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Theorem numclwwlk5 31831
Description: Huneke: "Let p be a prime divisor of k-1; then f(p) = 1 (mod p) [for each vertex v]". (Contributed by Alexander van der Vekens, 7-Oct-2018.)
Hypotheses
Ref Expression
numclwwlk.c  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
numclwwlk.f  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
Assertion
Ref Expression
numclwwlk5  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( # `  ( X F P ) )  mod  P )  =  1 )
Distinct variable groups:    n, E    n, V    w, C, n, v    n, X, v, w    v, V    w, E    w, V    w, F    w, P    v, E    v, K, w    P, n, v
Allowed substitution hints:    F( v, n)    K( n)

Proof of Theorem numclwwlk5
Dummy variables  m  u  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl1 994 . . . . 5  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  2  e.  Prime  /\  2  ||  ( K  -  1 ) ) )  ->  <. V ,  E >. RegUSGrph  K
)
2 simpr3 999 . . . . 5  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  2  e.  Prime  /\  2  ||  ( K  -  1 ) ) )  -> 
2  ||  ( K  -  1 ) )
3 simpr1 997 . . . . 5  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  2  e.  Prime  /\  2  ||  ( K  -  1 ) ) )  ->  X  e.  V )
4 numclwwlk.c . . . . . 6  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
5 numclwwlk.f . . . . . 6  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
64, 5numclwwlk5lem 31830 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  2  ||  ( K  -  1 )  /\  X  e.  V )  ->  ( ( # `  ( X F 2 ) )  mod  2 )  =  1 )
71, 2, 3, 6syl3anc 1223 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  2  e.  Prime  /\  2  ||  ( K  -  1 ) ) )  -> 
( ( # `  ( X F 2 ) )  mod  2 )  =  1 )
87a1i 11 . . 3  |-  ( P  =  2  ->  (
( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  2  e.  Prime  /\  2  ||  ( K  -  1 ) ) )  ->  ( ( # `
 ( X F 2 ) )  mod  2 )  =  1 ) )
9 eleq1 2532 . . . . 5  |-  ( P  =  2  ->  ( P  e.  Prime  <->  2  e.  Prime ) )
10 breq1 4443 . . . . 5  |-  ( P  =  2  ->  ( P  ||  ( K  - 
1 )  <->  2  ||  ( K  -  1
) ) )
119, 103anbi23d 1297 . . . 4  |-  ( P  =  2  ->  (
( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) )  <-> 
( X  e.  V  /\  2  e.  Prime  /\  2  ||  ( K  -  1 ) ) ) )
1211anbi2d 703 . . 3  |-  ( P  =  2  ->  (
( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  <->  ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  2  e.  Prime  /\  2  ||  ( K  -  1 ) ) ) ) )
13 oveq2 6283 . . . . . 6  |-  ( P  =  2  ->  ( X F P )  =  ( X F 2 ) )
1413fveq2d 5861 . . . . 5  |-  ( P  =  2  ->  ( # `
 ( X F P ) )  =  ( # `  ( X F 2 ) ) )
15 id 22 . . . . 5  |-  ( P  =  2  ->  P  =  2 )
1614, 15oveq12d 6293 . . . 4  |-  ( P  =  2  ->  (
( # `  ( X F P ) )  mod  P )  =  ( ( # `  ( X F 2 ) )  mod  2 ) )
1716eqeq1d 2462 . . 3  |-  ( P  =  2  ->  (
( ( # `  ( X F P ) )  mod  P )  =  1  <->  ( ( # `  ( X F 2 ) )  mod  2
