Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  numclwwlk7 Structured version   Unicode version

Theorem numclwwlk7 30712
Description: Huneke: "The total number of closed walks of length p [in a friendship graph] is (k(k-1)+1)f(p)=1 (mod p)", since the number of vertices in a friendship graph is (k(k-1)+1), see frgregordn0 30668 or frrusgraord 30669, and p divides (k-1), i.e. (k-1) mod p = 0 => k(k-1) mod p = 0 => k(k-1)+1 mod p = 1. Since the empty graph is a friendship graph, see frgra0 30591, as well as k-regular (for any k), see 0vgrargra 30555, but has no closed walk, see clwlk0 30430, this theorem would be false:  ( ( # `
 ( C `  P ) )  mod 
P )  =  0  =/=  1, so this case must be excluded. ( (Contributed by Alexander van der Vekens, 1-Sep-2018.)
Assertion
Ref Expression
numclwwlk7  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( # `  (
( V ClWWalksN  E ) `  P ) )  mod 
P )  =  1 )

Proof of Theorem numclwwlk7
Dummy variables  m  n  p  q  v  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prmnn 13771 . . . . . . . . 9  |-  ( P  e.  Prime  ->  P  e.  NN )
21nnnn0d 10641 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e. 
NN0 )
32adantr 465 . . . . . . 7  |-  ( ( P  e.  Prime  /\  P  ||  ( K  -  1 ) )  ->  P  e.  NN0 )
433ad2ant3 1011 . . . . . 6  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  P  e.  NN0 )
5 eqid 2443 . . . . . . 7  |-  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n
) )  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n
) )
65numclwwlkfvc 30675 . . . . . 6  |-  ( P  e.  NN0  ->  ( ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n
) ) `  P
)  =  ( ( V ClWWalksN  E ) `  P
) )
74, 6syl 16 . . . . 5  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( n  e. 
NN0  |->  ( ( V ClWWalksN  E ) `  n
) ) `  P
)  =  ( ( V ClWWalksN  E ) `  P
) )
87eqcomd 2448 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( V ClWWalksN  E ) `
 P )  =  ( ( n  e. 
NN0  |->  ( ( V ClWWalksN  E ) `  n
) ) `  P
) )
98fveq2d 5700 . . 3  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( # `  ( ( V ClWWalksN  E ) `  P
) )  =  (
# `  ( (
n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n
) ) `  P
) ) )
109oveq1d 6111 . 2  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( # `  (
( V ClWWalksN  E ) `  P ) )  mod 
P )  =  ( ( # `  (
( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) ) `
 P ) )  mod  P ) )
11 simpr 461 . . . . . 6  |-  ( ( V  =/=  (/)  /\  V  e.  Fin )  ->  V  e.  Fin )
1211anim2i 569 . . . . 5  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin ) )  -> 
( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  V  e.  Fin ) )
13 df-3an 967 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  <->  ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  V  e.  Fin )
)
1412, 13sylibr 212 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin ) )  -> 
( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin ) )
15143adant3 1008 . . 3  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin ) )
16 simp3 990 . . 3  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )
17 fveq2 5696 . . . . 5  |-  ( n  =  m  ->  (
( V ClWWalksN  E ) `  n )  =  ( ( V ClWWalksN  E ) `  m ) )
1817cbvmptv 4388 . . . 4  |-  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n
) )  =  ( m  e.  NN0  |->  ( ( V ClWWalksN  E ) `  m
) )
19 fveq1 5695 . . . . . . . 8  |-  ( p  =  q  ->  (
p `  0 )  =  ( q ` 
0 ) )
2019eqeq1d 2451 . . . . . . 7  |-  ( p  =  q  ->  (
( p `  0
)  =  v  <->  ( q `  0 )  =  v ) )
2120cbvrabv 2976 . . . . . 6  |-  { p  e.  ( ( n  e. 
