Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  vfwlkniswwlkn Structured version   Unicode version

Theorem vfwlkniswwlkn 30508
Description: If the edge function of a walk has length n, then the vertex function of the walk is a word representing the walk as word of length n. (Contributed by Alexander van der Vekens, 25-Aug-2018.)
Assertion
Ref Expression
vfwlkniswwlkn  |-  ( ( N  e.  NN0  /\  ( W  e.  ( V Walks  E )  /\  ( # `
 ( 1st `  W
) )  =  N ) )  ->  ( 2nd `  W )  e.  ( ( V WWalksN  E
) `  N )
)

Proof of Theorem vfwlkniswwlkn
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 wlkcpr 30458 . . . . 5  |-  ( W  e.  ( V Walks  E
)  <->  ( 1st `  W
) ( V Walks  E
) ( 2nd `  W
) )
2 wlkn0 30447 . . . . 5  |-  ( ( 1st `  W ) ( V Walks  E ) ( 2nd `  W
)  ->  ( 2nd `  W )  =/=  (/) )
31, 2sylbi 195 . . . 4  |-  ( W  e.  ( V Walks  E
)  ->  ( 2nd `  W )  =/=  (/) )
4 wlkelwrd 30448 . . . . 5  |-  ( W  e.  ( V Walks  E
)  ->  ( ( 1st `  W )  e. Word  dom  E  /\  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W
) ) ) --> V ) )
5 ffz0iswrd 12376 . . . . . 6  |-  ( ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V  ->  ( 2nd `  W )  e. Word  V )
65adantl 466 . . . . 5  |-  ( ( ( 1st `  W
)  e. Word  dom  E  /\  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V )  -> 
( 2nd `  W
)  e. Word  V )
74, 6syl 16 . . . 4  |-  ( W  e.  ( V Walks  E
)  ->  ( 2nd `  W )  e. Word  V
)
8 edgwlk 30462 . . . . . 6  |-  ( ( 1st `  W ) ( V Walks  E ) ( 2nd `  W
)  ->  A. i  e.  ( 0..^ ( # `  ( 1st `  W
) ) ) { ( ( 2nd `  W
) `  i ) ,  ( ( 2nd `  W ) `  (
i  +  1 ) ) }  e.  ran  E )
9 wlklenfislenpm1 30452 . . . . . . . 8  |-  ( ( 1st `  W ) ( V Walks  E ) ( 2nd `  W
)  ->  ( # `  ( 1st `  W ) )  =  ( ( # `  ( 2nd `  W
) )  -  1 ) )
109oveq2d 6219 . . . . . . 7  |-  ( ( 1st `  W ) ( V Walks  E ) ( 2nd `  W
)  ->  ( 0..^ ( # `  ( 1st `  W ) ) )  =  ( 0..^ ( ( # `  ( 2nd `  W ) )  -  1 ) ) )
1110raleqdv 3029 . . . . . 6  |-  ( ( 1st `  W ) ( V Walks  E ) ( 2nd `  W
)  ->  ( A. i  e.  ( 0..^ ( # `  ( 1st `  W ) ) ) { ( ( 2nd `  W ) `
 i ) ,  ( ( 2nd `  W
) `  ( i  +  1 ) ) }  e.  ran  E  <->  A. i  e.  ( 0..^ ( ( # `  ( 2nd `  W ) )  -  1 ) ) { ( ( 2nd `  W ) `  i
) ,  ( ( 2nd `  W ) `
 ( i  +  1 ) ) }  e.  ran  E ) )
128, 11mpbid 210 . . . . 5  |-  ( ( 1st `  W ) ( V Walks  E ) ( 2nd `  W
)  ->  A. i  e.  ( 0..^ ( (
# `  ( 2nd `  W ) )  - 
1 ) ) { ( ( 2nd `  W
) `  i ) ,  ( ( 2nd `  W ) `  (
i  +  1 ) ) }  e.  ran  E )
131, 12sylbi 195 . . . 4  |-  ( W  e.  ( V Walks  E
)  ->  A. i  e.  ( 0..^ ( (
# `  ( 2nd `  W ) )  - 
1 ) ) { ( ( 2nd `  W
) `  i ) ,  ( ( 2nd `  W ) `  (
i  +  1 ) ) }  e.  ran  E )
143, 7, 133jca 1168 . . 3  |-  ( W  e.  ( V Walks  E
)  ->  ( ( 2nd `  W )  =/=  (/)  /\  ( 2nd `  W
)  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  ( 2nd `  W ) )  -  1 ) ) { ( ( 2nd `  W ) `  i
) ,  ( ( 2nd `  W ) `
 ( i  +  1 ) ) }  e.  ran  E ) )
1514ad2antrl 727 . 2  |-  ( ( N  e.  NN0  /\  ( W  e.  ( V Walks  E )  /\  ( # `
 ( 1st `  W
) )  =  N ) )  ->  (
( 2nd `  W
)  =/=  (/)  /\  ( 2nd `  W )  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  ( 2nd `  W ) )  - 
1 ) ) { ( ( 2nd `  W
) `  i ) ,  ( ( 2nd `  W ) `  (
i  +  1 ) ) }  e.  ran  E ) )
16 id 22 . . . . . 6  |-  ( N  e.  NN0  ->  N  e. 