)  =  1 ) )
188, 12, 173imtr4d 268 . 2  |-  ( P  =  2  ->  (
( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( ( # `
 ( X F P ) )  mod 
P )  =  1 ) )
19 3simpa 988 . . . . . . . 8  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E ) )
2019adantr 465 . . . . . . 7  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E ) )
2120adantl 466 . . . . . 6  |-  ( ( P  =/=  2  /\  ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) ) )  ->  ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E ) )
22 simprl3 1038 . . . . . 6  |-  ( ( P  =/=  2  /\  ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) ) )  ->  V  e.  Fin )
23 simprr1 1039 . . . . . 6  |-  ( ( P  =/=  2  /\  ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) ) )  ->  X  e.  V )
24 prmn2uzge3 14085 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  P  =/=  2 )  ->  P  e.  ( ZZ>= `  3 )
)
2524ex 434 . . . . . . . . 9  |-  ( P  e.  Prime  ->  ( P  =/=  2  ->  P  e.  ( ZZ>= `  3 )
) )
26253ad2ant2 1013 . . . . . . . 8  |-  ( ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) )  ->  ( P  =/=  2  ->  P  e.  ( ZZ>= `  3 )
) )
2726adantl 466 . . . . . . 7  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( P  =/=  2  ->  P  e.  ( ZZ>= ` 
3 ) ) )
2827impcom 430 . . . . . 6  |-  ( ( P  =/=  2  /\  ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) ) )  ->  P  e.  ( ZZ>= `  3 )
)
29 fveq1 5856 . . . . . . . . . . . 12  |-  ( u  =  w  ->  (
u `  0 )  =  ( w ` 
0 ) )
3029eqeq1d 2462 . . . . . . . . . . 11  |-  ( u  =  w  ->  (
( u `  0
)  =  v  <->  ( w `  0 )  =  v ) )
31 fveq1 5856 . . . . . . . . . . . 12  |-  ( u  =  w  ->  (
u `  ( n  -  2 ) )  =  ( w `  ( n  -  2
) ) )
3231, 29eqeq12d 2482 . . . . . . . . . . 11  |-  ( u  =  w  ->  (
( u `  (
n  -  2 ) )  =  ( u `
 0 )  <->  ( w `  ( n  -  2 ) )  =  ( w `  0 ) ) )
3330, 32anbi12d 710 . . . . . . . . . 10  |-  ( u  =  w  ->  (
( ( u ` 
0 )  =  v  /\  ( u `  ( n  -  2
) )  =  ( u `  0 ) )  <->  ( ( w `
 0 )  =  v  /\  ( w `
 ( n  - 
2 ) )  =  ( w `  0
) ) ) )
3433cbvrabv 3105 . . . . . . . . 9  |-  { u  e.  ( C `  n
)  |  ( ( u `  0 )  =  v  /\  (
u `  ( n  -  2 ) )  =  ( u ` 
0 ) ) }  =  { w  e.  ( C `  n
)  |  ( ( w `  0 )  =  v  /\  (
w `  ( n  -  2 ) )  =  ( w ` 
0 ) ) }
3534a1i 11 . . . . . . . 8  |-  ( ( v  e.  V  /\  n  e.  ( ZZ>= ` 
2 ) )  ->  { u  e.  ( C `  n )  |  ( ( u `
 0 )  =  v  /\  ( u `
 ( n  - 
2 ) )  =  ( u `  0
) ) }  =  { w  e.  ( C `  n )  |  ( ( w `
 0 )  =  v  /\  ( w `
 ( n  - 
2 ) )  =  ( w `  0
) ) } )
3635mpt2eq3ia 6337 . . . . . . 7  |-  ( v  e.  V ,  n  e.  ( ZZ>= `  2 )  |->  { u  e.  ( C `  n )  |  ( ( u `
 0 )  =  v  /\  ( u `
 ( n  - 
2 ) )  =  ( u `  0
) ) } )  =  ( v  e.  V ,  n  e.  ( ZZ>= `  2 )  |->  { w  e.  ( C `  n )  |  ( ( w `
 0 )  =  v  /\  ( w `
 ( n  - 
2 ) )  =  ( w `  0
) ) } )
37 fveq2 5857 . . . . . . . . . . . 12  |-  ( u  =  w  ->  ( lastS  `  u )  =  ( lastS  `  w ) )
