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Theorem numclwwlk7 25687
Description: Statement 14 in [Huneke] p. 2: "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 25643 or frrusgraord 25644, 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 25567, as well as k-regular (for any k), see 0vgrargra 25510, but has no closed walk, see clwlk0 25335, 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 14596 . . . . . . . . 9  |-  ( P  e.  Prime  ->  P  e.  NN )
21nnnn0d 10925 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e. 
NN0 )
32adantr 466 . . . . . . 7  |-  ( ( P  e.  Prime  /\  P  ||  ( K  -  1 ) )  ->  P  e.  NN0 )
433ad2ant3 1028 . . . . . 6  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  P  e.  NN0 )
5 eqid 2429 . . . . . . 7  |-  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n
) )  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n
) )
65numclwwlkfvc 25650 . . . . . 6  |-  ( P  e.  NN0  ->  ( ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n
) ) `  P
)  =  ( ( V ClWWalksN  E ) `  P
) )
74, 6syl 17 . . . . 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 2437 . . . 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 5885 . . 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 6320 . 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 462 . . . . 5  |-  ( ( V  =/=  (/)  /\  V  e.  Fin )  ->  V  e.  Fin )
1211anim2i 571 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin ) )  -> 
( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  V  e.  Fin ) )
13 df-3an 984 . . . 4  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  <->  ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  V  e.  Fin )
)
1412, 13sylibr 215 . . 3  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin ) )  -> 
( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin ) )
15 fveq2 5881 . . . . 5  |-  ( n  =  m  ->  (
( V ClWWalksN  E ) `  n )  =  ( ( V ClWWalksN  E ) `  m ) )
1615cbvmptv 4518 . . . 4  |-  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n
) )  =  ( m  e.  NN0  |->  ( ( V ClWWalksN  E ) `  m
) )
17 fveq1 5880 . . . . . . . 8  |-  ( p  =  q  ->  (
p `  0 )  =  ( q ` 
0 ) )
1817eqeq1d 2431 . . . . . . 7  |-  ( p  =  q  ->  (
( p `  0
)  =  v  <->  ( q `  0 )  =  v ) )
1918cbvrabv 3086 . . . . . 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 }
2019a1i 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 } )
2120mpt2eq3ia 6370 . . . 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 } )
2216, 21numclwwlk6 25686 . . 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 )
)
2314, 22stoic3 1656 . 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 )
)
24 simp2 1006 . . . . . 6  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( V  =/=  (/)  /\  V  e.  Fin ) )
2524ancomd 452 . . . . 5  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( V  e.  Fin  /\  V  =/=  (/) ) )
26 simp1 1005 . . . . . 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 ) )
2726ancomd 452 . . . . 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 ) )
28 frrusgraord 25644 . . . . 5  |-  ( ( V  e.  Fin  /\  V  =/=  (/) )  ->  (
( V FriendGrph  E  /\  <. V ,  E >. RegUSGrph  K )  ->  ( # `  V
)  =  ( ( K  x.  ( K  -  1 ) )  +  1 ) ) )
2925, 27, 28sylc 62 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( # `  V )  =  ( ( K  x.  ( K  - 
1 ) )  +  1 ) )
3029oveq1d 6320 . . 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 ) )
31 rusgraprop 25502 . . . . . . 7  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. x  e.  V  (
( V VDeg  E ) `  x )  =  K ) )
32 nn0cn 10879 . . . . . . . . . . . . . . . 16  |-  ( K  e.  NN0  ->  K  e.  CC )
33 peano2cnm 9939 . . . . . . . . . . . . . . . . 17  |-  ( K  e.  CC  ->  ( K  -  1 )  e.  CC )
3432, 33syl 17 . . . . . . . . . . . . . . . 16  |-  ( K  e.  NN0  ->  ( K  -  1 )  e.  CC )
3532, 34mulcomd 9663 . . . . . . . . . . . . . . 15  |-  ( K  e.  NN0  ->  ( K  x.  ( K  - 
1 ) )  =  ( ( K  - 
1 )  x.  K
) )
3635oveq1d 6320 . . . . . . . . . . . . . 14  |-  ( K  e.  NN0  ->  ( ( K  x.  ( K  -  1 ) )  mod  P )  =  ( ( ( K  -  1 )  x.  K )  mod  P
) )
3736adantr 466 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( ( K  x.  ( K  -  1 ) )  mod  P )  =  ( ( ( K  -  1 )  x.  K )  mod  P
) )
381ad2antrl 732 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  P  e.  NN )
39 nn0z 10960 . . . . . . . . . . . . . . . . 17  |-  ( K  e.  NN0  ->  K  e.  ZZ )
40 peano2zm 10980 . . . . . . . . . . . . . . . . 17  |-  ( K  e.  ZZ  ->  ( K  -  1 )  e.  ZZ )
4139, 40syl 17 . . . . . . . . . . . . . . . 16  |-  ( K  e.  NN0  ->  ( K  -  1 )  e.  ZZ )
4241adantr 466 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( K  -  1 )  e.  ZZ )
4339adantr 466 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  K  e.  ZZ )
4438, 42, 433jca 1185 . . . . . . . . . . . . . 14  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( P  e.  NN  /\  ( K  -  1 )  e.  ZZ  /\  K  e.  ZZ ) )
45 simprr 764 . . . . . . . . . . . . . 14  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  P  ||  ( K  -  1 ) )
46 mulmoddvds 14341 . . . . . . . . . . . . . 14  |-  ( ( P  e.  NN  /\  ( K  -  1
