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Theorem numclwwlk5 25840
Description: Statement 13 in [Huneke] p. 2: "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 1011 . . . . 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 1016 . . . . 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 1014 . . . . 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 25839 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  2  ||  ( K  -  1 )  /\  X  e.  V )  ->  ( ( # `  ( X F 2 ) )  mod  2 )  =  1 )
71, 2, 3, 6syl3anc 1268 . . . 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 2517 . . . . 5  |-  ( P  =  2  ->  ( P  e.  Prime  <->  2  e.  Prime ) )
10 breq1 4405 . . . . 5  |-  ( P  =  2  ->  ( P  ||  ( K  - 
1 )  <->  2  ||  ( K  -  1
) ) )
119, 103anbi23d 1342 . . . 4  |-  ( P  =  2  ->  (
( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) )  <-> 
( X  e.  V  /\  2  e.  Prime  /\  2  ||  ( K  -  1 ) ) ) )
1211anbi2d 710 . . 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 6298 . . . . . 6  |-  ( P  =  2  ->  ( X F P )  =  ( X F 2 ) )
1413fveq2d 5869 . . . . 5  |-  ( P  =  2  ->  ( # `
 ( X F P ) )  =  ( # `  ( X F 2 ) ) )
15 id 22 . . . . 5  |-  ( P  =  2  ->  P  =  2 )
1614, 15oveq12d 6308 . . . 4  |-  ( P  =  2  ->  (
( # `  ( X F P ) )  mod  P )  =  ( ( # `  ( X F 2 ) )  mod  2 ) )
1716eqeq1d 2453 . . 3  |-  ( P  =  2  ->  (
( ( # `  ( X F P ) )  mod  P )  =  1  <->  ( ( # `  ( X F 2 ) )  mod  2
)  =  1 ) )
188, 12, 173imtr4d 272 . 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 1005 . . . . . . . 8  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E ) )
2019adantr 467 . . . . . . 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 468 . . . . . 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 1055 . . . . . 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 1056 . . . . . 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 14644 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  P  =/=  2 )  ->  P  e.  ( ZZ>= `  3 )
)
2524ex 436 . . . . . . . . 9  |-  ( P  e.  Prime  ->  ( P  =/=  2  ->  P  e.  ( ZZ>= `  3 )
) )
26253ad2ant2 1030 . . . . . . . 8  |-  ( ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) )  ->  ( P  =/=  2  ->  P  e.  ( ZZ>= `  3 )
) )
2726adantl 468 . . . . . . 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 432 . . . . . 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 5864 . . . . . . . . . . . 12  |-  ( u  =  w  ->  (
u `  0 )  =  ( w ` 
0 ) )
3029eqeq1d 2453 . . . . . . . . . . 11  |-  ( u  =  w  ->  (
( u `  0
)  =  v  <->  ( w `  0 )  =  v ) )
31 fveq1 5864 . . . . . . . . . . . 12  |-  ( u  =  w  ->  (
u `  ( n  -  2 ) )  =  ( w `  ( n  -  2
) ) )
3231, 29eqeq12d 2466 . . . . . . . . . . 11  |-  ( u  =  w  ->  (
( u `  (
n  -  2 ) )  =  ( u `
 0 )  <->  ( w `  ( n  -  2 ) )  =  ( w `  0 ) ) )
3330, 32anbi12d 717 . . . . . . . . . 10  |-  ( u  =  w  ->  (
( ( u ` 
0 )  =  v  /\  ( u `  ( n  -  2
) )  =  ( u `  0 ) )  <->  ( ( w `
 0 )  =  v  /\  ( w `
 ( n  - 
2 ) )  =  ( w `  0
) ) ) )
3433cbvrabv 3044 . . . . . . . . 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 6356 . . . . . . 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 5865 . . . . . . . . . . . 12  |-  ( u  =  w  ->  ( lastS  `  u )  =  ( lastS  `  w ) )
3837neeq1d 2683 . . . . . . . . . . 11  |-  ( u  =  w  ->  (
( lastS  `  u )  =/=  v  <->  ( lastS  `  w )  =/=  v ) )
3930, 38anbi12d 717 . . . . . . . . . 10  |-  ( u  =  w  ->  (
( ( u ` 
0 )  =  v  /\  ( lastS  `  u
)  =/=  v )  <-> 
( ( w ` 
0 )  =  v  /\  ( lastS  `  w
)  =/=  v ) ) )
4039cbvrabv 3044 . . . . . . . . 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 6356 . . . . . . 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 2462 . . . . . . . . . 10  |-  ( z  =  v  ->  (
( u `  0
)  =  z  <->  ( u `  0 )  =  v ) )
4443anbi1d 711 . . . . . . . . 9  |-  ( z  =  v  ->  (
( ( u ` 