3837neeq1d 2737 . . . . . . . . . . 11  |-  ( u  =  w  ->  (
( lastS  `  u )  =/=  v  <->  ( lastS  `  w )  =/=  v ) )
3930, 38anbi12d 710 . . . . . . . . . 10  |-  ( u  =  w  ->  (
( ( u ` 
0 )  =  v  /\  ( lastS  `  u
)  =/=  v )  <-> 
( ( w ` 
0 )  =  v  /\  ( lastS  `  w
)  =/=  v ) ) )
4039cbvrabv 3105 . . . . . . . . 9  |-  { u  e.  ( ( V WWalksN  E
) `  n )  |  ( ( u `
 0 )  =  v  /\  ( lastS  `  u
)  =/=  v ) }  =  { w  e.  ( ( V WWalksN  E
) `  n )  |  ( ( w `
 0 )  =  v  /\  ( lastS  `  w
)  =/=  v ) }
4140a1i 11 . . . . . . . 8  |-  ( ( v  e.  V  /\  n  e.  NN0 )  ->  { u  e.  (
( V WWalksN  E ) `  n )  |  ( ( u `  0
)  =  v  /\  ( lastS  `  u )  =/=  v ) }  =  { w  e.  (
( V WWalksN  E ) `  n )  |  ( ( w `  0
)  =  v  /\  ( lastS  `  w )  =/=  v ) } )
4241mpt2eq3ia 6337 . . . . . . 7  |-  ( v  e.  V ,  n  e.  NN0  |->  { u  e.  ( ( V WWalksN  E
) `  n )  |  ( ( u `
 0 )  =  v  /\  ( lastS  `  u
)  =/=  v ) } )  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( ( V WWalksN  E
) `  n )  |  ( ( w `
 0 )  =  v  /\  ( lastS  `  w
)  =/=  v ) } )
43 eqeq2 2475 . . . . . . . . . 10  |-  ( z  =  v  ->  (
( u `  0
)  =  z  <->  ( u `  0 )  =  v ) )
4443anbi1d 704 . . . . . . . . 9  |-  ( z  =  v  ->  (
( ( u ` 
0 )  =  z  /\  ( u `  ( m  -  2
) )  =/=  (
u `  0 )
)  <->  ( ( u `
 0 )  =  v  /\  ( u `
 ( m  - 
2 ) )  =/=  ( u `  0
) ) ) )
4544rabbidv 3098 . . . . . . . 8  |-  ( z  =  v  ->  { u  e.  ( C `  m
)  |  ( ( u `  0 )  =  z  /\  (
u `  ( m  -  2 ) )  =/=  ( u ` 
0 ) ) }  =  { u  e.  ( C `  m
)  |  ( ( u `  0 )  =  v  /\  (
u `  ( m  -  2 ) )  =/=  ( u ` 
0 ) ) } )
46 fveq2 5857 . . . . . . . . . 10  |-  ( m  =  n  ->  ( C `  m )  =  ( C `  n ) )
47 oveq1 6282 . . . . . . . . . . . . 13  |-  ( m  =  n  ->  (
m  -  2 )  =  ( n  - 
2 ) )
4847fveq2d 5861 . . . . . . . . . . . 12  |-  ( m  =  n  ->  (
u `  ( m  -  2 ) )  =  ( u `  ( n  -  2
) ) )
4948neeq1d 2737 . . . . . . . . . . 11  |-  ( m  =  n  ->  (
( u `  (
m  -  2 ) )  =/=  ( u `
 0 )  <->  ( u `  ( n  -  2 ) )  =/=  (
u `  0 )
) )
5049anbi2d 703 . . . . . . . . . 10  |-  ( m  =  n  ->  (
( ( u ` 
0 )  =  v  /\  ( u `  ( m  -  2
) )  =/=  (
u `  0 )
)  <->  ( ( u `
 0 )  =  v  /\  ( u `
 ( n  - 
2 ) )  =/=  ( u `  0
) ) ) )
5146, 50rabeqbidv 3101 . . . . . . . . 9  |-  ( m  =  n  ->  { u  e.  ( C `  m
)  |  ( ( u `  0 )  =  v  /\  (
u `  ( m  -  2 ) )  =/=  ( u ` 
0 ) ) }  =  { u  e.  ( C `  n
)  |  ( ( u `  0 )  =  v  /\  (
u `  ( n  -  2 ) )  =/=  ( u ` 
0 ) ) } )
5231, 29neeq12d 2739 . . . . . . . . . . 11  |-  ( u  =  w  ->  (
( u `  (
n  -  2 ) )  =/=  ( u `
 0 )  <->  ( w `  ( n  -  2 ) )  =/=  (
w `  0 )
) )
5330, 52anbi12d 710 . . . . . . . . . 10  |-  ( u  =  w  ->  (
( ( u ` 
0 )  =  v  /\  ( u `  ( n  -  2
) )  =/=  (
u `  0 )
)  <->  ( ( w `
 0 )  =  v  /\  ( w `
 ( n  - 
2 ) )  =/=  ( w `  0
) ) ) )
5453cbvrabv 3105 . . . . . . . . 9  |-  { u  e.  ( C `  n