)  e.  ZZ  /\  K  e.  ZZ )  ->  ( P  ||  ( K  -  1 )  ->  ( ( ( K  -  1 )  x.  K )  mod 
P )  =  0 ) )
4744, 45, 46sylc 62 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( (
( K  -  1 )  x.  K )  mod  P )  =  0 )
4837, 47eqtrd 2470 . . . . . . . . . . . 12  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( ( K  x.  ( K  -  1 ) )  mod  P )  =  0 )
491nnred 10624 . . . . . . . . . . . . . . 15  |-  ( P  e.  Prime  ->  P  e.  RR )
50 prmgt1 14614 . . . . . . . . . . . . . . 15  |-  ( P  e.  Prime  ->  1  < 
P )
5149, 50jca 534 . . . . . . . . . . . . . 14  |-  ( P  e.  Prime  ->  ( P  e.  RR  /\  1  <  P ) )
5251ad2antrl 732 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( P  e.  RR  /\  1  < 
P ) )
53 1mod 12126 . . . . . . . . . . . . 13  |-  ( ( P  e.  RR  /\  1  <  P )  -> 
( 1  mod  P
)  =  1 )
5452, 53syl 17 . . . . . . . . . . . 12  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( 1  mod  P )  =  1 )
5548, 54oveq12d 6323 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( (
( K  x.  ( K  -  1 ) )  mod  P )  +  ( 1  mod 
P ) )  =  ( 0  +  1 ) )
5655oveq1d 6320 . . . . . . . . . 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
) )
57 nn0re 10878 . . . . . . . . . . . . 13  |-  ( K  e.  NN0  ->  K  e.  RR )
58 peano2rem 9940 . . . . . . . . . . . . . 14  |-  ( K  e.  RR  ->  ( K  -  1 )  e.  RR )
5957, 58syl 17 . . . . . . . . . . . . 13  |-  ( K  e.  NN0  ->  ( K  -  1 )  e.  RR )
6057, 59remulcld 9670 . . . . . . . . . . . 12  |-  ( K  e.  NN0  ->  ( K  x.  ( K  - 
1 ) )  e.  RR )
6160adantr 466 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( K  x.  ( K  -  1 ) )  e.  RR )
62 1red 9657 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  1  e.  RR )
631nnrpd 11339 . . . . . . . . . . . 12  |-  ( P  e.  Prime  ->  P  e.  RR+ )
6463ad2antrl 732 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  P  e.  RR+ )
65 modaddabs 12132 . . . . . . . . . . 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 ) )
6661, 62, 64, 65syl3anc 1264 . . . . . . . . . 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
) )
67 0p1e1 10721 . . . . . . . . . . . 12  |-  ( 0  +  1 )  =  1
6867oveq1i 6315 . . . . . . . . . . 11  |-  ( ( 0  +  1 )  mod  P )  =  ( 1  mod  P
)
6949, 50, 53syl2anc 665 . . . . . . . . . . . 12  |-  ( P  e.  Prime  ->  ( 1  mod  P )  =  1 )
7069ad2antrl 732 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( 1  mod  P )  =  1 )
7168, 70syl5eq 2482 . . . . . . . . . 10  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( (
0  +  1 )  mod  P )  =  1 )
7256, 66, 713eqtr3d 2478 . . . . . . . . 9  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( (
( K  x.  ( K  -  1 ) )  +  1 )  mod  P )  =  1 )
7372ex 435 . . . . . . . 8  |-  ( K  e.  NN0  ->  ( ( P  e.  Prime  /\  P  ||  ( K  -  1 ) )  ->  (
( ( K  x.  ( K  -  1
) )  +  1 )  mod  P )  =  1 ) )
74733ad2ant2 1027 . . . . . . 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 ) )
7531, 74syl 17 . . . . . 6  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( ( P  e. 
Prime  /\  P  ||  ( K  -  1 ) )  ->  ( (
( K  x.  ( K  -  1 ) )  +  1 )  mod  P )  =  1 ) )
7675adantr 466 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  -> 
( ( P  e. 
Prime  /\  P  ||  ( K  -  1 ) )  ->  ( (
( K  x.  ( K  -  1 ) )  +  1 )  mod  P )  =  1 ) )
7776imp 430 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( (
( K  x.  ( K  -  1 ) )  +  1 )  mod  P )  =  1 )
78773adant2 1024 . . 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 )
7930, 78eqtrd 2470 . 2  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( # `  V
)  mod  P )  =  1 )
8010, 23, 793eqtrd 2474 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 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   {crab 2786   (/)c0 3767   <.cop 4008   class class class wbr 4426    |-> cmpt 4484   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   Fincfn 7577   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539    + caddc 9541    x. cmul 9543    < clt 9674    - cmin 9859   NNcn 10609   NN0cn0 10869   ZZcz 10937   RR+crp 11302    mod cmo 12093   #chash 12512    || cdvds 14283   Primecprime 14593   USGrph cusg 24903   ClWWalksN cclwwlkn 25322   VDeg cvdg 25466   RegUSGrph crusgra 25496   FriendGrph cfrgra 25561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-ot 4011  df-uni 4223  df-int 4259  df-iun 4304  df-disj 4398  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  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 11783  df-fzo 11914  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-hash 12513  df-word 12651  df-lsw 12652  df-concat 12653  df-s1 12654  df-substr 12655  df-s2 12929  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-sum 13731  df-dvds 14284  df-gcd 14443  df-prm 14594  df-phi 14683  df-usgra 24906  df-nbgra 24993  df-wlk 25081  df-trail 25082  df-pth 25083  df-spth 25084  df-wlkon 25087  df-spthon 25090  df-wwlk 25252  df-wwlkn 25253  df-clwwlk 25324  df-clwwlkn 25325  df-2wlkonot 25431  df-2spthonot 25433  df-2spthsot 25434  df-vdgr 25467  df-rgra 25497  df-rusgra 25498  df-frgra 25562
This theorem is referenced by:  frgrareggt1  25689
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