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Theorem wilthlem3 23100
Description: Lemma for wilth 23101. Here we round out the argument of wilthlem2 23099 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 14094 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
2 uz2m1nn 11156 . . . . . . . 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 2565 . . . . . 6  |-  ( P  e.  Prime  ->  ( P  -  1 )  e.  ( ZZ>= `  1 )
)
6 eluzfz2 11694 . . . . . 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 457 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  y  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  Prime )
9 elfzelz 11688 . . . . . . . . 9  |-  ( y  e.  ( 1 ... ( P  -  1 ) )  ->  y  e.  ZZ )
109adantl 466 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  y  e.  ( 1 ... ( P  -  1 ) ) )  ->  y  e.  ZZ )
11 prmnn 14079 . . . . . . . . 9  |-  ( P  e.  Prime  ->  P  e.  NN )
12 fzm1ndvds 13897 . . . . . . . . 9  |-  ( ( P  e.  NN  /\  y  e.  ( 1 ... ( P  - 
1 ) ) )  ->  -.  P  ||  y
)
1311, 12sylan 471 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  y  e.  ( 1 ... ( P  -  1 ) ) )  ->  -.  P  ||  y )
14 eqid 2467 . . . . . . . . 9  |-  ( ( y ^ ( P  -  2 ) )  mod  P )  =  ( ( y ^
( P  -  2 ) )  mod  P
)
1514prmdiv 14174 . . . . . . . 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 1228 . . . . . . 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 459 . . . . . 6  |-  ( ( P  e.  Prime  /\  y  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( y ^ ( P  -  2 ) )  mod  P )  e.  ( 1 ... ( P  -  1 ) ) )
1817ralrimiva 2878 . . . . 5  |-  ( P  e.  Prime  ->  A. y  e.  ( 1 ... ( P  -  1 ) ) ( ( y ^ ( P  - 
2 ) )  mod 
P )  e.  ( 1 ... ( P  -  1 ) ) )
19 ovex 6309 . . . . . . 7  |-  ( 1 ... ( P  - 
1 ) )  e. 
_V
2019pwid 4024 . . . . . 6  |-  ( 1 ... ( P  - 
1 ) )  e. 
~P ( 1 ... ( P  -  1 ) )
21 eleq2 2540 . . . . . . . 8  |-  ( x  =  ( 1 ... ( P  -  1 ) )  ->  (
( P  -  1 )  e.  x  <->  ( P  -  1 )  e.  ( 1 ... ( P  -  1 ) ) ) )
22 eleq2 2540 . . . . . . . . 9  |-  ( x  =  ( 1 ... ( P  -  1 ) )  ->  (
( ( y ^
( P  -  2 ) )  mod  P
)  e.  x  <->  ( (
y ^ ( P  -  2 ) )  mod  P )  e.  ( 1 ... ( P  -  1 ) ) ) )
2322raleqbi1dv 3066 . . . . . . . 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 710 . . . . . . 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 3263 . . . . . 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 664 . . . 4  |-  ( P  e.  Prime  ->  ( 1 ... ( P  - 
1 ) )  e.  A )
29 fzfi 12050 . . . . 5  |-  ( 1 ... ( P  - 
1 ) )  e. 
Fin
30 eleq1 2539 . . . . . . . 8  |-  ( s  =  t  ->  (
s  e.  A  <->  t  e.  A ) )
31 reseq2 5268 . . . . . . . . . . 11  |-  ( s  =  t  ->  (  _I  |`  s )  =  (  _I  |`  t
) )
3231oveq2d 6300 . . . . . . . . . 10  |-  ( s  =  t  ->  ( T  gsumg  (  _I  |`  s
) )  =  ( T  gsumg  (  _I  |`  t
) ) )
3332oveq1d 6299 . . . . . . . . 9  |-  ( s  =  t  ->  (
( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( ( T  gsumg  (  _I  |`  t
) )  mod  P
) )
3433eqeq1d 2469 . . . . . . . 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 320 . . . . . . 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 316 . . . . . 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 2539 . . . . . . . 8  |-  ( s  =  ( 1 ... ( P  -  1 ) )  ->  (
s  e.  A  <->  ( 1 ... ( P  - 
1 ) )  e.  A ) )
38 reseq2 5268 . . . . . . . . . . 11  |-  ( s  =  ( 1 ... ( P  -  1 ) )  ->  (  _I  |`  s )  =  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )
3938oveq2d 6300 . . . . . . . . . 10  |-  ( s  =  ( 1 ... ( P  -  1 ) )  ->  ( T  gsumg  (  _I  |`  s
) )  =  ( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) )
4039oveq1d 6299 . . . . . . . . 9  |-  ( s  =  ( 1 ... ( P  -  1 ) )  ->  (
( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( ( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  mod  P ) )
