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Theorem wwlknfi 30538
Description: The number of walks represented by words of fixed length is finite if the number of vertices is finite (in the graph). (Contributed by Alexander van der Vekens, 30-Jul-2018.)
Assertion
Ref Expression
wwlknfi  |-  ( V  e.  Fin  ->  (
( V WWalksN  E ) `  N )  e.  Fin )

Proof of Theorem wwlknfi
Dummy variables  i  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wwlkn 30484 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  (
( V WWalksN  E ) `  N )  =  {
w  e.  ( V WWalks  E )  |  (
# `  w )  =  ( N  + 
1 ) } )
2 df-rab 2808 . . . . . . 7  |-  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( N  +  1 ) }  =  { w  |  ( w  e.  ( V WWalks  E )  /\  ( # `  w )  =  ( N  + 
1 ) ) }
32a1i 11 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( N  +  1 ) }  =  { w  |  ( w  e.  ( V WWalks  E )  /\  ( # `  w )  =  ( N  + 
1 ) ) } )
4 iswwlk 30485 . . . . . . . . . 10  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( w  e.  ( V WWalks  E )  <->  ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
) ) )
543adant3 1008 . . . . . . . . 9  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  (
w  e.  ( V WWalks  E )  <->  ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
) ) )
65anbi1d 704 . . . . . . . 8  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  (
( w  e.  ( V WWalks  E )  /\  ( # `  w )  =  ( N  + 
1 ) )  <->  ( (
w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( N  +  1 ) ) ) )
76abbidv 2590 . . . . . . 7  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  { w  |  ( w  e.  ( V WWalks  E )  /\  ( # `  w
)  =  ( N  +  1 ) ) }  =  { w  |  ( ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( N  +  1 ) ) } )
8 3anan12 978 . . . . . . . . . . 11  |-  ( ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  <->  ( w  e. Word  V  /\  ( w  =/=  (/)  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
) ) )
98anbi1i 695 . . . . . . . . . 10  |-  ( ( ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( N  +  1 ) )  <-> 
( ( w  e. Word  V  /\  ( w  =/=  (/)  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
) )  /\  ( # `
 w )  =  ( N  +  1 ) ) )
10 anass 649 . . . . . . . . . 10  |-  ( ( ( w  e. Word  V  /\  ( w  =/=  (/)  /\  A. i  e.  ( 0..^ ( ( # `  w
)  -  1 ) ) { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  E ) )  /\  ( # `
 w )  =  ( N  +  1 ) )  <->  ( w  e. Word  V  /\  ( ( w  =/=  (/)  /\  A. i  e.  ( 0..^ ( ( # `  w
)  -  1 ) ) { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  E )  /\  ( # `  w )  =  ( N  +  1 ) ) ) )
119, 10bitri 249 . . . . . . . . 9  |-  ( ( ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( N  +  1 ) )  <-> 
( w  e. Word  V  /\  ( ( w  =/=  (/)  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( N  +  1 ) ) ) )
1211abbii 2588 . . . . . . . 8  |-  { w  |  ( ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( N  +  1 ) ) }  =  { w  |  ( w  e. Word  V  /\  ( ( w  =/=  (/)  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( N  +  1 ) ) ) }
13 df-rab 2808 . . . . . . . 8  |-  { w  e. Word  V  |  ( ( w  =/=  (/)  /\  A. i  e.  ( 0..^ ( ( # `  w
)  -  1 ) ) { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  E )  /\  ( # `  w )  =  ( N  +  1 ) ) }  =  {
w  |  ( w  e. Word  V  /\  (
( w  =/=  (/)  /\  A. i  e.  ( 0..^ ( ( # `  w
)  -  1 ) ) { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  E )  /\  ( # `  w )  =  ( N  +  1 ) ) ) }
1412, 13eqtr4i 2486 . . . . . . 7  |-  { w  |  ( ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( N  +  1 ) ) }  =  { w  e. Word  V  |  ( ( w  =/=  (/)  /\  A. i  e.  ( 0..^ ( ( # `  w
)  -  1 ) ) { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  E )  /\  ( # `  w )  =  ( N  +  1 ) ) }
157, 14syl6eq 2511 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  { w  |  ( w  e.  ( V WWalks  E )  /\  ( # `  w
)  =  ( N  +  1 ) ) }  =  { w  e. Word  V  |  ( ( w  =/=  (/)  /\  A. i  e.  ( 0..^ ( ( # `  w
)  -  1 ) ) { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  E )  /\  ( # `  w )  =  ( N  +  1 ) ) } )
161, 3, 153eqtrd 2499 . . . . 5  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  (
( V WWalksN  E ) `  N )  =  {
w  e. Word  V  | 
( ( w  =/=  (/)  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( N  +  1 ) ) } )
1716adantr 465 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  /\  V  e.  Fin )  ->  ( ( V WWalksN  E
) `  N )  =  { w  e. Word  V  |  ( ( w  =/=  (/)  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( N  +  1 ) ) } )
18 peano2nn0 10734 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
19183ad2ant3 1011 . . . . . . . 8  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  ( N  +  1 )  e.  NN0 )
2019anim2i 569 . . . . . . 7  |-  ( ( V  e.  Fin  /\  ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 ) )  ->  ( V  e. 
Fin  /\  ( N  +  1 )  e. 