NN0  |->  ( ( V ClWWalksN  E ) `  n
) ) `  m
)  |  ( p `
 0 )  =  v }  =  {
q  e.  ( ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n
) ) `  m
)  |  ( q `
 0 )  =  v }
2221a1i 11 . . . . 5  |-  ( ( v  e.  V  /\  m  e.  NN0 )  ->  { p  e.  (
( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) ) `
 m )  |  ( p `  0
)  =  v }  =  { q  e.  ( ( n  e. 
NN0  |->  ( ( V ClWWalksN  E ) `  n
) ) `  m
)  |  ( q `
 0 )  =  v } )
2322mpt2eq3ia 6156 . . . 4  |-  ( v  e.  V ,  m  e.  NN0  |->  { p  e.  ( ( n  e. 
NN0  |->  ( ( V ClWWalksN  E ) `  n
) ) `  m
)  |  ( p `
 0 )  =  v } )  =  ( v  e.  V ,  m  e.  NN0  |->  { q  e.  ( ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) ) `
 m )  |  ( q `  0
)  =  v } )
2418, 23numclwwlk6 30711 . . 3  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( # `  (
( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) ) `
 P ) )  mod  P )  =  ( ( # `  V
)  mod  P )
)
2515, 16, 24syl2anc 661 . 2  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( # `  (
( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) ) `
 P ) )  mod  P )  =  ( ( # `  V
)  mod  P )
)
26 simp2 989 . . . . . 6  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( V  =/=  (/)  /\  V  e.  Fin ) )
2726ancomd 451 . . . . 5  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( V  e.  Fin  /\  V  =/=  (/) ) )
28 simp1 988 . . . . . 6  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E ) )
2928ancomd 451 . . . . 5  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( V FriendGrph  E  /\  <. V ,  E >. RegUSGrph  K ) )
30 frrusgraord 30669 . . . . 5  |-  ( ( V  e.  Fin  /\  V  =/=  (/) )  ->  (
( V FriendGrph  E  /\  <. V ,  E >. RegUSGrph  K )  ->  ( # `  V
)  =  ( ( K  x.  ( K  -  1 ) )  +  1 ) ) )
3127, 29, 30sylc 60 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( # `  V )  =  ( ( K  x.  ( K  - 
1 ) )  +  1 ) )
3231oveq1d 6111 . . 3  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( # `  V
)  mod  P )  =  ( ( ( K  x.  ( K  -  1 ) )  +  1 )  mod 
P ) )
33 rusgraprop 30551 . . . . . . 7  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. x  e.  V  (
( V VDeg  E ) `  x )  =  K ) )
34 nn0cn 10594 . . . . . . . . . . . . . . . 16  |-  ( K  e.  NN0  ->  K  e.  CC )
35 cnm1cn 30172 . . . . . . . . . . . . . . . . 17  |-  ( K  e.  CC  ->  ( K  -  1 )  e.  CC )
3634, 35syl 16 . . . . . . . . . . . . . . . 16  |-  ( K  e.  NN0  ->  ( K  -  1 )  e.  CC )
3734, 36mulcomd 9412 . . . . . . . . . . . . . . 15  |-  ( K  e.  NN0  ->  ( K  x.  ( K  - 
1 ) )  =  ( ( K  - 
1 )  x.  K
) )
3837oveq1d 6111 . . . . . . . . . . . . . 14  |-  ( K  e.  NN0  ->  ( ( K  x.  ( K  -  1 ) )  mod  P )  =  ( ( ( K  -  1 )  x.  K )  mod  P
) )
3938adantr 465 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( ( K  x.  ( K  -  1 ) )  mod  P )  =  ( ( ( K  -  1 )  x.  K )  mod  P
) )
401ad2antrl 727 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  P  e.  NN )
41 nn0z 10674 . . . . . . . . . . . . . . . . 17  |-  ( K  e.  NN0  ->  K  e.  ZZ )
42 peano2zm 10693 . . . . . . . . . . . . . . . . 17  |-  ( K  e.  ZZ  ->  ( K  -  1 )  e.  ZZ )
4341, 42syl 16 . . . . . . . . . . . . . . . 16  |-  ( K  e.  NN0  ->  ( K  -  1 )  e.  ZZ )
4443adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( K  -  1 )  e.  ZZ )
4541adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  K  e.  ZZ )
4640, 44, 453jca 1168 . . . . . . . . . . . . . 14  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( P  e.  NN  /\  ( K  -  1 )  e.  ZZ  /\  K  e.  ZZ ) )