)  |  ( ( u `  0 )  =  v  /\  (
u `  ( n  -  2 ) )  =/=  ( u ` 
0 ) ) }  =  { w  e.  ( C `  n
)  |  ( ( w `  0 )  =  v  /\  (
w `  ( n  -  2 ) )  =/=  ( w ` 
0 ) ) }
5551, 54syl6eq 2517 . . . . . . . 8  |-  ( m  =  n  ->  { u  e.  ( C `  m
)  |  ( ( u `  0 )  =  v  /\  (
u `  ( m  -  2 ) )  =/=  ( u ` 
0 ) ) }  =  { w  e.  ( C `  n
)  |  ( ( w `  0 )  =  v  /\  (
w `  ( n  -  2 ) )  =/=  ( w ` 
0 ) ) } )
5645, 55cbvmpt2v 6352 . . . . . . 7  |-  ( z  e.  V ,  m  e.  NN0  |->  { u  e.  ( C `  m
)  |  ( ( u `  0 )  =  z  /\  (
u `  ( m  -  2 ) )  =/=  ( u ` 
0 ) ) } )  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n
)  |  ( ( w `  0 )  =  v  /\  (
w `  ( n  -  2 ) )  =/=  ( w ` 
0 ) ) } )
574, 5, 36, 42, 56numclwwlk3 31828 . . . . . 6  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  e.  Fin  /\  X  e.  V  /\  P  e.  ( ZZ>= ` 
3 ) ) )  ->  ( # `  ( X F P ) )  =  ( ( ( K  -  1 )  x.  ( # `  ( X F ( P  - 
2 ) ) ) )  +  ( K ^ ( P  - 
2 ) ) ) )
5821, 22, 23, 28, 57syl13anc 1225 . . . . 5  |-  ( ( P  =/=  2  /\  ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) ) )  ->  ( # `
 ( X F P ) )  =  ( ( ( K  -  1 )  x.  ( # `  ( X F ( P  - 
2 ) ) ) )  +  ( K ^ ( P  - 
2 ) ) ) )
5958oveq1d 6290 . . . 4  |-  ( ( P  =/=  2  /\  ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) ) )  ->  (
( # `  ( X F P ) )  mod  P )  =  ( ( ( ( K  -  1 )  x.  ( # `  ( X F ( P  - 
2 ) ) ) )  +  ( K ^ ( P  - 
2 ) ) )  mod  P ) )
60 rusgraprop 24591 . . . . . . . . . . . . 13  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K ) )
61 nn0z 10876 . . . . . . . . . . . . . 14  |-  ( K  e.  NN0  ->  K  e.  ZZ )
62613ad2ant2 1013 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  K  e. 
NN0  /\  A. v  e.  V  ( ( V VDeg  E ) `  v
)  =  K )  ->  K  e.  ZZ )
6360, 62syl 16 . . . . . . . . . . . 12  |-  ( <. V ,  E >. RegUSGrph  K  ->  K  e.  ZZ )
6463zred 10955 . . . . . . . . . . 11  |-  ( <. V ,  E >. RegUSGrph  K  ->  K  e.  RR )
65 peano2rem 9875 . . . . . . . . . . 11  |-  ( K  e.  RR  ->  ( K  -  1 )  e.  RR )
6664, 65syl 16 . . . . . . . . . 10  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( K  -  1 )  e.  RR )
67663ad2ant1 1012 . . . . . . . . 9  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  ( K  -  1 )  e.  RR )
6867adantr 465 . . . . . . . 8  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( K  -  1 )  e.  RR )
69 rusisusgra 24593 . . . . . . . . . . . . . . 15  |-  ( <. V ,  E >. RegUSGrph  K  ->  V USGrph  E )
70 usgrav 24001 . . . . . . . . . . . . . . . 16  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
7170simprd 463 . . . . . . . . . . . . . . 15  |-  ( V USGrph  E  ->  E  e.  _V )
7269, 71syl 16 . . . . . . . . . . . . . 14  |-  ( <. V ,  E >. RegUSGrph  K  ->  E  e.  _V )
7372anim1i 568 . . . . . . . . . . . . 13  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V  e.  Fin )  ->  ( E  e. 
