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Theorem numclwwlk5 30845
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 991 . . . . 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 996 . . . . 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 994 . . . . 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 30844 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  2  ||  ( K  -  1 )  /\  X  e.  V )  ->  ( ( # `  ( X F 2 ) )  mod  2 )  =  1 )
71, 2, 3, 6syl3anc 1219 . . . 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 2523 . . . . 5  |-  ( P  =  2  ->  ( P  e.  Prime  <->  2  e.  Prime ) )
10 breq1 4395 . . . . 5  |-  ( P  =  2  ->  ( P  ||  ( K  - 
1 )  <->  2  ||  ( K  -  1
) ) )
119, 103anbi23d 1293 . . . 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 6200 . . . . . 6  |-  ( P  =  2  ->  ( X F P )  =  ( X F 2 ) )
1413fveq2d 5795 . . . . 5  |-  ( P  =  2  ->  ( # `
 ( X F P ) )  =  ( # `  ( X F 2 ) ) )
15 id 22 . . . . 5  |-  ( P  =  2  ->  P  =  2 )
1614, 15oveq12d 6210 . . . 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 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 985 . . . . . . . 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 1035 . . . . . 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 1036 . . . . . 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 30389 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  P  =/=  2 )  ->  P  e.  ( ZZ>= `  3 )
)
2524ex 434 . . . . . . . . 9  |-  ( P  e.  Prime  ->  ( P  =/=  2  ->  P  e.  ( ZZ>= `  3 )
) )
26253ad2ant2 1010 . . . . . . . 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 5790 . . . . . . . . . . . 12  |-  ( u  =  w  ->  (
u `  0 )  =  ( w ` 
0 ) )
3029eqeq1d 2453 . . . . . . . . . . 11  |-  ( u  =  w  ->  (
( u `  0
)  =  v  <->  ( w `  0 )  =  v ) )
31 fveq1 5790 . . . . . . . . . . . 12  |-  ( u  =  w  ->  (
u `  ( n  -  2 ) )  =  ( w `  ( n  -  2
) ) )
3231, 29eqeq12d 2473 . . . . . . . . . . 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 3069 . . . . . . . . 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 6252 . . . . . . 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 5791 . . . . . . . . . . . 12  |-  ( u  =  w  ->  ( lastS  `  u )  =  ( lastS  `  w ) )
3837neeq1d 2725 . . . . . . . . . . 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 3069 . . . . . . . . 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 6252 . . . . . . 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 2466 . . . . . . . . . 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 3062 . . . . . . . 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 5791 . . . . . . . . . 10  |-  ( m  =  n  ->  ( C `  m )  =  ( C `  n ) )
47 oveq1 6199 . . . . . . . . . . . . 13  |-  ( m  =  n  ->  (
m  -  2 )  =  ( n  - 
2 ) )
4847fveq2d 5795 . . . . . . . . . . . 12  |-  ( m  =  n  ->  (
u `  ( m  -  2 ) )  =  ( u `  ( n  -  2
) ) )
4948neeq1d 2725 . . . . . . . . . . 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 3065 . . . . . . . . 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 2727 . . . . . . . . . . 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 3069 . . . . . . . . 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 2508 . . . . . . . 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 6267 . . . . . . 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 30842 . . . . . 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 1221 . . . . 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 6207 . . . 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 30686 . . . . . . . . . . . . 13  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K ) )
61 nn0z 10772 . . . . . . . . . . . . . 14  |-  ( K  e.  NN0  ->  K  e.  ZZ )
62613ad2ant2 1010 . . . . . . . . . . . . 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 10850 . . . . . . . . . . 11  |-  ( <. V ,  E >. RegUSGrph  K  ->  K  e.  RR )
65 peano2rem 9778 . . . . . . . . . . 11  |-  ( K  e.  RR  ->  ( K  -  1 )  e.  RR )
6664, 65syl 16 . . . . . . . . . 10  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( K  -  1 )  e.  RR )
67663ad2ant1 1009 . . . . . . . . 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 30688 . . . . . . . . . . . . . . 15  |-  ( <. V ,  E >. RegUSGrph  K  ->  V USGrph  E )
70 usgrav 23407 . . . . . . . . . . . . . . . 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 1007 . . . . . . . . . . 11  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  ( V  e.  Fin  /\  E  e.  _V ) )
76 prmm2nn0 13887 . . . . . . . . . . . . 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 1008 . . . . . . . . . . 11  |-  ( ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) )  ->  ( X  e.  V  /\  ( P  -  2
)  e.  NN0 )
)
