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Theorem wwlknextbi 24851
Description: Extension of a walk (as word) by adding an edge/vertex. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
Assertion
Ref Expression
wwlknextbi  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T ++  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  ( W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  <-> 
T  e.  ( ( V WWalksN  E ) `  N
) ) )

Proof of Theorem wwlknextbi
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 wwlknimp 24813 . . . 4  |-  ( W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  ->  ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 )  /\  A. i  e.  ( 0..^ ( N  +  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E
) )
2 wwlknred 24849 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  ->  ( W substr  <.
0 ,  ( N  +  1 ) >.
)  e.  ( ( V WWalksN  E ) `  N
) ) )
32ad2antrr 725 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T ++  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  ( W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  ->  ( W substr  <. 0 ,  ( N  + 
1 ) >. )  e.  ( ( V WWalksN  E
) `  N )
) )
4 fveq2 5872 . . . . . . . . . . . . . . . . 17  |-  ( W  =  ( T ++  <" S "> )  ->  ( # `  W
)  =  ( # `  ( T ++  <" S "> ) ) )
54eqeq1d 2459 . . . . . . . . . . . . . . . 16  |-  ( W  =  ( T ++  <" S "> )  ->  ( ( # `  W
)  =  ( ( N  +  1 )  +  1 )  <->  ( # `  ( T ++  <" S "> ) )  =  ( ( N  +  1 )  +  1 ) ) )
653ad2ant2 1018 . . . . . . . . . . . . . . 15  |-  ( ( T  e. Word  V  /\  W  =  ( T ++  <" S "> )  /\  { ( lastS  `  T
) ,  S }  e.  ran  E )  -> 
( ( # `  W
)  =  ( ( N  +  1 )  +  1 )  <->  ( # `  ( T ++  <" S "> ) )  =  ( ( N  +  1 )  +  1 ) ) )
76adantl 466 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T ++  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  ( ( # `
 W )  =  ( ( N  + 
1 )  +  1 )  <->  ( # `  ( T ++  <" S "> ) )  =  ( ( N  +  1 )  +  1 ) ) )
8 s1cl 12622 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( S  e.  V  ->  <" S ">  e. Word  V )
98adantl 466 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( N  e.  NN0  /\  S  e.  V )  ->  <" S ">  e. Word  V )
109anim2i 569 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( T  e. Word  V  /\  ( N  e.  NN0  /\  S  e.  V ) )  ->  ( T  e. Word  V  /\  <" S ">  e. Word  V )
)
1110ancoms 453 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( T  e. Word  V  /\  <" S ">  e. Word  V )
)
12 ccatlen 12602 . . . . . . . . . . . . . . . . . . 19  |-  ( ( T  e. Word  V  /\  <" S ">  e. Word  V )  ->  ( # `
 ( T ++  <" S "> )
)  =  ( (
# `  T )  +  ( # `  <" S "> )
) )
1311, 12syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( # `  ( T ++  <" S "> ) )  =  ( ( # `  T
)  +  ( # `  <" S "> ) ) )
1413eqeq1d 2459 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( ( # `
 ( T ++  <" S "> )
)  =  ( ( N  +  1 )  +  1 )  <->  ( ( # `
 T )  +  ( # `  <" S "> )
)  =  ( ( N  +  1 )  +  1 ) ) )
15 s1len 12625 . . . . . . . . . . . . . . . . . . . 20  |-  ( # `  <" S "> )  =  1
1615a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( # `  <" S "> )  =  1 )
1716oveq2d 6312 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( ( # `
 T )  +  ( # `  <" S "> )
)  =  ( (
# `  T )  +  1 ) )
1817eqeq1d 2459 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( (
( # `  T )  +  ( # `  <" S "> )
)  =  ( ( N  +  1 )  +  1 )  <->  ( ( # `
 T )  +  1 )  =  ( ( N  +  1 )  +  1 ) ) )
19 lencl 12568 . . . . . . . . . . . . . . . . . . . 20  |-  ( T  e. Word  V  ->  ( # `
 T )  e. 