_V  /\  V  e.  Fin ) )
7473ancomd 451 . . . . . . . . . . . 12  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V  e.  Fin )  ->  ( V  e. 
Fin  /\  E  e.  _V ) )
75743adant2 1010 . . . . . . . . . . 11  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  ( V  e.  Fin  /\  E  e.  _V ) )
76 prmm2nn0 14086 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  ( P  -  2 )  e. 
NN0 )
7776anim2i 569 . . . . . . . . . . . 12  |-  ( ( X  e.  V  /\  P  e.  Prime )  -> 
( X  e.  V  /\  ( P  -  2 )  e.  NN0 )
)
78773adant3 1011 . . . . . . . . . . 11  |-  ( ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) )  ->  ( X  e.  V  /\  ( P  -  2
)  e.  NN0 )
)
794, 5numclwwlkffin 31801 . . . . . . . . . . 11  |-  ( ( ( V  e.  Fin  /\  E  e.  _V )  /\  ( X  e.  V  /\  ( P  -  2 )  e.  NN0 )
)  ->  ( X F ( P  - 
2 ) )  e. 
Fin )
8075, 78, 79syl2an 477 . . . . . . . . . 10  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( X F ( P  -  2 ) )  e.  Fin )
81 hashcl 12383 . . . . . . . . . 10  |-  ( ( X F ( P  -  2 ) )  e.  Fin  ->  ( # `
 ( X F ( P  -  2 ) ) )  e. 
NN0 )
8280, 81syl 16 . . . . . . . . 9  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( # `  ( X F ( P  - 
2 ) ) )  e.  NN0 )
8382nn0red 10842 . . . . . . . 8  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( # `  ( X F ( P  - 
2 ) ) )  e.  RR )
8468, 83remulcld 9613 . . . . . . 7  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( K  - 
1 )  x.  ( # `
 ( X F ( P  -  2 ) ) ) )  e.  RR )
85643ad2ant1 1012 . . . . . . . 8  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  K  e.  RR )
86763ad2ant2 1013 . . . . . . . 8  |-  ( ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) )  ->  ( P  -  2 )  e.  NN0 )
87 reexpcl 12139 . . . . . . . 8  |-  ( ( K  e.  RR  /\  ( P  -  2
)  e.  NN0 )  ->  ( K ^ ( P  -  2 ) )  e.  RR )
8885, 86, 87syl2an 477 . . . . . . 7  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( K ^ ( P  -  2 ) )  e.  RR )
89 prmnn 14068 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
9089nnrpd 11244 . . . . . . . . 9  |-  ( P  e.  Prime  ->  P  e.  RR+ )
91903ad2ant2 1013 . . . . . . . 8  |-  ( ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) )  ->  P  e.  RR+ )
9291adantl 466 . . . . . . 7  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  P  e.  RR+ )
9384, 88, 923jca 1171 . . . . . 6  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( ( K  -  1 )  x.  ( # `  ( X F ( P  - 
2 ) ) ) )  e.  RR  /\  ( K ^ ( P  -  2 ) )  e.  RR  /\  P  e.  RR+ ) )
9493adantl 466 . . . . 5  |-  ( ( P  =/=  2  /\  ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) ) )  ->  (
( ( K  - 
1 )  x.  ( # `
 ( X F ( P  -  2 ) ) ) )  e.  RR  /\  ( K ^ ( P  - 
2 ) )  e.  RR  /\  P  e.  RR+ ) )
95 modaddabs 11990 . . . . . 6  |-  ( ( ( ( K  - 
1 )  x.  ( # `
 ( X F ( P  -  2 ) ) ) )  e.  RR  /\  ( K ^ ( P  - 
2 ) )  e.  RR  /\  P  e.  RR+ )  ->  ( ( ( ( ( K  -  1 )  x.  ( # `  ( X F ( P  - 
2 ) ) ) )  mod  P )  +  ( ( K ^ ( P  - 
2 ) )  mod 
P ) )  mod 
P )  =  ( ( ( ( K  -  1 )  x.  ( # `  ( X F ( P  - 
2 ) ) ) )  +  ( K ^ ( P  - 
2 ) ) )  mod  P ) )
9695eqcomd 2468 . . . . 5  |-  ( ( ( ( K  - 
1 )  x.  ( # `