NN0 ) )
2120ancoms 453 . . . . . 6  |-  ( ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  /\  V  e.  Fin )  ->  ( V  e.  Fin  /\  ( N  +  1 )  e.  NN0 )
)
22 wrdlenfi 12379 . . . . . 6  |-  ( ( V  e.  Fin  /\  ( N  +  1
)  e.  NN0 )  ->  { w  e. Word  V  |  ( # `  w
)  =  ( N  +  1 ) }  e.  Fin )
2321, 22syl 16 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  /\  V  e.  Fin )  ->  { w  e. Word  V  |  ( # `  w
)  =  ( N  +  1 ) }  e.  Fin )
24 simpr 461 . . . . . . 7  |-  ( ( ( w  =/=  (/)  /\  A. i  e.  ( 0..^ ( ( # `  w
)  -  1 ) ) { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  E )  /\  ( # `  w )  =  ( N  +  1 ) )  ->  ( # `  w
)  =  ( N  +  1 ) )
2524rgenw 2901 . . . . . 6  |-  A. w  e. Word  V ( ( ( w  =/=  (/)  /\  A. i  e.  ( 0..^ ( ( # `  w
)  -  1 ) ) { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  E )  /\  ( # `  w )  =  ( N  +  1 ) )  ->  ( # `  w
)  =  ( N  +  1 ) )
26 ss2rab 3539 . . . . . 6  |-  ( { w  e. Word  V  | 
( ( w  =/=  (/)  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( N  +  1 ) ) }  C_  { w  e. Word  V  |  ( # `  w )  =  ( N  +  1 ) }  <->  A. w  e. Word  V
( ( ( w  =/=  (/)  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( N  +  1 ) )  ->  ( # `  w
)  =  ( N  +  1 ) ) )
2725, 26mpbir 209 . . . . 5  |-  { w  e. Word  V  |  ( ( w  =/=  (/)  /\  A. i  e.  ( 0..^ ( ( # `  w
)  -  1 ) ) { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  E )  /\  ( # `  w )  =  ( N  +  1 ) ) }  C_  { w  e. Word  V  |  ( # `  w )  =  ( N  +  1 ) }
28 ssfi 7647 . . . . 5  |-  ( ( { w  e. Word  V  |  ( # `  w
)  =  ( N  +  1 ) }  e.  Fin  /\  {
w  e. Word  V  | 
( ( w  =/=  (/)  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( N  +  1 ) ) }  C_  { w  e. Word  V  |  ( # `  w )  =  ( N  +  1 ) } )  ->  { w  e. Word  V  |  ( ( w  =/=  (/)  /\  A. i  e.  ( 0..^ ( ( # `  w
)  -  1 ) ) { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  E )  /\  ( # `  w )  =  ( N  +  1 ) ) }  e.  Fin )
2923, 27, 28sylancl 662 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  /\  V  e.  Fin )  ->  { w  e. Word  V  |  ( ( w  =/=  (/)  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  /\  ( # `  w
)  =  ( N  +  1 ) ) }  e.  Fin )
3017, 29eqeltrd 2542 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  /\  V  e.  Fin )  ->  ( ( V WWalksN  E
) `  N )  e.  Fin )
3130ex 434 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  ( V  e.  Fin  ->  (
( V WWalksN  E ) `  N )  e.  Fin ) )
32 wwlknndef 30537 . . . . 5  |-  ( ( V  e/  _V  \/  E  e/  _V  \/  N  e/  NN0 )  ->  (
( V WWalksN  E ) `  N )  =  (/) )
33 3ioran 983 . . . . . 6  |-  ( -.  ( V  e/  _V  \/  E  e/  _V  \/  N  e/  NN0 )  <->  ( -.  V  e/  _V  /\  -.  E  e/  _V  /\  -.  N  e/  NN0 ) )
34 nnel 2797 . . . . . . 7  |-  ( -.  V  e/  _V  <->  V  e.  _V )
35 nnel 2797 . . . . . . 7  |-  ( -.  E  e/  _V  <->  E  e.  _V )
36 nnel 2797 . . . . . . 7  |-  ( -.  N  e/  NN0  <->  N  e.  NN0 )
3734, 35, 363anbi123i 1177 . . . . . 6  |-  ( ( -.  V  e/  _V  /\ 
-.  E  e/  _V  /\ 
-.  N  e/  NN0 ) 
<->  ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 ) )
3833, 37sylbb 197 . . . . 5  |-  ( -.  ( V  e/  _V  \/  E  e/  _V  \/  N  e/  NN0 )  -> 
( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 ) )
3932, 38nsyl4 142 . . . 4  |-  ( -.  ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  -> 
( ( V WWalksN  E
) `  N )  =  (/) )
40 0fin 7654 . . . . 5  |-  (/)  e.  Fin
4140a1i 11 . . . 4  |-  ( -.  ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  (/) 
e.  Fin )
4239, 41eqeltrd 2542 . . 3  |-  ( -.  ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  -> 
( ( V WWalksN  E
) `  N )  e.  Fin )
4342a1d 25 . 2  |-  ( -.  ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  -> 
( V  e.  Fin  ->  ( ( V WWalksN  E
) `  N )  e.  Fin ) )
4431, 43pm2.61i 164 1  |-  ( V  e.  Fin  ->  (
( V WWalksN  E ) `  N )  e.  Fin )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 964    /\ w3a 965    = wceq 1370    e. wcel 1758   {cab 2439    =/= wne 2648    e/ wnel 2649   A.wral 2799   {crab 2803   _Vcvv 3078    C_ wss 3439   (/)c0 3748   {cpr 3990   ran crn 4952   ` cfv 5529  (class class class)co 6203   Fincfn 7423   0cc0 9396   1c1 9397    + caddc 9399    - cmin 9709   NN0cn0 10693  ..^cfzo 11668   #chash 12223  Word cword 12342   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-2o 7034  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-wwlk 30481  df-wwlkn 30482
This theorem is referenced by:  wlknfi  30539  hashwwlkext  30733  rusgranumwlks  30742  clwlknclwlkdifnum  30747
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