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Theorem wilthlem3 20853
Description: Lemma for wilth 20854. Here we round out the argument of wilthlem2 20852 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.)
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 13097 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
2 uz2m1nn 10550 . . . . . . . 8  |-  ( P  e.  ( ZZ>= `  2
)  ->  ( P  -  1 )  e.  NN )
31, 2syl 16 . . . . . . 7  |-  ( P  e.  Prime  ->  ( P  -  1 )  e.  NN )
4 nnuz 10521 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
53, 4syl6eleq 2526 . . . . . 6  |-  ( P  e.  Prime  ->  ( P  -  1 )  e.  ( ZZ>= `  1 )
)
6 eluzfz2 11065 . . . . . 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 444 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  y  e.  ( 1 ... ( P  -  1 ) ) )  ->  P  e.  Prime )
9 elfzelz 11059 . . . . . . . . 9  |-  ( y  e.  ( 1 ... ( P  -  1 ) )  ->  y  e.  ZZ )
109adantl 453 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  y  e.  ( 1 ... ( P  -  1 ) ) )  ->  y  e.  ZZ )
11 prmnn 13082 . . . . . . . . 9  |-  ( P  e.  Prime  ->  P  e.  NN )
12 fzm1ndvds 12901 . . . . . . . . 9  |-  ( ( P  e.  NN  /\  y  e.  ( 1 ... ( P  - 
1 ) ) )  ->  -.  P  ||  y
)
1311, 12sylan 458 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  y  e.  ( 1 ... ( P  -  1 ) ) )  ->  -.  P  ||  y )
14 eqid 2436 . . . . . . . . 9  |-  ( ( y ^ ( P  -  2 ) )  mod  P )  =  ( ( y ^
( P  -  2 ) )  mod  P
)
1514prmdiv 13174 . . . . . . . 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 1184 . . . . . . 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 446 . . . . . 6  |-  ( ( P  e.  Prime  /\  y  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( y ^ ( P  -  2 ) )  mod  P )  e.  ( 1 ... ( P  -  1 ) ) )
1817ralrimiva 2789 . . . . 5  |-  ( P  e.  Prime  ->  A. y  e.  ( 1 ... ( P  -  1 ) ) ( ( y ^ ( P  - 
2 ) )  mod 
P )  e.  ( 1 ... ( P  -  1 ) ) )
19 ovex 6106 . . . . . . 7  |-  ( 1 ... ( P  - 
1 ) )  e. 
_V
2019pwid 3812 . . . . . 6  |-  ( 1 ... ( P  - 
1 ) )  e. 
~P ( 1 ... ( P  -  1 ) )
21 eleq2 2497 . . . . . . . 8  |-  ( x  =  ( 1 ... ( P  -  1 ) )  ->  (
( P  -  1 )  e.  x  <->  ( P  -  1 )  e.  ( 1 ... ( P  -  1 ) ) ) )
22 eleq2 2497 . . . . . . . . 9  |-  ( x  =  ( 1 ... ( P  -  1 ) )  ->  (
( ( y ^
( P  -  2 ) )  mod  P
)  e.  x  <->  ( (
y ^ ( P  -  2 ) )  mod  P )  e.  ( 1 ... ( P  -  1 ) ) ) )
2322raleqbi1dv 2912 . . . . . . . 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 692 . . . . . . 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 3094 . . . . . 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 885 . . . . 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 646 . . . 4  |-  ( P  e.  Prime  ->  ( 1 ... ( P  - 
1 ) )  e.  A )
29 fzfi 11311 . . . . 5  |-  ( 1 ... ( P  - 
1 ) )  e. 