4140eqeq1d 2469 . . . . . . . 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 320 . . . . . . 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 316 . . . . . 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 bi2.04 361 . . . . . . . . . . . 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 1685 . . . . . . . . . 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 2819 . . . . . . . . . 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 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 )  ->  P  e.  Prime )
54 simp3 998 . . . . . . . . . 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 997 . . . . . . . . . 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 23099 . . . . . . . . 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 1195 . . . . . . . 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 7763 . . . . 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 18242 . . . . . 6  |-  1  =  ( 1r ` fld )
6452, 63rngidval 16957 . . . . 5  |-  1  =  ( 0g `  T )
65 cncrng 18238 . . . . . 6  |-fld  e.  CRing
6652crngmgp 17008 . . . . . 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 18267 . . . . . 6  |-  ZZ  e.  (SubRing ` fld )
7052subrgsubm 17242 . . . . . 6  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubMnd `  T ) )
7169, 70mp1i 12 . . . . 5  |-  ( P  e.  Prime  ->  ZZ  e.  (SubMnd `  T ) )
72 f1oi 5851 . . . . . . . 8  |-  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) : ( 1 ... ( P  -  1 ) ) -1-1-onto-> ( 1 ... ( P  -  1 ) )
73 f1of 5816 . . . . . . . 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 3508 . . . . . . 7  |-  ( 1 ... ( P  - 
1 ) )  C_  ZZ
76 fss 5739 . . . . . . 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 672 . . . . . 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 9591 . . . . . . 7  |-  1  e.  _V
8079a1i 11 . . . . . 6  |-  ( P  e.  Prime  ->  1  e. 
_V )
8178, 68, 80fdmfifsupp 7839 . . . . 5  |-  ( P  e.  Prime  ->  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) finSupp  1 )
8264, 67, 68, 71, 78, 81gsumsubmcl 16733 . . . 4  |-  ( P  e.  Prime  ->  ( T 
gsumg  (  _I  |`  ( 1 ... ( P  - 
1 ) ) ) )  e.  ZZ )
83 1z 10894 . . . . 5  |-  1  e.  ZZ
84 znegcl 10898 . . . . 5  |-  ( 1  e.  ZZ  ->  -u 1  e.  ZZ )
8583, 84mp1i 12 . . . 4  |-  ( P  e.  Prime  ->  -u 1  e.  ZZ )
86 moddvds 13854 . . . 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 1228 . . 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 5759 . . . . . . . . . 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 5867 . . . . . . . 8  |-  ( ( (  _I  |`  (
1 ... ( P  - 
1 ) ) )  o.  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) `  k )  =  ( (  _I  |`  ( 1 ... ( P  -  1 ) ) ) `  k
)
92 fvres 5880 . . . . . . . 8  |-  ( k  e.  ( 1 ... ( P  -  1 ) )  ->  (
(  _I  |`  (
1 ... ( P  - 
1 ) ) ) `
 k )  =  (  _I  `  k
) )
9391, 92syl5eq 2520 . . . . . . 7  |-  ( k  e.  ( 1 ... ( P  -  1 ) )  ->  (
( (  _I  |`  (
1 ... ( P  - 
1 ) ) )  o.  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) `  k )  =  (  _I  `  k ) )
9493adantl 466 . . . . . 6  |-  ( ( P  e.  Prime  /\  k  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( (  _I  |`  (
1 ... ( P  - 
1 ) ) )  o.  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) `  k )  =  (  _I  `  k ) )
955, 94seqfveq 12099 . . . . 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 18223 . . . . . . 7  |-  CC  =  ( Base ` fld )
9752, 96mgpbas 16949 . . . . . 6  |-  CC  =  ( Base `  T )
98 cnfldmul 18225 . . . . . . 7  |-  x.  =  ( .r ` fld )
9952, 98mgpplusg 16947 . . . . . 6  |-  x.  =  ( +g  `  T )
100 eqid 2467 . . . . . 6  |-  (Cntz `  T )  =  (Cntz `  T )
101 cnrng 18239 . . . . . . 7  |-fld  e.  Ring
10252rngmgp 17006 . . . . . . 7  |-  (fld  e.  Ring  ->  T  e.  Mnd )
103101, 102mp1i 12 . . . . . 6  |-  ( P  e.  Prime  ->  T  e. 
