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Theorem 2pthon 23499
Description: A path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
Assertion
Ref Expression
2pthon  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( i  =/=  j  /\  ( E `
 i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C }
)  ->  { <. 0 ,  i >. ,  <. 1 ,  j >. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) )

Proof of Theorem 2pthon
StepHypRef Expression
1 simp2 989 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  A  =/=  C )
2 vex 2973 . . . . . . . . . 10  |-  i  e. 
_V
3 vex 2973 . . . . . . . . . 10  |-  j  e. 
_V
42, 3pm3.2i 455 . . . . . . . . 9  |-  ( i  e.  _V  /\  j  e.  _V )
5 eqid 2441 . . . . . . . . 9  |-  { <. 0 ,  i >. , 
<. 1 ,  j
>. }  =  { <. 0 ,  i >. , 
<. 1 ,  j
>. }
6 eqid 2441 . . . . . . . . 9  |-  { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }  =  { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }
74, 5, 6constr2trl 23496 . . . . . . . 8  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  -> 
( ( i  =/=  j  /\  ( E `
 i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C }
)  ->  { <. 0 ,  i >. ,  <. 1 ,  j >. }  ( V Trails  E ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) )
873adant3 1008 . . . . . . 7  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  A  =/=  C )  ->  (
( i  =/=  j  /\  ( E `  i
)  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } )  ->  { <. 0 ,  i >. , 
<. 1 ,  j
>. }  ( V Trails  E
) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) )
91, 8syl3an3 1253 . . . . . 6  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( i  =/=  j  /\  ( E `
 i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C }
)  ->  { <. 0 ,  i >. ,  <. 1 ,  j >. }  ( V Trails  E ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) )
109imp 429 . . . . 5  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  ->  { <. 0 ,  i
>. ,  <. 1 ,  j >. }  ( V Trails  E ) { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. } )
11 trliswlk 23436 . . . . 5  |-  ( {
<. 0 ,  i
>. ,  <. 1 ,  j >. }  ( V Trails  E ) { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }  ->  { <. 0 ,  i >. ,  <. 1 ,  j >. }  ( V Walks  E ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } )
1210, 11syl 16 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  ->  { <. 0 ,  i
>. ,  <. 1 ,  j >. }  ( V Walks 
E ) { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. } )
13 c0ex 9378 . . . . . . . . 9  |-  0  e.  _V
1413jctl 541 . . . . . . . 8  |-  ( A  e.  V  ->  (
0  e.  _V  /\  A  e.  V )
)
15143ad2ant1 1009 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( 0  e.  _V  /\  A  e.  V ) )
16 0ne1 10387 . . . . . . . 8  |-  0  =/=  1
17 0ne2 10531 . . . . . . . 8  |-  0  =/=  2
1816, 17pm3.2i 455 . . . . . . 7  |-  ( 0  =/=  1  /\  0  =/=  2 )
19 fvtp1g 5926 . . . . . . 7  |-  ( ( ( 0  e.  _V  /\  A  e.  V )  /\  ( 0  =/=  1  /\  0  =/=  2 ) )  -> 
( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  0
)  =  A )
2015, 18, 19sylancl 662 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  0
)  =  A )
21203ad2ant2 1010 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  0
)  =  A )
2221adantr 465 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  -> 
( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  0
)  =  A )
23 ax-1ne0 9349 . . . . . . . . 9  |-  1  =/=  0
2423nesymi 2646 . . . . . . . 8  |-  -.  0  =  1
2513, 2opth1 4563 . . . . . . . . 9  |-  ( <.
0 ,  i >.  =  <. 1 ,  j
>.  ->  0  =  1 )
2625necon3bi 2650 . . . . . . . 8  |-  ( -.  0  =  1  ->  <. 0 ,  i >.  =/=  <. 1 ,  j
>. )
2724, 26ax-mp 5 . . . . . . 7  |-  <. 0 ,  i >.  =/=  <. 1 ,  j >.
28 opex 4554 . . . . . . . 8  |-  <. 0 ,  i >.  e.  _V
29 opex 4554 . . . . . . . 8  |-  <. 1 ,  j >.  e.  _V
30 hashprg 12153 . . . . . . . 8  |-  ( (
<. 0 ,  i
>.  e.  _V  /\  <. 1 ,  j >.  e. 
_V )  ->  ( <. 0 ,  i >.  =/=  <. 1 ,  j
>. 
<->  ( # `  { <. 0 ,  i >. ,  <. 1 ,  j
>. } )  =  2 ) )
3128, 29, 30mp2an 672 . . . . . . 7  |-  ( <.
0 ,  i >.  =/=  <. 1 ,  j
>. 