0 )  =  z  /\  ( u `  ( m  -  2
) )  =/=  (
u `  0 )
)  <->  ( ( u `
 0 )  =  v  /\  ( u `
 ( m  - 
2 ) )  =/=  ( u `  0
) ) ) )
4544rabbidv 3036 . . . . . . . 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 5865 . . . . . . . . . 10  |-  ( m  =  n  ->  ( C `  m )  =  ( C `  n ) )
47 oveq1 6297 . . . . . . . . . . . . 13  |-  ( m  =  n  ->  (
m  -  2 )  =  ( n  - 
2 ) )
4847fveq2d 5869 . . . . . . . . . . . 12  |-  ( m  =  n  ->  (
u `  ( m  -  2 ) )  =  ( u `  ( n  -  2
) ) )
4948neeq1d 2683 . . . . . . . . . . 11  |-  ( m  =  n  ->  (
( u `  (
m  -  2 ) )  =/=  ( u `
 0 )  <->  ( u `  ( n  -  2 ) )  =/=  (
u `  0 )
) )
5049anbi2d 710 . . . . . . . . . 10  |-  ( m  =  n  ->  (
( ( u ` 
0 )  =  v  /\  ( u `  ( m  -  2
) )  =/=  (
u `  0 )
)  <->  ( ( u `
 0 )  =  v  /\  ( u `
 ( n  - 
2 ) )  =/=  ( u `  0
) ) ) )
5146, 50rabeqbidv 3040 . . . . . . . . 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 2685 . . . . . . . . . . 11  |-  ( u  =  w  ->  (
( u `  (
n  -  2 ) )  =/=  ( u `
 0 )  <->  ( w `  ( n  -  2 ) )  =/=  (
w `  0 )
) )
5330, 52anbi12d 717 . . . . . . . . . 10  |-  ( u  =  w  ->  (
( ( u ` 
0 )  =  v  /\  ( u `  ( n  -  2
) )  =/=  (
u `  0 )
)  <->  ( ( w `
 0 )  =  v  /\  ( w `
 ( n  - 
2 ) )  =/=  ( w `  0
) ) ) )
5453cbvrabv 3044 . . . . . . . . 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 2501 . . . . . . . 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 6371 . . . . . . 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 25837 . . . . . 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 1270 . . . . 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 6305 . . . 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 25657 . . . . . . . . . . . . 13  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K ) )
61 nn0z 10960 . . . . . . . . . . . . . 14  |-  ( K  e.  NN0  ->  K  e.  ZZ )
62613ad2ant2 1030 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  K  e. 
NN0  /\  A. v  e.  V  ( ( V VDeg  E ) `  v
)  =  K )  ->  K  e.  ZZ )
6360, 62syl 17 . . . . . . . . . . . 12  |-  ( <. V ,  E >. RegUSGrph  K  ->  K  e.  ZZ )
6463zred 11040 . . . . . . . . . . 11  |-  ( <. V ,  E >. RegUSGrph  K  ->  K  e.  RR )
65 peano2rem 9941 . . . . . . . . . . 11  |-  ( K  e.  RR  ->  ( K  -  1 )  e.  RR )
6664, 65syl 17 . . . . . . . . . 10  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( K  -  1 )  e.  RR )
67663ad2ant1 1029 . . . . . . . . 9  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  ( K  -  1 )  e.  RR )
6867adantr 467 . . . . . . . 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 25659 . . . . . . . . . . . . . . 15  |-  ( <. V ,  E >. RegUSGrph  K  ->  V USGrph  E )
70 usgrav 25065 . . . . . . . . . . . . . . . 16  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
7170simprd 465 . . . . . . . . . . . . . . 15  |-  ( V USGrph  E  ->  E  e.  _V )
7269, 71syl 17 . . . . . . . . . . . . . 14  |-  ( <. V ,  E >. RegUSGrph  K  ->  E  e.  _V )
7372anim1i 572 . . . . . . . . . . . . 13  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V  e.  Fin )  ->  ( E  e. 
_V  /\  V  e.  Fin ) )
7473ancomd 453 . . . . . . . . . . . 12  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V  e.  Fin )  ->  ( V  e. 
Fin  /\  E  e.  _V ) )
75743adant2 1027 . . . . . . . . . . 11  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  ( V  e.  Fin  /\  E  e.  _V ) )
76 prmm2nn0 14645 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  ( P  -  2 )  e. 
NN0 )
7776anim2i 573 . . . . . . . . . . . 12  |-  ( ( X  e.  V  /\  P  e.  Prime )  -> 
( X  e.  V  /\  ( P  -  2 )  e.  NN0 )
)
78773adant3 1028 . . . . . . . . . . 11  |-  ( ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) )  ->  ( X  e.  V  /\  ( P  -  2
)  e.  NN0 )
)
794, 5numclwwlkffin 25810 . . . . . . . . . . 11  |-  ( ( ( V  e.  Fin  /\  E  e.  _V )  /\  ( X  e.  V  /\  ( P  -  2 )  e.  NN0 )
)  ->  ( X F ( P  - 
2 ) )  e. 