NN0 )
17 oveq2 6211 . . . . . . . . . . 11  |-  ( (
# `  ( 1st `  W ) )  =  N  ->  ( 0 ... ( # `  ( 1st `  W ) ) )  =  ( 0 ... N ) )
1817adantl 466 . . . . . . . . . 10  |-  ( ( ( 1st `  W
)  e. Word  dom  E  /\  ( # `  ( 1st `  W ) )  =  N )  ->  (
0 ... ( # `  ( 1st `  W ) ) )  =  ( 0 ... N ) )
1918feq2d 5658 . . . . . . . . 9  |-  ( ( ( 1st `  W
)  e. Word  dom  E  /\  ( # `  ( 1st `  W ) )  =  N )  ->  (
( 2nd `  W
) : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V  <->  ( 2nd `  W ) : ( 0 ... N ) --> V ) )
2019biimpd 207 . . . . . . . 8  |-  ( ( ( 1st `  W
)  e. Word  dom  E  /\  ( # `  ( 1st `  W ) )  =  N )  ->  (
( 2nd `  W
) : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V  ->  ( 2nd `  W ) : ( 0 ... N
) --> V ) )
2120impancom 440 . . . . . . 7  |-  ( ( ( 1st `  W
)  e. Word  dom  E  /\  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V )  -> 
( ( # `  ( 1st `  W ) )  =  N  ->  ( 2nd `  W ) : ( 0 ... N
) --> V ) )
2221imp 429 . . . . . 6  |-  ( ( ( ( 1st `  W
)  e. Word  dom  E  /\  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V )  /\  ( # `  ( 1st `  W ) )  =  N )  ->  ( 2nd `  W ) : ( 0 ... N
) --> V )
23 hashfzdm 12323 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( 2nd `  W ) : ( 0 ... N ) --> V )  ->  ( # `  ( 2nd `  W ) )  =  ( N  + 
1 ) )
2416, 22, 23syl2anr 478 . . . . 5  |-  ( ( ( ( ( 1st `  W )  e. Word  dom  E  /\  ( 2nd `  W
) : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V )  /\  ( # `  ( 1st `  W ) )  =  N )  /\  N  e.  NN0 )  ->  ( # `
 ( 2nd `  W
) )  =  ( N  +  1 ) )
2524ex 434 . . . 4  |-  ( ( ( ( 1st `  W
)  e. Word  dom  E  /\  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V )  /\  ( # `  ( 1st `  W ) )  =  N )  ->  ( N  e.  NN0  ->  ( # `
 ( 2nd `  W
) )  =  ( N  +  1 ) ) )
264, 25sylan 471 . . 3  |-  ( ( W  e.  ( V Walks 
E )  /\  ( # `
 ( 1st `  W
) )  =  N )  ->  ( N  e.  NN0  ->  ( # `  ( 2nd `  W ) )  =  ( N  + 
1 ) ) )
2726impcom 430 . 2  |-  ( ( N  e.  NN0  /\  ( W  e.  ( V Walks  E )  /\  ( # `
 ( 1st `  W
) )  =  N ) )  ->  ( # `
 ( 2nd `  W
) )  =  ( N  +  1 ) )
28 wlkbprop 23605 . . . . . . . 8  |-  ( ( 1st `  W ) ( V Walks  E ) ( 2nd `  W
)  ->  ( ( # `
 ( 1st `  W
) )  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( ( 1st `  W
)  e.  _V  /\  ( 2nd `  W )  e.  _V ) ) )
2928simp2d 1001 . . . . . . 7  |-  ( ( 1st `  W ) ( V Walks  E ) ( 2nd `  W
)  ->  ( V  e.  _V  /\  E  e. 
_V ) )
301, 29sylbi 195 . . . . . 6  |-  ( W  e.  ( V Walks  E
)  ->  ( V  e.  _V  /\  E  e. 
_V ) )
31 simpll 753 . . . . . . . 8  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 )  ->  V  e.  _V )
32 simpr 461 . . . . . . . . 9  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  E  e.  _V )
3332adantr 465 . . . . . . . 8  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 )  ->  E  e.  _V )
34 simpr 461 . . . . . . . 8  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 )  ->  N  e.  NN0 )
3531, 33, 343jca 1168 . . . . . . 7  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 )  ->  ( V  e. 
_V  /\  E  e.  _V  /\  N  e.  NN0 ) )
3635ex 434 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( N  e.  NN0  ->  ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 ) ) )
3730, 36syl 16 . . . . 5  |-  ( W  e.  ( V Walks  E
)  ->  ( N  e.  NN0  ->  ( V  e.  _V  /\  E  e. 