<->  ( # `  { <. 0 ,  i >. ,  <. 1 ,  j
>. } )  =  2 )
3227, 31mpbi 208 . . . . . 6  |-  ( # `  { <. 0 ,  i
>. ,  <. 1 ,  j >. } )  =  2
3332fveq2i 5692 . . . . 5  |-  ( {
<. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  ( # `  { <. 0 ,  i
>. ,  <. 1 ,  j >. } ) )  =  ( { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. } `  2 )
34 2z 10676 . . . . . . . . . 10  |-  2  e.  ZZ
3534jctl 541 . . . . . . . . 9  |-  ( C  e.  V  ->  (
2  e.  ZZ  /\  C  e.  V )
)
36353ad2ant3 1011 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( 2  e.  ZZ  /\  C  e.  V ) )
37 1ne2 10532 . . . . . . . . 9  |-  1  =/=  2
3817, 37pm3.2i 455 . . . . . . . 8  |-  ( 0  =/=  2  /\  1  =/=  2 )
39 fvtp3g 5928 . . . . . . . 8  |-  ( ( ( 2  e.  ZZ  /\  C  e.  V )  /\  ( 0  =/=  2  /\  1  =/=  2 ) )  -> 
( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  2
)  =  C )
4036, 38, 39sylancl 662 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  2
)  =  C )
41403ad2ant2 1010 . . . . . 6  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  2
)  =  C )
4241adantr 465 . . . . 5  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  -> 
( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  2
)  =  C )
4333, 42syl5eq 2485 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  -> 
( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  ( # `
 { <. 0 ,  i >. ,  <. 1 ,  j >. } ) )  =  C )
44 simpl1 991 . . . . . 6  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  -> 
( V  e.  X  /\  E  e.  Y
) )
45 prex 4532 . . . . . . . 8  |-  { <. 0 ,  i >. , 
<. 1 ,  j
>. }  e.  _V
46 tpex 6377 . . . . . . . 8  |-  { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }  e.  _V
4745, 46pm3.2i 455 . . . . . . 7  |-  ( {
<. 0 ,  i
>. ,  <. 1 ,  j >. }  e.  _V  /\ 
{ <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }  e.  _V )
4847a1i 11 . . . . . 6  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  -> 
( { <. 0 ,  i >. ,  <. 1 ,  j >. }  e.  _V  /\  { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }  e.  _V )
)
49 3simpb 986 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( A  e.  V  /\  C  e.  V
) )
50493ad2ant2 1010 . . . . . . 7  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( A  e.  V  /\  C  e.  V
) )
5150adantr 465 . . . . . 6  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  -> 
( A  e.  V  /\  C  e.  V
) )
5244, 48, 513jca 1168 . . . . 5  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  -> 
( ( V  e.  X  /\  E  e.  Y )  /\  ( { <. 0 ,  i
>. ,  <. 1 ,  j >. }  e.  _V  /\ 
{ <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) ) )
53 iswlkon 23428 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( { <. 0 ,  i >. ,  <. 1 ,  j
>. }  e.  _V  /\  {
<. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( { <. 0 ,  i
>. ,  <. 1 ,  j >. }  ( A ( V WalkOn  E ) C ) { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }  <->  ( { <. 0 ,  i >. , 
<. 1 ,  j
>. }  ( V Walks  E
) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }  /\  ( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  0 )  =  A  /\  ( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  ( # `  { <. 0 ,  i
>. ,  <. 1 ,  j >. } ) )  =  C ) ) )
5452, 53syl 16 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  -> 
( { <. 0 ,  i >. ,  <. 1 ,  j >. }  ( A ( V WalkOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }  <->  ( { <. 0 ,  i >. , 
<. 1 ,  j
>. }  ( V Walks  E
) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }  /\  ( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  0 )  =  A  /\  ( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  ( # `  { <. 0 ,  i
>. ,  <. 1 ,  j >. } ) )  =  C ) ) )
5512, 22, 43, 54mpbir3and 1171 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  ->  { <. 0 ,  i
>. ,  <. 1 ,  j >. }  ( A ( V WalkOn  E ) C ) { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. } )
564, 5, 6constr2pth 23498 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( i  =/=  j  /\  ( E `
 i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C }
)  ->  { <. 0 ,  i >. ,  <. 1 ,  j >. }  ( V Paths  E ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) )
5756imp 429 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  ->  { <. 0 ,  i
>. ,  <. 1 ,  j >. }  ( V Paths 
E ) { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. } )
58 ispthon 23473 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( { <. 0 ,  i >. ,  <. 1 ,  j
>. }  e.  _V  /\  {
<. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( { <. 0 ,  i
>. ,  <. 1 ,  j >. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }  <->  ( { <. 0 ,  i >. , 
<. 1 ,  j
>. }  ( A ( V WalkOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }  /\  { <. 0 ,  i >. ,  <. 1 ,  j
>. }  ( V Paths  E
) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) ) )
5952, 58syl 16 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  -> 
( { <. 0 ,  i >. ,  <. 1 ,  j >. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }  <->  ( { <. 0 ,  i >. , 
<. 1 ,  j
>. }  ( A ( V WalkOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }  /\  { <. 0 ,  i >. ,  <. 1 ,  j
>. }  ( V Paths  E
) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) ) )
6055, 57, 59mpbir2and 913 . 2  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  ->  { <. 0 ,  i
>. ,  <. 1 ,  j >. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. } )
6160ex 434 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( i  =/=  j  /\  ( E `
 i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C }
)  ->  { <. 0 ,  i >. ,  <. 1 ,  j >. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2604   _Vcvv 2970   {cpr 3877   {ctp 3879   <.cop 3881   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   0cc0 9280   1c1 9281   2c2 10369   ZZcz 10644   #chash 12101   Walks cwalk 23403   Trails ctrail 23404   Paths cpath 23405   WalkOn cwlkon 23407   PathOn cpthon 23409
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 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-map 7214  df-pm 7215  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-card 8107  df-cda 8335  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436  df-fzo 11547  df-hash 12102  df-word 12227  df-wlk 23413  df-trail 23414  df-pth 23415  df-wlkon 23419  df-pthon 23421
This theorem is referenced by:  2pthoncl  23500
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