Fin )
8075, 78, 79syl2an 480 . . . . . . . . . 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 12538 . . . . . . . . . 10  |-  ( ( X F ( P  -  2 ) )  e.  Fin  ->  ( # `
 ( X F ( P  -  2 ) ) )  e. 
NN0 )
8280, 81syl 17 . . . . . . . . 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 10926 . . . . . . . 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 9671 . . . . . . 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 1029 . . . . . . . 8  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  K  e.  RR )
86763ad2ant2 1030 . . . . . . . 8  |-  ( ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) )  ->  ( P  -  2 )  e.  NN0 )
87 reexpcl 12289 . . . . . . . 8  |-  ( ( K  e.  RR  /\  ( P  -  2
)  e.  NN0 )  ->  ( K ^ ( P  -  2 ) )  e.  RR )
8885, 86, 87syl2an 480 . . . . . . 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 14625 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
9089nnrpd 11339 . . . . . . . . 9  |-  ( P  e.  Prime  ->  P  e.  RR+ )
91903ad2ant2 1030 . . . . . . . 8  |-  ( ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) )  ->  P  e.  RR+ )
9291adantl 468 . . . . . . 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 1188 . . . . . 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 468 . . . . 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 12135 . . . . . 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 2457 . . . . 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 17 . . . 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 1030 . . . . . . . . . . 11  |-  ( ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) )  ->  P  e.  NN )
9998adantl 468 . . . . . . . . . 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 10980 . . . . . . . . . . . . 13  |-  ( K  e.  ZZ  ->  ( K  -  1 )  e.  ZZ )
10163, 100syl 17 . . . . . . . . . . . 12  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( K  -  1 )  e.  ZZ )
1021013ad2ant1 1029 . . . . . . . . . . 11  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  ( K  -  1 )  e.  ZZ )
103102adantr 467 . . . . . . . . . 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 11038 . . . . . . . . . 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 1188 . . . . . . . . 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 1016 . . . . . . . . 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 14363 . . . . . . . . 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 62 . . . . . . . 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 1029 . . . . . . . . . 10  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  K  e.  ZZ )
110 simp2 1009 . . . . . . . . . 10  |-  ( ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) )  ->  P  e.  Prime )
111109, 110anim12ci 571 . . . . . . . . 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 14754 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  K  e.  ZZ )  ->  ( P  ||  ( K  - 
1 )  ->  (
( K ^ ( P  -  2 ) )  mod  P )  =  1 ) )
113111, 106, 112sylc 62 . . . . . . . 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 6308 . . . . . . 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 6305 . . . . . 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 10721 . . . . . . . . . 10  |-  ( 0  +  1 )  =  1
117116oveq1i 6300 . . . . . . . . 9  |-  ( ( 0  +  1 )  mod  P )  =  ( 1  mod  P
)
11889nnred 10624 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  RR )
119 prmgt1 14643 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  1  < 
P )
120 1mod 12129 . . . . . . . . . 10  |-  ( ( P  e.  RR  /\  1  <  P )  -> 
( 1  mod  P
)  =  1 )
121118, 119, 120syl2anc 667 . . . . . . . . 9  |-  ( P  e.  Prime  ->  ( 1  mod  P )  =  1 )
122117, 121syl5eq 2497 . . . . . . . 8  |-  ( P  e.  Prime  ->  ( ( 0  +  1 )  mod  P )  =  1 )
1231223ad2ant2 1030 . . . . . . 7  |-  ( ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) )  ->  (
( 0  +  1 )  mod  P )  =  1 )
124123adantl 468 . . . . . 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 2485 . . . . 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 468 . . . 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 2489 . . 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 436 . 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 2707 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 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   {crab 2741   _Vcvv 3045   <.cop 3974   class class class wbr 4402    |-> cmpt 4461   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292   Fincfn 7569   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544    < clt 9675    - cmin 9860   NNcn 10609   2c2 10659   3c3 10660   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   RR+crp 11302    mod cmo 12096   ^cexp 12272   #chash 12515   lastS clsw 12657    || cdvds 14305   Primecprime 14622   USGrph cusg 25057   WWalksN cwwlkn 25406   ClWWalksN cclwwlkn 25477   VDeg cvdg 25621   RegUSGrph crusgra 25651   FriendGrph cfrgra 25716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-disj 4374  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-xadd 11410  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-hash 12516  df-word 12664  df-lsw 12665  df-concat 12666  df-s1 12667  df-substr 12668  df-s2 12944  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753  df-dvds 14306  df-gcd 14469  df-prm 14623  df-phi 14714  df-usgra 25060  df-nbgra 25148  df-wlk 25236  df-wwlk 25407  df-wwlkn 25408  df-clwwlk 25479  df-clwwlkn 25480  df-vdgr 25622  df-rgra 25652  df-rusgra 25653  df-frgra 25717
This theorem is referenced by:  numclwwlk6  25841
  Copyright terms: Public domain W3C validator