Fin
30 eleq1 2496 . . . . . . . 8  |-  ( s  =  t  ->  (
s  e.  A  <->  t  e.  A ) )
31 reseq2 5141 . . . . . . . . . . 11  |-  ( s  =  t  ->  (  _I  |`  s )  =  (  _I  |`  t
) )
3231oveq2d 6097 . . . . . . . . . 10  |-  ( s  =  t  ->  ( T  gsumg  (  _I  |`  s
) )  =  ( T  gsumg  (  _I  |`  t
) ) )
3332oveq1d 6096 . . . . . . . . 9  |-  ( s  =  t  ->  (
( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( ( T  gsumg  (  _I  |`  t
) )  mod  P
) )
3433eqeq1d 2444 . . . . . . . 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 312 . . . . . . 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 308 . . . . . 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 2496 . . . . . . . 8  |-  ( s  =  ( 1 ... ( P  -  1 ) )  ->  (
s  e.  A  <->  ( 1 ... ( P  - 
1 ) )  e.  A ) )
38 reseq2 5141 . . . . . . . . . . 11  |-  ( s  =  ( 1 ... ( P  -  1 ) )  ->  (  _I  |`  s )  =  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )
3938oveq2d 6097 . . . . . . . . . 10  |-  ( s  =  ( 1 ... ( P  -  1 ) )  ->  ( T  gsumg  (  _I  |`  s
) )  =  ( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) )
4039oveq1d 6096 . . . . . . . . 9  |-  ( s  =  ( 1 ... ( P  -  1 ) )  ->  (
( T  gsumg  (  _I  |`  s
) )  mod  P
)  =  ( ( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  mod  P ) )
4140eqeq1d 2444 . . . . . . . 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 312 . . . . . . 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 308 . . . . . 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 bi2.04 351 . . . . . . . . . . . 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 37 . . . . . . . . . . . . 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 79 . . . . . . . . . . . 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 209 . . . . . . . . . . 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 1631 . . . . . . . . . 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 2710 . . . . . . . . . 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 219 . . . . . . . . 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 29 . . . . . . . 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 957 . . . . . . . . . 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 959 . . . . . . . . . 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 958 . . . . . . . . . 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 20852 . . . . . . . . 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 1152 . . . . . . . 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 27 . . . . . . 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 7350 . . . . 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 8 . . . 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 16726 . . . . . 6  |-  1  =  ( 1r ` fld )
6452, 63rngidval 15666 . . . . 5  |-  1  =  ( 0g `  T )
65 cncrng 16722 . . . . . 6  |-fld  e.  CRing
6652crngmgp 15672 . . . . . 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 16752 . . . . . 6  |-  ZZ  e.  (SubRing ` fld )
7052subrgsubm 15881 . . . . . 6  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubMnd `  T ) )
7169, 70mp1i 12 . . . . 5  |-  ( P  e.  Prime  ->  ZZ  e.  (SubMnd `  T ) )
72 f1oi 5713 . . . . . . . 8  |-  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) : ( 1 ... ( P  -  1 ) ) -1-1-onto-> ( 1 ... ( P  -  1 ) )
73 f1of 5674 . . . . . . . 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 8 . . . . . . 7  |-  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) : ( 1 ... ( P  -  1 ) ) --> ( 1 ... ( P  -  1 ) )
759ssriv 3352 . . . . . . 7  |-  ( 1 ... ( P  - 
1 ) )  C_  ZZ
76 fss 5599 . . . . . . 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 654 . . . . . 6  |-  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) : ( 1 ... ( P  -  1 ) ) --> ZZ
7877a1i 11 . . . . 5  |-  ( P  e.  Prime  ->  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) : ( 1 ... ( P  -  1 ) ) --> ZZ )
7968, 78fisuppfi 14773 . . . . 5  |-  ( P  e.  Prime  ->  ( `' (  _I  |`  (