NN0 )
2019nn0cnd 10875 . . . . . . . . . . . . . . . . . . 19  |-  ( T  e. Word  V  ->  ( # `
 T )  e.  CC )
2120adantl 466 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( # `  T
)  e.  CC )
22 peano2nn0 10857 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
2322nn0cnd 10875 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  CC )
2423ad2antrr 725 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( N  +  1 )  e.  CC )
25 1cnd 9629 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  1  e.  CC )
2621, 24, 25addcan2d 9801 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( (
( # `  T )  +  1 )  =  ( ( N  + 
1 )  +  1 )  <->  ( # `  T
)  =  ( N  +  1 ) ) )
2714, 18, 263bitrd 279 . . . . . . . . . . . . . . . 16  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( ( # `
 ( T ++  <" S "> )
)  =  ( ( N  +  1 )  +  1 )  <->  ( # `  T
)  =  ( N  +  1 ) ) )
28 opeq2 4220 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  +  1 )  =  ( # `  T
)  ->  <. 0 ,  ( N  +  1 ) >.  =  <. 0 ,  ( # `  T
) >. )
2928eqcoms 2469 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  T )  =  ( N  + 
1 )  ->  <. 0 ,  ( N  + 
1 ) >.  =  <. 0 ,  ( # `  T
) >. )
3029oveq2d 6312 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  T )  =  ( N  + 
1 )  ->  (
( T ++  <" S "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  ( ( T ++  <" S "> ) substr  <. 0 ,  ( # `  T
) >. ) )
31 swrdccat1 12693 . . . . . . . . . . . . . . . . . . 19  |-  ( ( T  e. Word  V  /\  <" S ">  e. Word  V )  ->  (
( T ++  <" S "> ) substr  <. 0 ,  ( # `  T
) >. )  =  T )
3211, 31syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( ( T ++  <" S "> ) substr  <. 0 ,  ( # `  T
) >. )  =  T )
3330, 32sylan9eqr 2520 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( N  e. 
NN0  /\  S  e.  V )  /\  T  e. Word  V )  /\  ( # `
 T )  =  ( N  +  1 ) )  ->  (
( T ++  <" S "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  T )
3433ex 434 . . . . . . . . . . . . . . . 16  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( ( # `
 T )  =  ( N  +  1 )  ->  ( ( T ++  <" S "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  T ) )
3527, 34sylbid 215 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( ( # `
 ( T ++  <" S "> )
)  =  ( ( N  +  1 )  +  1 )  -> 
( ( T ++  <" S "> ) substr  <.
0 ,  ( N  +  1 ) >.
)  =  T ) )
36353ad2antr1 1161 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T ++  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  ( ( # `
 ( T ++  <" S "> )
)  =  ( ( N  +  1 )  +  1 )  -> 
( ( T ++  <" S "> ) substr  <.
0 ,  ( N  +  1 ) >.
)  =  T ) )
377, 36sylbid 215 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T ++  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  ( ( # `
 W )  =  ( ( N  + 
1 )  +  1 )  ->  ( ( T ++  <" S "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  T ) )
3837imp 429 . . . . . . . . . . . 12  |-  ( ( ( ( N  e. 
NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T ++  <" S "> )  /\  { ( lastS  `  T
) ,  S }  e.  ran  E ) )  /\  ( # `  W
)  =  ( ( N  +  1 )  +  1 ) )  ->  ( ( T ++ 
<" S "> ) substr  <. 0 ,  ( N  +  1 )
>. )  =  T
)
39 oveq1 6303 . . . . . . . . . . . . . . 15  |-  ( W  =  ( T ++  <" S "> )  ->  ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  ( ( T ++ 
<" S "> ) substr  <. 0 ,  ( N  +  1 )
>. ) )
4039eqeq1d 2459 . . . . . . . . . . . . . 14  |-  ( W  =  ( T ++  <" S "> )  ->  ( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  T  <->  ( ( T ++ 
<" S "> ) substr  <. 0 ,  ( N  +  1 )
>. )  =  T
) )
41403ad2ant2 1018 . . . . . . . . . . . . 13  |-  ( ( T  e. Word  V  /\  W  =  ( T ++  <" S "> )  /\  { ( lastS  `  T
) ,  S }  e.  ran  E )  -> 
( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  T  <->  ( ( T ++ 
<" S "> ) substr  <. 0 ,  ( N  +  1 )
>. )  =  T
) )
4241ad2antlr 726 . . . . . . . . . . . 12  |-  ( ( ( ( N  e. 
NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T ++  <" S "> )  /\  { ( lastS  `  T
) ,  S }  e.  ran  E ) )  /\  ( # `  W
)  =  ( ( N  +  1 )  +  1 ) )  ->  ( ( W substr  <. 0 ,  ( N  +  1 ) >.
)  =  T  <->  ( ( T ++  <" S "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  T ) )
4338, 42mpbird 232 . . . . . . . . . . 11  |-  ( ( ( ( N  e. 
NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T ++  <" S "> )  /\  { ( lastS  `  T
) ,  S }  e.  ran  E ) )  /\  ( # `  W
)  =  ( ( N  +  1 )  +  1 ) )  ->  ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  T )
4443eleq1d 2526 . . . . . . . . . 10  |-  ( ( ( ( N  e. 
NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T ++  <" S "> )  /\  { ( lastS  `  T
) ,  S }  e.  ran  E ) )  /\  ( # `  W
)  =  ( ( N  +  1 )  +  1 ) )  ->  ( ( W substr  <. 0 ,  ( N  +  1 ) >.
)  e.  ( ( V WWalksN  E ) `  N
)  <->  T  e.  (
( V WWalksN  E ) `  N ) ) )
4544biimpd 207 . . . . . . . . 9  |-  ( ( ( ( N  e. 
NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T ++  <" S "> )  /\  { ( lastS  `  T
) ,  S }  e.  ran  E ) )  /\  ( # `  W
)  =  ( ( N  +  1 )  +  1 ) )  ->  ( ( W substr  <. 0 ,  ( N  +  1 ) >.
)  e.  ( ( V WWalksN  E ) `  N
)  ->  T  e.  ( ( V WWalksN  E
) `  N )
) )
4645ex 434 . . . . . . . 8  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T ++  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  ( ( # `
 W )  =  ( ( N  + 
1 )  +  1 )  ->  ( ( W substr  <. 0 ,  ( N  +  1 )
>. )  e.  (
( V WWalksN  E ) `  N )  ->  T  e.  ( ( V WWalksN  E
) `  N )
) ) )
4746com23 78 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T ++  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  ( ( W substr  <. 0 ,  ( N  +  1 )
>. )  e.  (
( V WWalksN  E ) `  N )  ->  (
( # `  W )  =  ( ( N  +  1 )  +  1 )  ->  T  e.  ( ( V WWalksN  E
) `  N )
) ) )
483, 47syld 44 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T ++  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  ( W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  ->  ( ( # `  W )  =  ( ( N  +  1 )  +  1 )  ->  T  e.  ( ( V WWalksN  E ) `  N ) ) ) )
4948com13 80 . . . . 5  |-  ( (
# `  W )  =  ( ( N  +  1 )  +  1 )  ->  ( W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  ->  ( (
( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T ++  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  T  e.  ( ( V WWalksN  E
) `  N )
) ) )
50493ad2ant2 1018 . . . 4  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 )  /\  A. i  e.  ( 0..^ ( N  +  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E )  ->  ( W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  ->  ( (
( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T ++  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  T  e.  ( ( V WWalksN  E
) `  N )
) ) )
511, 50mpcom 36 . . 3  |-  ( W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  ->  ( (
( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T ++  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  T  e.  ( ( V WWalksN  E
) `  N )
) )
5251com12 31 . 2  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T ++  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  ( W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  ->  T  e.  ( ( V WWalksN  E ) `  N ) ) )
53 wwlknext 24850 . . . . . . . . . . 11  |-  ( ( T  e.  ( ( V WWalksN  E ) `  N
