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Theorem numclwwlk7 25921
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 25877 or frrusgraord 25878, 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 25801, as well as k-regular (for any k), see 0vgrargra 25744, but has no closed walk, see clwlk0 25569, 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 14704 . . . . . . . . 9  |-  ( P  e.  Prime  ->  P  e.  NN )
21nnnn0d 10949 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e. 
NN0 )
32adantr 472 . . . . . . 7  |-  ( ( P  e.  Prime  /\  P  ||  ( K  -  1 ) )  ->  P  e.  NN0 )
433ad2ant3 1053 . . . . . 6  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  P  e.  NN0 )
5 eqid 2471 . . . . . . 7  |-  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n
) )  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n
) )
65numclwwlkfvc 25884 . . . . . 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 2477 . . . 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 5883 . . 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 6323 . 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 468 . . . . 5  |-  ( ( V  =/=  (/)  /\  V  e.  Fin )  ->  V  e.  Fin )
1211anim2i 579 . . . 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 1009 . . . 4  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  <->  ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  V  e.  Fin )
)
1412, 13sylibr 217 . . 3  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin ) )  -> 
( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin ) )
15 fveq2 5879 . . . . 5  |-  ( n  =  m  ->  (
( V ClWWalksN  E ) `  n )  =  ( ( V ClWWalksN  E ) `  m ) )
1615cbvmptv 4488 . . . 4  |-  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n
) )  =  ( m  e.  NN0  |->  ( ( V ClWWalksN  E ) `  m
) )
17 fveq1 5878 . . . . . . . 8  |-  ( p  =  q  ->  (
p `  0 )  =  ( q ` 
0 ) )
1817eqeq1d 2473 . . . . . . 7  |-  ( p  =  q  ->  (
( p `  0
)  =  v  <->  ( q `  0 )  =  v ) )
1918cbvrabv 3030 . . . . . 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 6375 . . . 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 25920 . . 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 1668 . 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 1031 . . . . . 6  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( V  =/=  (/)  /\  V  e.  Fin ) )
2524ancomd 458 . . . . 5  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( V  e.  Fin  /\  V  =/=  (/) ) )
26 simp1 1030 . . . . . 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 458 . . . . 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 25878 . . . . 5  |-  ( ( V  e.  Fin  /\  V  =/=  (/) )  ->  (
( V FriendGrph  E  /\  <. V ,  E >. RegUSGrph  K )  ->  ( # `  V
)  =  ( ( K  x.  ( K  -  1 ) )  +  1 ) ) )
2925, 27, 28sylc 61 . . . 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 6323 . . 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 25736 . . . . . . 7  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. x  e.  V  (
( V VDeg  E ) `  x )  =  K ) )
32 nn0cn 10903 . . . . . . . . . . . . . . . 16  |-  ( K  e.  NN0  ->  K  e.  CC )
33 peano2cnm 9960 . . . . . . . . . . . . . . . . 17  |-  ( K  e.  CC  ->  ( K  -  1 )  e.  CC )
3432, 33syl 17 . . . . . . . . . . . . . . . 16  |-  ( K  e.  NN0  ->  ( K  -  1 )  e.  CC )
3532, 34mulcomd 9682 . . . . . . . . . . . . . . 15  |-  ( K  e.  NN0  ->  ( K  x.  ( K  - 
1 ) )  =  ( ( K  - 
1 )  x.  K
) )
3635oveq1d 6323 . . . . . . . . . . . . . 14  |-  ( K  e.  NN0  ->  ( ( K  x.  ( K  -  1 ) )  mod  P )  =  ( ( ( K  -  1 )  x.  K )  mod  P
) )
3736adantr 472 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( ( K  x.  ( K  -  1 ) )  mod  P )  =  ( ( ( K  -  1 )  x.  K )  mod  P
) )
381ad2antrl 742 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  P  e.  NN )
39 nn0z 10984 . . . . . . . . . . . . . . . . 17  |-  ( K  e.  NN0  ->  K  e.  ZZ )
40 peano2zm 11004 . . . . . . . . . . . . . . . . 17  |-  ( K  e.  ZZ  ->  ( K  -  1 )  e.  ZZ )
4139, 40syl 17 . . . . . . . . . . . . . . . 16  |-  ( K  e.  NN0  ->  ( K  -  1 )  e.  ZZ )
4241adantr 472 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( K  -  1 )  e.  ZZ )
4339adantr 472 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  K  e.  ZZ )
4438, 42, 433jca 1210 . . . . . . . . . . . . . 14  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( P  e.  NN  /\  ( K  -  1 )  e.  ZZ  /\  K  e.  ZZ ) )
45 simprr 774 . . . . . . . . . . . . . 14  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  P  ||  ( K  -  1 ) )
46 mulmoddvds 14441 . . . . . . . . . . . . . 14  |-  ( ( P  e.  NN  /\  ( K  -  1
