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Theorem wwlknfi 25142
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 25086 . . . . . 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 2762 . . . . . . 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 25087 . . . . . . . . . 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 1017 . . . . . . . . 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 703 . . . . . . . 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 2538 . . . . . . 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 987 . . . . . . . . . . 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 693 . . . . . . . . . 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 647 . . . . . . . . . 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 2536 . . . . . . . 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 2762 . . . . . . . 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 2434 . . . . . . 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 2459 . . . . . 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 2447 . . . . 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 463 . . . 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 10876 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
19183ad2ant3 1020 . . . . . . . 8  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  ( N  +  1 )  e.  NN0 )
2019anim2i 567 . . . . . . 7  |-  ( ( V  e.  Fin  /\  ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 ) )  ->  ( V  e. 
Fin  /\  ( N  +  1 )  e. 
NN0 ) )
2120ancoms 451 . . . . . 6  |-  ( ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  /\  V  e.  Fin )  ->  ( V  e.  Fin  /\  ( N  +  1 )  e.  NN0 )
)
22 wrdnfi 12625 . . . . . 6  |-  ( ( V  e.  Fin  /\  ( N  +  1
)  e.  NN0 )  ->  { w  e. Word  V  |  ( # `  w
)  =  ( N  +  1 ) }  e.  Fin )
2321, 22syl 17 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  /\  V  e.  Fin )  ->  { w  e. Word  V  |  ( # `  w
)  =  ( N  +  1 ) }  e.  Fin )
24 simpr 459 . . . . . . 7  |-  ( ( ( w  =/=  (/)  /\  A. i  e.  ( 0..^ ( ( # `  w
)  -  1 ) ) { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  E )  /\  ( # `  w )  =  ( N  +  1 ) )  ->  ( # `  w
)  =  ( N  +  1 ) )
2524rgenw 2764 . . . . . 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 3514 . . . . . 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 7774 . . . . 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 660 . . . 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 2490 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  /\  V  e.  Fin )  ->  ( ( V WWalksN  E
) `  N )  e.  Fin )
3130ex 432 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  ( V  e.  Fin  ->  (
( V WWalksN  E ) `  N )  e.  Fin ) )
32 wwlknndef 25141 . . . . 5  |-  ( ( V  e/  _V  \/  E  e/  _V  \/  N  e/  NN0 )  ->  (
( V WWalksN  E ) `  N )  =  (/) )
33 3ioran 992 . . . . . 6  |-  ( -.  ( V  e/  _V  \/  E  e/  _V  \/  N  e/  NN0 )  <->  ( -.  V  e/  _V  /\  -.  E  e/  _V  /\  -.  N  e/  NN0 ) )
34 nnel 2748 . . . . . . 7  |-  ( -.  V  e/  _V  <->  V  e.  _V )
35 nnel 2748 . . . . . . 7  |-  ( -.  E  e/  _V  <->  E  e.  _V )
36 nnel 2748 . . . . . . 7  |-  ( -.  N  e/  NN0  <->  N  e.  NN0 )
3734, 35, 363anbi123i 1186 . . . . . 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 7781 . . . . 5  |-  (/)  e.  Fin
4140a1i 11 . . . 4  |-  ( -.  ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  (/) 
e.  Fin )
4239, 41eqeltrd 2490 . . 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 367    \/ w3o 973    /\ w3a 974    = wceq 1405    e. wcel 1842   {cab 2387    =/= wne 2598    e/ wnel 2599   A.wral 2753   {crab 2757   _Vcvv 3058    C_ wss 3413   (/)c0 3737   {cpr 3973   ran crn 4823   ` cfv 5568  (class class class)co 6277   Fincfn 7553   0cc0 9521   1c1 9522    + caddc 9524    - cmin 9840   NN0cn0 10835  ..^cfzo 11852   #chash 12450  Word cword 12581   WWalks cwwlk 25081   WWalksN cwwlkn 25082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-2o 7167  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-fzo 11853  df-seq 12150  df-exp 12209  df-hash 12451  df-word 12589  df-wwlk 25083  df-wwlkn 25084
This theorem is referenced by:  wlknfi  25143  hashwwlkext  25150  rusgranumwlks  25360  clwlknclwlkdifnum  25365
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