)  /\  S  e.  V  /\  { ( lastS  `  T
) ,  S }  e.  ran  E )  -> 
( T ++  <" S "> )  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )
54 eleq1 2529 . . . . . . . . . . 11  |-  ( W  =  ( T ++  <" S "> )  ->  ( W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  <->  ( T ++  <" S "> )  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) ) )
5553, 54syl5ibrcom 222 . . . . . . . . . 10  |-  ( ( T  e.  ( ( V WWalksN  E ) `  N
)  /\  S  e.  V  /\  { ( lastS  `  T
) ,  S }  e.  ran  E )  -> 
( W  =  ( T ++  <" S "> )  ->  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) )
56553exp 1195 . . . . . . . . 9  |-  ( T  e.  ( ( V WWalksN  E ) `  N
)  ->  ( S  e.  V  ->  ( { ( lastS  `  T ) ,  S }  e.  ran  E  ->  ( W  =  ( T ++  <" S "> )  ->  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) ) ) )
5756com23 78 . . . . . . . 8  |-  ( T  e.  ( ( V WWalksN  E ) `  N
)  ->  ( {
( lastS  `  T ) ,  S }  e.  ran  E  ->  ( S  e.  V  ->  ( W  =  ( T ++  <" S "> )  ->  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) ) ) ) )
5857com14 88 . . . . . . 7  |-  ( W  =  ( T ++  <" S "> )  ->  ( { ( lastS  `  T
) ,  S }  e.  ran  E  ->  ( S  e.  V  ->  ( T  e.  ( ( V WWalksN  E ) `  N
)  ->  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) ) ) )
5958imp 429 . . . . . 6  |-  ( ( W  =  ( T ++ 
<" S "> )  /\  { ( lastS  `  T
) ,  S }  e.  ran  E )  -> 
( S  e.  V  ->  ( T  e.  ( ( V WWalksN  E ) `  N )  ->  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) ) )
60593adant1 1014 . . . . 5  |-  ( ( T  e. Word  V  /\  W  =  ( T ++  <" S "> )  /\  { ( lastS  `  T
) ,  S }  e.  ran  E )  -> 
( S  e.  V  ->  ( T  e.  ( ( V WWalksN  E ) `  N )  ->  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) ) )
6160com12 31 . . . 4  |-  ( S  e.  V  ->  (
( T  e. Word  V  /\  W  =  ( T ++  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E )  ->  ( T  e.  ( ( V WWalksN  E
) `  N )  ->  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) ) ) )
6261adantl 466 . . 3  |-  ( ( N  e.  NN0  /\  S  e.  V )  ->  ( ( T  e. Word  V  /\  W  =  ( T ++  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E )  ->  ( T  e.  ( ( V WWalksN  E
) `  N )  ->  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) ) ) )
6362imp 429 . 2  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T ++  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  ( T  e.  ( ( V WWalksN  E
) `  N )  ->  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) ) )
6452, 63impbid 191 1  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T ++  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  ( W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  <-> 
T  e.  ( ( V WWalksN  E ) `  N
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   {cpr 4034   <.cop 4038   ran crn 5009   ` cfv 5594  (class class class)co 6296   CCcc 9507   0cc0 9509   1c1 9510    + caddc 9512   NN0cn0 10816  ..^cfzo 11820   #chash 12407  Word cword 12537   lastS clsw 12538   ++ cconcat 12539   <"cs1 12540   substr csubstr 12541   WWalksN cwwlkn 24804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11821  df-hash 12408  df-word 12545  df-lsw 12546  df-concat 12547  df-s1 12548  df-substr 12549  df-wwlk 24805  df-wwlkn 24806
This theorem is referenced by:  wwlkextwrd  24854
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