)  e.  ZZ  /\  K  e.  ZZ )  ->  ( P  ||  ( K  -  1 )  ->  ( ( ( K  -  1 )  x.  K )  mod 
P )  =  0 ) )
4744, 45, 46sylc 61 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( (
( K  -  1 )  x.  K )  mod  P )  =  0 )
4837, 47eqtrd 2505 . . . . . . . . . . . 12  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( ( K  x.  ( K  -  1 ) )  mod  P )  =  0 )
491nnred 10646 . . . . . . . . . . . . . . 15  |-  ( P  e.  Prime  ->  P  e.  RR )
50 prmgt1 14722 . . . . . . . . . . . . . . 15  |-  ( P  e.  Prime  ->  1  < 
P )
5149, 50jca 541 . . . . . . . . . . . . . 14  |-  ( P  e.  Prime  ->  ( P  e.  RR  /\  1  <  P ) )
5251ad2antrl 742 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( P  e.  RR  /\  1  < 
P ) )
53 1mod 12162 . . . . . . . . . . . . 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 6326 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( (
( K  x.  ( K  -  1 ) )  mod  P )  +  ( 1  mod 
P ) )  =  ( 0  +  1 ) )
5655oveq1d 6323 . . . . . . . . . 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 10902 . . . . . . . . . . . . 13  |-  ( K  e.  NN0  ->  K  e.  RR )
58 peano2rem 9961 . . . . . . . . . . . . . 14  |-  ( K  e.  RR  ->  ( K  -  1 )  e.  RR )
5957, 58syl 17 . . . . . . . . . . . . 13  |-  ( K  e.  NN0  ->  ( K  -  1 )  e.  RR )
6057, 59remulcld 9689 . . . . . . . . . . . 12  |-  ( K  e.  NN0  ->  ( K  x.  ( K  - 
1 ) )  e.  RR )
6160adantr 472 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( K  x.  ( K  -  1 ) )  e.  RR )
62 1red 9676 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  1  e.  RR )
631nnrpd 11362 . . . . . . . . . . . 12  |-  ( P  e.  Prime  ->  P  e.  RR+ )
6463ad2antrl 742 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  P  e.  RR+ )
65 modaddabs 12168 . . . . . . . . . . 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 1292 . . . . . . . . . 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 10743 . . . . . . . . . . . 12  |-  ( 0  +  1 )  =  1
6867oveq1i 6318 . . . . . . . . . . 11  |-  ( ( 0  +  1 )  mod  P )  =  ( 1  mod  P
)
6949, 50, 53syl2anc 673 . . . . . . . . . . . 12  |-  ( P  e.  Prime  ->  ( 1  mod  P )  =  1 )
7069ad2antrl 742 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( 1  mod  P )  =  1 )
7168, 70syl5eq 2517 . . . . . . . . . 10  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( (
0  +  1 )  mod  P )  =  1 )
7256, 66, 713eqtr3d 2513 . . . . . . . . 9  |-  ( ( K  e.  NN0  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( (
( K  x.  ( K  -  1 ) )  +  1 )  mod  P )  =  1 )
7372ex 441 . . . . . . . 8  |-  ( K  e.  NN0  ->  ( ( P  e.  Prime  /\  P  ||  ( K  -  1 ) )  ->  (
( ( K  x.  ( K  -  1
) )  +  1 )  mod  P )  =  1 ) )
74733ad2ant2 1052 . . . . . . 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 472 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  -> 
( ( P  e. 
Prime  /\  P  ||  ( K  -  1 ) )  ->  ( (
( K  x.  ( K  -  1 ) )  +  1 )  mod  P )  =  1 ) )
7776imp 436 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( (
( K  x.  ( K  -  1 ) )  +  1 )  mod  P )  =  1 )
78773adant2 1049 . . 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 2505 . 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 2509 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 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   {crab 2760   (/)c0 3722   <.cop 3965   class class class wbr 4395    |-> cmpt 4454   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   Fincfn 7587   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562    < clt 9693    - cmin 9880   NNcn 10631   NN0cn0 10893   ZZcz 10961   RR+crp 11325    mod cmo 12129   #chash 12553    || cdvds 14382   Primecprime 14701   USGrph cusg 25136   ClWWalksN cclwwlkn 25556   VDeg cvdg 25700   RegUSGrph crusgra 25730   FriendGrph cfrgra 25795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-ot 3968  df-uni 4191  df-int 4227  df-iun 4271  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-xadd 11433  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-word 12711  df-lsw 12712  df-concat 12713  df-s1 12714  df-substr 12715  df-s2 13003  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-dvds 14383  df-gcd 14548  df-prm 14702  df-phi 14793  df-usgra 25139  df-nbgra 25227  df-wlk 25315  df-trail 25316  df-pth 25317  df-spth 25318  df-wlkon 25321  df-spthon 25324  df-wwlk 25486  df-wwlkn 25487  df-clwwlk 25558  df-clwwlkn 25559  df-2wlkonot 25665  df-2spthonot 25667  df-2spthsot 25668  df-vdgr 25701  df-rgra 25731  df-rusgra 25732  df-frgra 25796
This theorem is referenced by:  frgrareggt1  25923
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