47 simprr 756 . . . . . . . . . . . . . 14  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  P  ||  ( K  -  1 ) )
48 mulmoddvds 30251 . . . . . . . . . . . . . 14  |-  ( ( P  e.  NN  /\  ( K  -  1
)  e.  ZZ  /\  K  e.  ZZ )  ->  ( P  ||  ( K  -  1 )  ->  ( ( ( K  -  1 )  x.  K )  mod 
P )  =  0 ) )
4946, 47, 48sylc 60 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( (
( K  -  1 )  x.  K )  mod  P )  =  0 )
5039, 49eqtrd 2475 . . . . . . . . . . . 12  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( ( K  x.  ( K  -  1 ) )  mod  P )  =  0 )
511nnred 10342 . . . . . . . . . . . . . . 15  |-  ( P  e.  Prime  ->  P  e.  RR )
52 prmgt1 13787 . . . . . . . . . . . . . . 15  |-  ( P  e.  Prime  ->  1  < 
P )
5351, 52jca 532 . . . . . . . . . . . . . 14  |-  ( P  e.  Prime  ->  ( P  e.  RR  /\  1  <  P ) )
5453ad2antrl 727 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( P  e.  RR  /\  1  < 
P ) )
55 1mod 11745 . . . . . . . . . . . . 13  |-  ( ( P  e.  RR  /\  1  <  P )  -> 
( 1  mod  P
)  =  1 )
5654, 55syl 16 . . . . . . . . . . . 12  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( 1  mod  P )  =  1 )
5750, 56oveq12d 6114 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( (
( K  x.  ( K  -  1 ) )  mod  P )  +  ( 1  mod 
P ) )  =  ( 0  +  1 ) )
5857oveq1d 6111 . . . . . . . . . 10  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( (
( ( K  x.  ( K  -  1
) )  mod  P
)  +  ( 1  mod  P ) )  mod  P )  =  ( ( 0  +  1 )  mod  P
) )
59 nn0re 10593 . . . . . . . . . . . . 13  |-  ( K  e.  NN0  ->  K  e.  RR )
60 peano2rem 9680 . . . . . . . . . . . . . 14  |-  ( K  e.  RR  ->  ( K  -  1 )  e.  RR )
6159, 60syl 16 . . . . . . . . . . . . 13  |-  ( K  e.  NN0  ->  ( K  -  1 )  e.  RR )
6259, 61remulcld 9419 . . . . . . . . . . . 12  |-  ( K  e.  NN0  ->  ( K  x.  ( K  - 
1 ) )  e.  RR )
6362adantr 465 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( K  x.  ( K  -  1 ) )  e.  RR )
64 1re 9390 . . . . . . . . . . . 12  |-  1  e.  RR
6564a1i 11 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  1  e.  RR )
661nnrpd 11031 . . . . . . . . . . . 12  |-  ( P  e.  Prime  ->  P  e.  RR+ )
6766ad2antrl 727 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  P  e.  RR+ )
68 modaddabs 11751 . . . . . . . . . . 11  |-  ( ( ( K  x.  ( K  -  1 ) )  e.  RR  /\  1  e.  RR  /\  P  e.  RR+ )  ->  (
( ( ( K  x.  ( K  - 
1 ) )  mod 
P )  +  ( 1  mod  P ) )  mod  P )  =  ( ( ( K  x.  ( K  -  1 ) )  +  1 )  mod 
P ) )
6963, 65, 67, 68syl3anc 1218 . . . . . . . . . 10  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( (
( ( K  x.  ( K  -  1
) )  mod  P
)  +  ( 1  mod  P ) )  mod  P )  =  ( ( ( K  x.  ( K  - 
1 ) )  +  1 )  mod  P
) )
70 0p1e1 10438 . . . . . . . . . . . 12  |-  ( 0  +  1 )  =  1
7170oveq1i 6106 . . . . . . . . . . 11  |-  ( ( 0  +  1 )  mod  P )  =  ( 1  mod  P
)
7251, 52, 55syl2anc 661 . . . . . . . . . . . 12  |-  ( P  e.  Prime  ->  ( 1  mod  P )  =  1 )
7372ad2antrl 727 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( 1  mod  P )  =  1 )
7471, 73syl5eq 2487 . . . . . . . . . 10  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( (
0  +  1 )  mod  P )  =  1 )
7558, 69, 743eqtr3d 2483 . . . . . . . . 9  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( (
( K  x.  ( K  -  1 ) )  +  1 )  mod  P )  =  1 )
7675ex 434 . . . . . . . 8  |-  ( K  e.  NN0  ->  ( ( P  e.  Prime  /\  P  ||  ( K  -  1 ) )  ->  (
( ( K  x.  ( K  -  1
) )  +  1 )  mod  P )  =  1 ) )
77763ad2ant2 1010 . . . . . . 7  |-  ( ( V USGrph  E  /\  K  e. 