794, 5numclwwlkffin 30815 . . . . . . . . . . 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 12229 . . . . . . . . . 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 10740 . . . . . . . 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 9517 . . . . . . 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 1009 . . . . . . . 8  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  K  e.  RR )
86763ad2ant2 1010 . . . . . . . 8  |-  ( ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) )  ->  ( P  -  2 )  e.  NN0 )
87 reexpcl 11985 . . . . . . . 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 13870 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
9089nnrpd 11129 . . . . . . . . 9  |-  ( P  e.  Prime  ->  P  e.  RR+ )
91903ad2ant2 1010 . . . . . . . 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 1168 . . . . . 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 11849 . . . . . 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 2459 . . . . 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 1010 . . . . . . . . . . 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 10791 . . . . . . . . . . . . 13  |-  ( K  e.  ZZ  ->  ( K  -  1 )  e.  ZZ )
10163, 100syl 16 . . . . . . . . . . . 12  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( K  -  1 )  e.  ZZ )
1021013ad2ant1 1009 . . . . . . . . . . 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 10848 . . . . . . . . . 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 1168 . . . . . . . . 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 996 . . . . . . . . 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 30386 . . . . . . . . 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 1009 . . . . . . . . . 10  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  K  e.  ZZ )
110 simp2 989 . . . . . . . . . 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 30388 . . . . . . . . 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 6210 . . . . . . 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 6207 . . . . . 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 10536 . . . . . . . . . 10  |-  ( 0  +  1 )  =  1
117116oveq1i 6202 . . . . . . . . 9  |-  ( ( 0  +  1 )  mod  P )  =  ( 1  mod  P
)
11889nnred 10440 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  RR )
119 prmgt1 13886 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  1  < 
P )
120 1mod 11843 . . . . . . . . . 10  |-  ( ( P  e.  RR  /\  1  <  P )  -> 
( 1  mod  P
)  =  1 )
121118, 119, 120syl2anc 661 . . . . . . . . 9  |-  ( P  e.  Prime  ->  ( 1  mod  P )  =  1 )
122117, 121syl5eq 2504 . . . . . . . 8  |-  ( P  e.  Prime  ->  ( ( 0  +  1 )  mod  P )  =  1 )
1231223ad2ant2 1010 . . . . . . 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 2492 . . . . 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 2496 . . 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 2761 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 965    = wceq 1370    e. wcel 1758    =/= wne 2644   A.wral 2795   {crab 2799   _Vcvv 3070   <.cop 3983   class class class wbr 4392    |-> cmpt 4450   ` cfv 5518  (class class class)co 6192    |-> cmpt2 6194   Fincfn 7412   RRcr 9384   0cc0 9385   1c1 9386    + caddc 9388    x. cmul 9390    < clt 9521    - cmin 9698   NNcn 10425   2c2 10474   3c3 10475   NN0cn0 10682   ZZcz 10749   ZZ>=cuz 10964   RR+crp 11094    mod cmo 11811   ^cexp 11968   #chash 12206   lastS clsw 12326    || cdivides 13639   Primecprime 13867   USGrph cusg 23401   VDeg cvdg 23700   WWalksN cwwlkn 30452   ClWWalksN cclwwlkn 30554   RegUSGrph crusgra 30680   FriendGrph cfrgra 30720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-inf2 7950  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-pre-sup 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-disj 4363  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-isom 5527  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-2o 7023  df-oadd 7026  df-er 7203  df-map 7318  df-pm 7319  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-sup 7794  df-oi 7827  df-card 8212  df-cda 8440  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-div 10097  df-nn 10426  df-2 10483  df-3 10484  df-n0 10683  df-z 10750  df-uz 10965  df-rp 11095  df-xadd 11193  df-fz 11541  df-fzo 11652  df-fl 11745  df-mod 11812  df-seq 11910  df-exp 11969  df-hash 12207  df-word 12333  df-lsw 12334  df-concat 12335  df-s1 12336  df-substr 12337  df-s2 12579  df-cj 12692  df-re 12693  df-im 12694  df-sqr 12828  df-abs 12829  df-clim 13070  df-sum 13268  df-dvds 13640  df-gcd 13795  df-prm 13868  df-phi 13945  df-usgra 23403  df-nbgra 23469  df-wlk 23552  df-vdgr 23701  df-wwlk 30453  df-wwlkn 30454  df-clwwlk 30556  df-clwwlkn 30557  df-rgra 30681  df-rusgra 30682  df-frgra 30721
This theorem is referenced by:  numclwwlk6  30846
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