 ( X F ( P  -  2 ) ) ) )  e.  RR  /\  ( K ^ ( P  - 
2 ) )  e.  RR  /\  P  e.  RR+ )  ->  ( ( ( ( K  - 
1 )  x.  ( # `
 ( X F ( P  -  2 ) ) ) )  +  ( K ^
( P  -  2 ) ) )  mod 
P )  =  ( ( ( ( ( K  -  1 )  x.  ( # `  ( X F ( P  - 
2 ) ) ) )  mod  P )  +  ( ( K ^ ( P  - 
2 ) )  mod 
P ) )  mod 
P ) )
9794, 96syl 16 . . . 4  |-  ( ( P  =/=  2  /\  ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) ) )  ->  (
( ( ( K  -  1 )  x.  ( # `  ( X F ( P  - 
2 ) ) ) )  +  ( K ^ ( P  - 
2 ) ) )  mod  P )  =  ( ( ( ( ( K  -  1 )  x.  ( # `  ( X F ( P  -  2 ) ) ) )  mod 
P )  +  ( ( K ^ ( P  -  2 ) )  mod  P ) )  mod  P ) )
98893ad2ant2 1013 . . . . . . . . . . 11  |-  ( ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) )  ->  P  e.  NN )
9998adantl 466 . . . . . . . . . 10  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  P  e.  NN )
100 peano2zm 10895 . . . . . . . . . . . . 13  |-  ( K  e.  ZZ  ->  ( K  -  1 )  e.  ZZ )
10163, 100syl 16 . . . . . . . . . . . 12  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( K  -  1 )  e.  ZZ )
1021013ad2ant1 1012 . . . . . . . . . . 11  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  ( K  -  1 )  e.  ZZ )
103102adantr 465 . . . . . . . . . 10  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( K  -  1 )  e.  ZZ )
10482nn0zd 10953 . . . . . . . . . 10  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( # `  ( X F ( P  - 
2 ) ) )  e.  ZZ )
10599, 103, 1043jca 1171 . . . . . . . . 9  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( P  e.  NN  /\  ( K  -  1 )  e.  ZZ  /\  ( # `  ( X F ( P  - 
2 ) ) )  e.  ZZ ) )
106 simpr3 999 . . . . . . . . 9  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  P  ||  ( K  - 
1 ) )
107 mulmoddvds 13892 . . . . . . . . 9  |-  ( ( P  e.  NN  /\  ( K  -  1
)  e.  ZZ  /\  ( # `  ( X F ( P  - 
2 ) ) )  e.  ZZ )  -> 
( P  ||  ( K  -  1 )  ->  ( ( ( K  -  1 )  x.  ( # `  ( X F ( P  - 
2 ) ) ) )  mod  P )  =  0 ) )
108105, 106, 107sylc 60 . . . . . . . 8  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( ( K  -  1 )  x.  ( # `  ( X F ( P  - 
2 ) ) ) )  mod  P )  =  0 )
109633ad2ant1 1012 . . . . . . . . . 10  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  K  e.  ZZ )
110 simp2 992 . . . . . . . . . 10  |-  ( ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) )  ->  P  e.  Prime )
111109, 110anim12ci 567 . . . . . . . . 9  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( P  e.  Prime  /\  K  e.  ZZ ) )
112 powm2modprm 14176 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  ZZ )  ->  ( P  ||  ( K  - 
1 )  ->  (
( K ^ ( P  -  2 ) )  mod  P )  =  1 ) )
113111, 106, 112sylc 60 . . . . . . . 8  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( K ^
( P  -  2 ) )  mod  P
)  =  1 )
114108, 113oveq12d 6293 . . . . . . 7  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( ( ( K  -  1 )  x.  ( # `  ( X F ( P  - 
2 ) ) ) )  mod  P )  +  ( ( K ^ ( P  - 
2 ) )  mod 
P ) )  =  ( 0  +  1 ) )
115114oveq1d 6290 . . . . . 6  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( ( ( ( K  -  1 )  x.  ( # `  ( X F ( P  -  2 ) ) ) )  mod 
P )  +  ( ( K ^ ( P  -  2 ) )  mod  P ) )  mod  P )  =  ( ( 0  +  1 )  mod 
