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Theorem numclwwlk7 24819
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 24775 or frrusgraord 24776, 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 24698, as well as k-regular (for any k), see 0vgrargra 24641, but has no closed walk, see clwlk0 24466, 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 14079 . . . . . . . . 9  |-  ( P  e.  Prime  ->  P  e.  NN )
21nnnn0d 10852 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e. 
NN0 )
32adantr 465 . . . . . . 7  |-  ( ( P  e.  Prime  /\  P  ||  ( K  -  1 ) )  ->  P  e.  NN0 )
433ad2ant3 1019 . . . . . 6  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  P  e.  NN0 )
5 eqid 2467 . . . . . . 7  |-  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n
) )  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n
) )
65numclwwlkfvc 24782 . . . . . 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 2475 . . . 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 5870 . . 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 6299 . 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 975 . . . . 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 1016 . . 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 998 . . 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 5866 . . . . 5  |-  ( n  =  m  ->  (
( V ClWWalksN  E ) `  n )  =  ( ( V ClWWalksN  E ) `  m ) )
1817cbvmptv 4538 . . . 4  |-  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n
) )  =  ( m  e.  NN0  |->  ( ( V ClWWalksN  E ) `  m
) )
19 fveq1 5865 . . . . . . . 8  |-  ( p  =  q  ->  (
p `  0 )  =  ( q ` 
0 ) )
2019eqeq1d 2469 . . . . . . 7  |-  ( p  =  q  ->  (
( p `  0
)  =  v  <->  ( q `  0 )  =  v ) )
2120cbvrabv 3112 . . . . . 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 6346 . . . 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 24818 . . 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 997 . . . . . 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 996 . . . . . 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 24776 . . . . 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 6299 . . 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 24633 . . . . . . 7  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. x  e.  V  (
( V VDeg  E ) `  x )  =  K ) )
34 nn0cn 10805 . . . . . . . . . . . . . . . 16  |-  ( K  e.  NN0  ->  K  e.  CC )
35 peano2cnm 9885 . . . . . . . . . . . . . . . . 17  |-  ( K  e.  CC  ->  ( K  -  1 )  e.  CC )
3634, 35syl 16 . . . . . . . . . . . . . . . 16  |-  ( K  e.  NN0  ->  ( K  -  1 )  e.  CC )
3734, 36mulcomd 9617 . . . . . . . . . . . . . . 15  |-  ( K  e.  NN0  ->  ( K  x.  ( K  - 
1 ) )  =  ( ( K  - 
1 )  x.  K
) )
3837oveq1d 6299 . . . . . . . . . . . . . 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 10887 . . . . . . . . . . . . . . . . 17  |-  ( K  e.  NN0  ->  K  e.  ZZ )
42 peano2zm 10906 . . . . . . . . . . . . . . . . 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 1176 . . . . . . . . . . . . . 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 13903 . . . . . . . . . . . . . 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 2508 . . . . . . . . . . . 12  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( ( K  x.  ( K  -  1 ) )  mod  P )  =  0 )
511nnred 10551 . . . . . . . . . . . . . . 15  |-  ( P  e.  Prime  ->  P  e.  RR )
52 prmgt1 14095 . . . . . . . . . . . . . . 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 11996 . . . . . . . . . . . . 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 6302 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( (
( K  x.  ( K  -  1 ) )  mod  P )  +  ( 1  mod 
P ) )  =  ( 0  +  1 ) )
5857oveq1d 6299 . . . . . . . . . 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 10804 . . . . . . . . . . . . 13  |-  ( K  e.  NN0  ->  K  e.  RR )
60 peano2rem 9886 . . . . . . . . . . . . . 14  |-  ( K  e.  RR  ->  ( K  -  1 )  e.  RR )
6159, 60syl 16 . . . . . . . . . . . . 13  |-  ( K  e.  NN0  ->  ( K  -  1 )  e.  RR )
6259, 61remulcld 9624 . . . . . . . . . . . 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 9595 . . . . . . . . . . . 12  |-  1  e.  RR
6564a1i 11 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  1  e.  RR )
661nnrpd 11255 . . . . . . . . . . . 12  |-  ( P  e.  Prime  ->  P  e.  RR+ )
6766ad2antrl 727 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  P  e.  RR+ )
68 modaddabs 12002 . . . . . . . . . . 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 1228 . . . . . . . . . 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 10647 . . . . . . . . . . . 12  |-  ( 0  +  1 )  =  1
7170oveq1i 6294 . . . . . . . . . . 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 2520 . . . . . . . . . 10  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( (
0  +  1 )  mod  P )  =  1 )
7558, 69, 743eqtr3d 2516 . . . . . . . . 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 1018 . . . . . . 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 1015 . . 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 2508 . 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 2512 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   {crab 2818   (/)c0 3785   <.cop 4033   class class class wbr 4447    |-> cmpt 4505   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   Fincfn 7516   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493    + caddc 9495    x. cmul 9497    < clt 9628    - cmin 9805   NNcn 10536   NN0cn0 10795   ZZcz 10864   RR+crp 11220    mod cmo 11964   #chash 12373    || cdivides 13847   Primecprime 14076   USGrph cusg 24034   ClWWalksN cclwwlkn 24453   VDeg cvdg 24597   RegUSGrph crusgra 24627   FriendGrph cfrgra 24692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-xadd 11319  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-hash 12374  df-word 12508  df-lsw 12509  df-concat 12510  df-s1 12511  df-substr 12512  df-s2 12776  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-sum 13472  df-dvds 13848  df-gcd 14004  df-prm 14077  df-phi 14155  df-usgra 24037  df-nbgra 24124  df-wlk 24212  df-trail 24213  df-pth 24214  df-spth 24215  df-wlkon 24218  df-spthon 24221  df-wwlk 24383  df-wwlkn 24384  df-clwwlk 24455  df-clwwlkn 24456  df-2wlkonot 24562  df-2spthonot 24564  df-2spthsot 24565  df-vdgr 24598  df-rgra 24628  df-rusgra 24629  df-frgra 24693
This theorem is referenced by:  frgrareggt1  24821
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