NN0  /\  A. x  e.  V  ( ( V VDeg  E ) `  x
)  =  K )  ->  ( ( P  e.  Prime  /\  P  ||  ( K  -  1
) )  ->  (
( ( K  x.  ( K  -  1
) )  +  1 )  mod  P )  =  1 ) )
7833, 77syl 16 . . . . . 6  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( ( P  e. 
Prime  /\  P  ||  ( K  -  1 ) )  ->  ( (
( K  x.  ( K  -  1 ) )  +  1 )  mod  P )  =  1 ) )
7978adantr 465 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  -> 
( ( P  e. 
Prime  /\  P  ||  ( K  -  1 ) )  ->  ( (
( K  x.  ( K  -  1 ) )  +  1 )  mod  P )  =  1 ) )
8079imp 429 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( (
( K  x.  ( K  -  1 ) )  +  1 )  mod  P )  =  1 )
81803adant2 1007 . . 3  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( ( K  x.  ( K  - 
1 ) )  +  1 )  mod  P
)  =  1 )
8232, 81eqtrd 2475 . 2  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( # `  V
)  mod  P )  =  1 )
8310, 25, 823eqtrd 2479 1  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( # `  (
( V ClWWalksN  E ) `  P ) )  mod 
P )  =  1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   {crab 2724   (/)c0 3642   <.cop 3888   class class class wbr 4297    e. cmpt 4355   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   Fincfn 7315   CCcc 9285   RRcr 9286   0cc0 9287   1c1 9288    + caddc 9290    x. cmul 9292    < clt 9423    - cmin 9600   NNcn 10327   NN0cn0 10584   ZZcz 10651   RR+crp 10996    mod cmo 11713   #chash 12108    || cdivides 13540   Primecprime 13768   USGrph cusg 23269   VDeg cvdg 23568   ClWWalksN cclwwlkn 30419   RegUSGrph crusgra 30545   FriendGrph cfrgra 30585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-ot 3891  df-uni 4097  df-int 4134  df-iun 4178  df-disj 4268  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-oi 7729  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-xadd 11095  df-fz 11443  df-fzo 11554  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-hash 12109  df-word 12234  df-lsw 12235  df-concat 12236  df-s1 12237  df-substr 12238  df-s2 12480  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-sum 13169  df-dvds 13541  df-gcd 13696  df-prm 13769  df-phi 13846  df-usgra 23271  df-nbgra 23337  df-wlk 23420  df-trail 23421  df-pth 23422  df-spth 23423  df-wlkon 23426  df-spthon 23429  df-vdgr 23569  df-wwlk 30318  df-wwlkn 30319  df-2wlkonot 30382  df-2spthonot 30384  df-2spthsot 30385  df-clwwlk 30421  df-clwwlkn 30422  df-rgra 30546  df-rusgra 30547  df-frgra 30586
This theorem is referenced by:  frgrareggt1  30714
  Copyright terms: Public domain W3C validator