1 ... ( P  - 
1 ) ) )
" ( _V  \  { 1 } ) )  e.  Fin )
8064, 67, 68, 71, 78, 79gsumsubmcl 15524 . . . 4  |-  ( P  e.  Prime  ->  ( T 
gsumg  (  _I  |`  ( 1 ... ( P  - 
1 ) ) ) )  e.  ZZ )
81 1z 10311 . . . . 5  |-  1  e.  ZZ
82 znegcl 10313 . . . . 5  |-  ( 1  e.  ZZ  ->  -u 1  e.  ZZ )
8381, 82mp1i 12 . . . 4  |-  ( P  e.  Prime  ->  -u 1  e.  ZZ )
84 moddvds 12859 . . . 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
) ) )
8511, 80, 83, 84syl3anc 1184 . . 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
) ) )
8662, 85mpbid 202 . 2  |-  ( P  e.  Prime  ->  P  ||  ( ( T  gsumg  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) )  -  -u 1 ) )
87 fcoi1 5617 . . . . . . . . . 10  |-  ( (  _I  |`  ( 1 ... ( P  - 
1 ) ) ) : ( 1 ... ( P  -  1 ) ) --> ( 1 ... ( P  - 
1 ) )  -> 
( (  _I  |`  (
1 ... ( P  - 
1 ) ) )  o.  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  =  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) )
8874, 87ax-mp 8 . . . . . . . . 9  |-  ( (  _I  |`  ( 1 ... ( P  - 
1 ) ) )  o.  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  =  (  _I  |`  ( 1 ... ( P  -  1 ) ) )
8988fveq1i 5729 . . . . . . . 8  |-  ( ( (  _I  |`  (
1 ... ( P  - 
1 ) ) )  o.  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) `  k )  =  ( (  _I  |`  ( 1 ... ( P  -  1 ) ) ) `  k
)
90 fvres 5745 . . . . . . . 8  |-  ( k  e.  ( 1 ... ( P  -  1 ) )  ->  (
(  _I  |`  (
1 ... ( P  - 
1 ) ) ) `
 k )  =  (  _I  `  k
) )
9189, 90syl5eq 2480 . . . . . . 7  |-  ( k  e.  ( 1 ... ( P  -  1 ) )  ->  (
( (  _I  |`  (
1 ... ( P  - 
1 ) ) )  o.  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) `  k )  =  (  _I  `  k ) )
9291adantl 453 . . . . . 6  |-  ( ( P  e.  Prime  /\  k  e.  ( 1 ... ( P  -  1 ) ) )  ->  (
( (  _I  |`  (
1 ... ( P  - 
1 ) ) )  o.  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) `  k )  =  (  _I  `  k ) )
935, 92seqfveq 11347 . . . . 5  |-  ( P  e.  Prime  ->  (  seq  1 (  x.  , 
( (  _I  |`  (
1 ... ( P  - 
1 ) ) )  o.  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) ) `  ( P  -  1 ) )  =  (  seq  1 (  x.  ,  _I  ) `  ( P  -  1 ) ) )
94 cnfldbas 16707 . . . . . . 7  |-  CC  =  ( Base ` fld )
9552, 94mgpbas 15654 . . . . . 6  |-  CC  =  ( Base `  T )
96 cnfldmul 16709 . . . . . . 7  |-  x.  =  ( .r ` fld )
9752, 96mgpplusg 15652 . . . . . 6  |-  x.  =  ( +g  `  T )
98 eqid 2436 . . . . . 6  |-  (Cntz `  T )  =  (Cntz `  T )
99 cnrng 16723 . . . . . . 7  |-fld  e.  Ring
10052rngmgp 15670 . . . . . . 7  |-  (fld  e.  Ring  ->  T  e.  Mnd )
10199, 100mp1i 12 . . . . . 6  |-  ( P  e.  Prime  ->  T  e. 
Mnd )
102 zsscn 10290 . . . . . . . 8  |-  ZZ  C_  CC
103 fss 5599 . . . . . . . 8  |-  ( ( (  _I  |`  (
1 ... ( P  - 
1 ) ) ) : ( 1 ... ( P  -  1 ) ) --> ZZ  /\  ZZ  C_  CC )  -> 
(  _I  |`  (
1 ... ( P  - 
1 ) ) ) : ( 1 ... ( P  -  1 ) ) --> CC )
10477, 102, 103mp2an 654 . . . . . . 7  |-  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) : ( 1 ... ( P  -  1 ) ) --> CC
105104a1i 11 . . . . . 6  |-  ( P  e.  Prime  ->  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) : ( 1 ... ( P  -  1 ) ) --> CC )
10695, 98, 67, 105cntzcmnf 15515 . . . . . 6  |-  ( P  e.  Prime  ->  ran  (  _I  |`  ( 1 ... ( P  -  1 ) ) )  C_  ( (Cntz `  T ) `  ran  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) )
107 f1of1 5673 . . . . . . 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 ) ) )
10872, 107mp1i 12 . . . . . 6  |-  ( P  e.  Prime  ->  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) : ( 1 ... ( P  -  1 ) )
-1-1-> ( 1 ... ( P  -  1 ) ) )
109 cnvimass 5224 . . . . . . . . 9  |-  ( `' (  _I  |`  (
1 ... ( P  - 
1 ) ) )
" ( _V  \  { 1 } ) )  C_  dom  (  _I  |`  ( 1 ... ( P  -  1 ) ) )
110 dmresi 5196 . . . . . . . . 9  |-  dom  (  _I  |`  ( 1 ... ( P  -  1 ) ) )  =  ( 1 ... ( P  -  1 ) )
111109, 110sseqtri 3380 . . . . . . . 8  |-  ( `' (  _I  |`  (
1 ... ( P  - 
1 ) ) )
" ( _V  \  { 1 } ) )  C_  ( 1 ... ( P  - 
1 ) )