Mnd )
104 zsscn 10872 . . . . . . . 8  |-  ZZ  C_  CC
105 fss 5739 . . . . . . . 8  |-  ( ( (  _I  |`  (
1 ... ( P  - 
1 ) ) ) : ( 1 ... ( P  -  1 ) ) --> ZZ  /\  ZZ  C_  CC )  -> 
(  _I  |`  (
1 ... ( P  - 
1 ) ) ) : ( 1 ... ( P  -  1 ) ) --> CC )
10677, 104, 105mp2an 672 . . . . . . 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 16654 . . . . . 6  |-  ( P  e.  Prime  ->  ran  (  _I  |`  ( 1 ... ( P  -  1 ) ) )  C_  ( (Cntz `  T ) `  ran  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) )
109 f1of1 5815 . . . . . . 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 6914 . . . . . . . . 9  |-  ( (  _I  |`  ( 1 ... ( P  - 
1 ) ) ) supp  1 )  C_  dom  (  _I  |`  ( 1 ... ( P  - 
1 ) ) )
112 dmresi 5329 . . . . . . . . 9  |-  dom  (  _I  |`  ( 1 ... ( P  -  1 ) ) )  =  ( 1 ... ( P  -  1 ) )
113111, 112sseqtri 3536 . . . . . . . 8  |-  ( (  _I  |`  ( 1 ... ( P  - 
1 ) ) ) supp  1 )  C_  (
1 ... ( P  - 
1 ) )
114 rnresi 5350 . . . . . . . 8  |-  ran  (  _I  |`  ( 1 ... ( P  -  1 ) ) )  =  ( 1 ... ( P  -  1 ) )
115113, 114sseqtr4i 3537 . . . . . . 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 2467 . . . . . 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 16714 . . . . 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 12323 . . . . . 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 2518 . . . 4  |-  ( P  e.  Prime  ->  ( T 
gsumg  (  _I  |`  ( 1 ... ( P  - 
1 ) ) ) )  =  ( ! `
 ( P  - 
1 ) ) )
122121oveq1d 6299 . . 3  |-  ( P  e.  Prime  ->  ( ( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  -  -u 1
)  =  ( ( ! `  ( P  -  1 ) )  -  -u 1 ) )
123 nnm1nn0 10837 . . . . . . 7  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
12411, 123syl 16 . . . . . 6  |-  ( P  e.  Prime  ->  ( P  -  1 )  e. 
NN0 )
125 faccl 12331 . . . . . 6  |-  ( ( P  -  1 )  e.  NN0  ->  ( ! `
 ( P  - 
1 ) )  e.  NN )
126124, 125syl 16 . . . . 5  |-  ( P  e.  Prime  ->  ( ! `
 ( P  - 
1 ) )  e.  NN )
127126nncnd 10552 . . . 4  |-  ( P  e.  Prime  ->  ( ! `
 ( P  - 
1 ) )  e.  CC )
128 ax-1cn 9550 . . . 4  |-  1  e.  CC
129 subneg 9868 . . . 4  |-  ( ( ( ! `  ( P  -  1 ) )  e.  CC  /\  1  e.  CC )  ->  ( ( ! `  ( P  -  1
) )  -  -u 1
)  =  ( ( ! `  ( P  -  1 ) )  +  1 ) )
130127, 128, 129sylancl 662 . . 3  |-  ( P  e.  Prime  ->  ( ( ! `  ( P  -  1 ) )  -  -u 1 )  =  ( ( ! `  ( P  -  1
) )  +  1 ) )
131122, 130eqtrd 2508 . 2  |-  ( P  e.  Prime  ->  ( ( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  -  -u 1
)  =  ( ( ! `  ( P  -  1 ) )  +  1 ) )
13288, 131breqtrd 4471 1  |-  ( P  e.  Prime  ->  P  ||  ( ( ! `  ( P  -  1
) )  +  1 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973   A.wal 1377    = wceq 1379    e. wcel 1767   A.wral 2814   {crab 2818   _Vcvv 3113    C_ wss 3476    C. wpss 3477   ~Pcpw 4010   class class class wbr 4447    _I cid 4790   dom cdm 4999   ran crn 5000    |` cres 5001    o. ccom 5003   -->wf 5584   -1-1->wf1 5585   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6284   supp csupp 6901   Fincfn 7516   CCcc 9490   1c1 9493    + caddc 9495    x. cmul 9497    - cmin 9805   -ucneg 9806   NNcn 10536   2c2 10585   NN0cn0 10795   ZZcz 10864   ZZ>=cuz 11082   ...cfz 11672    mod cmo 11964    seqcseq 12075   ^cexp 12134   !cfa 12321    || cdivides 13847   Primecprime 14076    gsumg cgsu 14696   Mndcmnd 15726  SubMndcsubmnd 15785  Cntzccntz 16158  CMndccmn 16604  mulGrpcmgp 16943   Ringcrg 17000   CRingccrg 17001  SubRingcsubrg 17225  ℂfldccnfld 18219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-rp 11221  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-fac 12322  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-dvds 13848  df-gcd 14004  df-prm 14077  df-phi 14155  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-0g 14697  df-gsum 14698  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-submnd 15787  df-grp 15867  df-minusg 15868  df-mulg 15870  df-subg 16003  df-cntz 16160  df-cmn 16606  df-mgp 16944  df-ur 16956  df-rng 17002  df-cring 17003  df-subrg 17227  df-cnfld 18220
This theorem is referenced by:  wilth  23101
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