_V  /\  N  e.  NN0 ) ) )
3837adantr 465 . . . 4  |-  ( ( W  e.  ( V Walks 
E )  /\  ( # `
 ( 1st `  W
) )  =  N )  ->  ( N  e.  NN0  ->  ( V  e.  _V  /\  E  e. 
_V  /\  N  e.  NN0 ) ) )
3938impcom 430 . . 3  |-  ( ( N  e.  NN0  /\  ( W  e.  ( V Walks  E )  /\  ( # `
 ( 1st `  W
) )  =  N ) )  ->  ( V  e.  _V  /\  E  e.  _V  /\  N  e. 
NN0 ) )
40 iswwlkn 30486 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  (
( 2nd `  W
)  e.  ( ( V WWalksN  E ) `  N
)  <->  ( ( 2nd `  W )  e.  ( V WWalks  E )  /\  ( # `  ( 2nd `  W ) )  =  ( N  +  1 ) ) ) )
41 iswwlk 30485 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( ( 2nd `  W
)  e.  ( V WWalks  E )  <->  ( ( 2nd `  W )  =/=  (/)  /\  ( 2nd `  W
)  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  ( 2nd `  W ) )  -  1 ) ) { ( ( 2nd `  W ) `  i
) ,  ( ( 2nd `  W ) `
 ( i  +  1 ) ) }  e.  ran  E ) ) )
42413adant3 1008 . . . . 5  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  (
( 2nd `  W
)  e.  ( V WWalks  E )  <->  ( ( 2nd `  W )  =/=  (/)  /\  ( 2nd `  W
)  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  ( 2nd `  W ) )  -  1 ) ) { ( ( 2nd `  W ) `  i
) ,  ( ( 2nd `  W ) `
 ( i  +  1 ) ) }  e.  ran  E ) ) )
4342anbi1d 704 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  (
( ( 2nd `  W
)  e.  ( V WWalks  E )  /\  ( # `
 ( 2nd `  W
) )  =  ( N  +  1 ) )  <->  ( ( ( 2nd `  W )  =/=  (/)  /\  ( 2nd `  W )  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  ( 2nd `  W
) )  -  1 ) ) { ( ( 2nd `  W
) `  i ) ,  ( ( 2nd `  W ) `  (
i  +  1 ) ) }  e.  ran  E )  /\  ( # `  ( 2nd `  W
) )  =  ( N  +  1 ) ) ) )
4440, 43bitrd 253 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  (
( 2nd `  W
)  e.  ( ( V WWalksN  E ) `  N
)  <->  ( ( ( 2nd `  W )  =/=  (/)  /\  ( 2nd `  W )  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  ( 2nd `  W
) )  -  1 ) ) { ( ( 2nd `  W
) `  i ) ,  ( ( 2nd `  W ) `  (
i  +  1 ) ) }  e.  ran  E )  /\  ( # `  ( 2nd `  W
) )  =  ( N  +  1 ) ) ) )
4539, 44syl 16 . 2  |-  ( ( N  e.  NN0  /\  ( W  e.  ( V Walks  E )  /\  ( # `
 ( 1st `  W
) )  =  N ) )  ->  (
( 2nd `  W
)  e.  ( ( V WWalksN  E ) `  N
)  <->  ( ( ( 2nd `  W )  =/=  (/)  /\  ( 2nd `  W )  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  ( 2nd `  W
) )  -  1 ) ) { ( ( 2nd `  W
) `  i ) ,  ( ( 2nd `  W ) `  (
i  +  1 ) ) }  e.  ran  E )  /\  ( # `  ( 2nd `  W
) )  =  ( N  +  1 ) ) ) )
4615, 27, 45mpbir2and 913 1  |-  ( ( N  e.  NN0  /\  ( W  e.  ( V Walks  E )  /\  ( # `
 ( 1st `  W
) )  =  N ) )  ->  ( 2nd `  W )  e.  ( ( V WWalksN  E
) `  N )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   _Vcvv 3078   (/)c0 3748   {cpr 3990   class class class wbr 4403   dom cdm 4951   ran crn 4952   -->wf 5525   ` cfv 5529  (class class class)co 6203   1stc1st 6688   2ndc2nd 6689   0cc0 9396   1c1 9397    + caddc 9399    - cmin 9709   NN0cn0 10693   ...cfz 11557  ..^cfzo 11668   #chash 12223  Word cword 12342   Walks cwalk 23577   WWalks cwwlk 30479   WWalksN cwwlkn 30480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8223  df-cda 8451  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-fzo 11669  df-hash 12224  df-word 12350  df-wlk 23587  df-wwlk 30481  df-wwlkn 30482
This theorem is referenced by:  wlknwwlknfun  30510  wlkiswwlkfun  30517
  Copyright terms: Public domain W3C validator