112 rnresi 5219 . . . . . . . 8  |-  ran  (  _I  |`  ( 1 ... ( P  -  1 ) ) )  =  ( 1 ... ( P  -  1 ) )
113111, 112sseqtr4i 3381 . . . . . . 7  |-  ( `' (  _I  |`  (
1 ... ( P  - 
1 ) ) )
" ( _V  \  { 1 } ) )  C_  ran  (  _I  |`  ( 1 ... ( P  -  1 ) ) )
114113a1i 11 . . . . . 6  |-  ( P  e.  Prime  ->  ( `' (  _I  |`  (
1 ... ( P  - 
1 ) ) )
" ( _V  \  { 1 } ) )  C_  ran  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) )
115 eqid 2436 . . . . . 6  |-  ( `' ( (  _I  |`  (
1 ... ( P  - 
1 ) ) )  o.  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) " ( _V 
\  { 1 } ) )  =  ( `' ( (  _I  |`  ( 1 ... ( P  -  1 ) ) )  o.  (  _I  |`  ( 1 ... ( P  -  1 ) ) ) )
" ( _V  \  { 1 } ) )
11695, 64, 97, 98, 101, 68, 105, 106, 3, 108, 114, 115gsumval3 15514 . . . . 5  |-  ( P  e.  Prime  ->  ( T 
gsumg  (  _I  |`  ( 1 ... ( P  - 
1 ) ) ) )  =  (  seq  1 (  x.  , 
( (  _I  |`  (
1 ... ( P  - 
1 ) ) )  o.  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) ) ) `  ( P  -  1 ) ) )
117 facnn 11568 . . . . . 6  |-  ( ( P  -  1 )  e.  NN  ->  ( ! `  ( P  -  1 ) )  =  (  seq  1
(  x.  ,  _I  ) `  ( P  -  1 ) ) )
1183, 117syl 16 . . . . 5  |-  ( P  e.  Prime  ->  ( ! `
 ( P  - 
1 ) )  =  (  seq  1 (  x.  ,  _I  ) `  ( P  -  1 ) ) )
11993, 116, 1183eqtr4d 2478 . . . 4  |-  ( P  e.  Prime  ->  ( T 
gsumg  (  _I  |`  ( 1 ... ( P  - 
1 ) ) ) )  =  ( ! `
 ( P  - 
1 ) ) )
120119oveq1d 6096 . . 3  |-  ( P  e.  Prime  ->  ( ( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  -  -u 1
)  =  ( ( ! `  ( P  -  1 ) )  -  -u 1 ) )
121 nnm1nn0 10261 . . . . . . 7  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
12211, 121syl 16 . . . . . 6  |-  ( P  e.  Prime  ->  ( P  -  1 )  e. 
NN0 )
123 faccl 11576 . . . . . 6  |-  ( ( P  -  1 )  e.  NN0  ->  ( ! `
 ( P  - 
1 ) )  e.  NN )
124122, 123syl 16 . . . . 5  |-  ( P  e.  Prime  ->  ( ! `
 ( P  - 
1 ) )  e.  NN )
125124nncnd 10016 . . . 4  |-  ( P  e.  Prime  ->  ( ! `
 ( P  - 
1 ) )  e.  CC )
126 ax-1cn 9048 . . . 4  |-  1  e.  CC
127 subneg 9350 . . . 4  |-  ( ( ( ! `  ( P  -  1 ) )  e.  CC  /\  1  e.  CC )  ->  ( ( ! `  ( P  -  1
) )  -  -u 1
)  =  ( ( ! `  ( P  -  1 ) )  +  1 ) )
128125, 126, 127sylancl 644 . . 3  |-  ( P  e.  Prime  ->  ( ( ! `  ( P  -  1 ) )  -  -u 1 )  =  ( ( ! `  ( P  -  1
) )  +  1 ) )
129120, 128eqtrd 2468 . 2  |-  ( P  e.  Prime  ->  ( ( T  gsumg  (  _I  |`  (
1 ... ( P  - 
1 ) ) ) )  -  -u 1
)  =  ( ( ! `  ( P  -  1 ) )  +  1 ) )
13086, 129breqtrd 4236 1  |-  ( P  e.  Prime  ->  P  ||  ( ( ! `  ( P  -  1
) )  +  1 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1549    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709   _Vcvv 2956    \ cdif 3317    C_ wss 3320    C. wpss 3321   ~Pcpw 3799   {csn 3814   class class class wbr 4212    _I cid 4493   `'ccnv 4877   dom cdm 4878   ran crn 4879    |` cres 4880   "cima 4881    o. ccom 4882   -->wf 5450   -1-1->wf1 5451   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081   Fincfn 7109   CCcc 8988   1c1 8991    + caddc 8993    x. cmul 8995    - cmin 9291   -ucneg 9292   NNcn 10000   2c2 10049   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488   ...cfz 11043    mod cmo 11250    seq cseq 11323   ^cexp 11382   !cfa 11566    || cdivides 12852   Primecprime 13079    gsumg cgsu 13724   Mndcmnd 14684  SubMndcsubmnd 14737  Cntzccntz 15114  CMndccmn 15412  mulGrpcmgp 15648   Ringcrg 15660   CRingccrg 15661  SubRingcsubrg 15864  ℂfldccnfld 16703
This theorem is referenced by:  wilth  20854
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-fac 11567  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-dvds 12853  df-gcd 13007  df-prm 13080  df-phi 13155  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-0g 13727  df-gsum 13728  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-submnd 14739  df-grp 14812  df-minusg 14813  df-mulg 14815  df-subg 14941  df-cntz 15116  df-cmn 15414  df-mgp 15649  df-rng 15663  df-cring 15664  df-ur 15665  df-subrg 15866  df-cnfld 16704
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