P ) )
116 0p1e1 10636 . . . . . . . . . 10  |-  ( 0  +  1 )  =  1
117116oveq1i 6285 . . . . . . . . 9  |-  ( ( 0  +  1 )  mod  P )  =  ( 1  mod  P
)
11889nnred 10540 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  RR )
119 prmgt1 14084 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  1  < 
P )
120 1mod 11984 . . . . . . . . . 10  |-  ( ( P  e.  RR  /\  1  <  P )  -> 
( 1  mod  P
)  =  1 )
121118, 119, 120syl2anc 661 . . . . . . . . 9  |-  ( P  e.  Prime  ->  ( 1  mod  P )  =  1 )
122117, 121syl5eq 2513 . . . . . . . 8  |-  ( P  e.  Prime  ->  ( ( 0  +  1 )  mod  P )  =  1 )
1231223ad2ant2 1013 . . . . . . 7  |-  ( ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) )  ->  (
( 0  +  1 )  mod  P )  =  1 )
124123adantl 466 . . . . . 6  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( 0  +  1 )  mod  P
)  =  1 )
125115, 124eqtrd 2501 . . . . 5  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( ( ( ( K  -  1 )  x.  ( # `  ( X F ( P  -  2 ) ) ) )  mod 
P )  +  ( ( K ^ ( P  -  2 ) )  mod  P ) )  mod  P )  =  1 )
126125adantl 466 . . . 4  |-  ( ( P  =/=  2  /\  ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) ) )  ->  (
( ( ( ( K  -  1 )  x.  ( # `  ( X F ( P  - 
2 ) ) ) )  mod  P )  +  ( ( K ^ ( P  - 
2 ) )  mod 
P ) )  mod 
P )  =  1 )
12759, 97, 1263eqtrd 2505 . . 3  |-  ( ( P  =/=  2  /\  ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) ) )  ->  (
( # `  ( X F P ) )  mod  P )  =  1 )
128127ex 434 . 2  |-  ( P  =/=  2  ->  (
( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( ( # `
 ( X F P ) )  mod 
P )  =  1 ) )
12918, 128pm2.61ine 2773 1  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( # `  ( X F P ) )  mod  P )  =  1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   A.wral 2807   {crab 2811   _Vcvv 3106   <.cop 4026   class class class wbr 4440    |-> cmpt 4498   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   Fincfn 7506   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    < clt 9617    - cmin 9794   NNcn 10525   2c2 10574   3c3 10575   NN0cn0 10784   ZZcz 10853   ZZ>=cuz 11071   RR+crp 11209    mod cmo 11952   ^cexp 12122   #chash 12360   lastS clsw 12488    || cdivides 13836   Primecprime 14065   USGrph cusg 23993   WWalksN cwwlkn 24340   ClWWalksN cclwwlkn 24411   VDeg cvdg 24555   RegUSGrph crusgra 24585   FriendGrph cfrgra 31706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  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
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-disj 4411  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-xadd 11308  df-fz 11662  df-fzo 11782  df-fl 11886  df-mod 11953  df-seq 12064  df-exp 12123  df-hash 12361  df-word 12495  df-lsw 12496  df-concat 12497  df-s1 12498  df-substr 12499  df-s2 12763  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-sum 13458  df-dvds 13837  df-gcd 13993  df-prm 14066  df-phi 14144  df-usgra 23996  df-nbgra 24082  df-wlk 24170  df-wwlk 24341  df-wwlkn 24342  df-clwwlk 24413  df-clwwlkn 24414  df-vdgr 24556  df-rgra 24586  df-rusgra 24587  df-frgra 31707
This theorem is referenced by:  numclwwlk6  31832
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