 . . 3  |-  ( P  e.  Prime  ->  ( ( ! `  ( P  -  1 ) )  -  -u 1 )  =  ( ( ! `  ( P  -  1
) )  +  1 ) )
131122, 130eqtrd 2505 . 2  |-  ( P  e.  Prime  ->  ( ( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  -  -u 1
)  =  ( ( ! `  ( P  -  1 ) )  +  1 ) )
13288, 131breqtrd 4420 1  |-  ( P  e.  Prime  ->  P  ||  ( ( ! `  ( P  -  1
) )  +  1 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007   A.wal 1450    = wceq 1452    e. wcel 1904   A.wral 2756   {crab 2760   _Vcvv 3031    C_ wss 3390    C. wpss 3391   ~Pcpw 3942   class class class wbr 4395    _I cid 4749   dom cdm 4839   ran crn 4840    |` cres 4841    o. ccom 4843   -->wf 5585   -1-1->wf1 5586   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308   supp csupp 6933   Fincfn 7587   CCcc 9555   1c1 9558    + caddc 9560    x. cmul 9562    - cmin 9880   -ucneg 9881   NNcn 10631   2c2 10681   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810    mod cmo 12129    seqcseq 12251   ^cexp 12310   !cfa 12497    || cdvds 14382   Primecprime 14701    gsumg cgsu 15417   Mndcmnd 16613  SubMndcsubmnd 16659  Cntzccntz 17047  CMndccmn 17508  mulGrpcmgp 17801   Ringcrg 17858   CRingccrg 17859  SubRingcsubrg 18082  ℂfldccnfld 19047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-dvds 14383  df-gcd 14548  df-prm 14702  df-phi 14793  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-0g 15418  df-gsum 15419  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-grp 16751  df-minusg 16752  df-mulg 16754  df-subg 16892  df-cntz 17049  df-cmn 17510  df-mgp 17802  df-ur 17814  df-ring 17860  df-cring 17861  df-subrg 18084  df-cnfld 19048
This theorem is referenced by:  wilth  24075
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