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Theorem wwlknprop 30444
Description: Properties of a walk of a fixed length (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 16-Jul-2018.)
Assertion
Ref Expression
wwlknprop  |-  ( P  e.  ( ( V WWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) ) )

Proof of Theorem wwlknprop
Dummy variables  e  n  v  w  i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-wwlkn 30438 . . 3  |- WWalksN  =  ( v  e.  _V , 
e  e.  _V  |->  ( n  e.  NN0  |->  { w  e.  ( v WWalks  e )  |  ( # `  w
)  =  ( n  +  1 ) } ) )
2 df-rab 2801 . . . . . . 7  |-  { w  e.  ( v WWalks  e )  |  ( # `  w
)  =  ( n  +  1 ) }  =  { w  |  ( w  e.  ( v WWalks  e )  /\  ( # `  w )  =  ( n  + 
1 ) ) }
3 iswwlk 30441 . . . . . . . . . 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
) ) )
43adantr 465 . . . . . . . . 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
) ) )
54anbi1d 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 ) ) ) )
65abbidv 2584 . . . . . . 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 ) ) } )
72, 6syl5eq 2502 . . . . . 6  |-  ( ( ( v  e.  _V  /\  e  e.  _V )  /\  n  e.  NN0 )  ->  { 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 . . . . . . . . . 10  |-  ( ( 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 . . . . . . . . 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 ) ) )
10 anass 649 . . . . . . . . 9  |-  ( ( ( 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 . . . . . . . 8  |-  ( ( ( 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 2582 . . . . . . 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  |  ( 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 2801 . . . . . . 7  |-  { 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 2481 . . . . . 6  |-  { 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 2506 . . . . 5  |-  ( ( ( v  e.  _V  /\  e  e.  _V )  /\  n  e.  NN0 )  ->  { 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 ) ) } )
1615mpteq2dva 4462 . . . 4  |-  ( ( v  e.  _V  /\  e  e.  _V )  ->  ( n  e.  NN0  |->  { w  e.  (
v WWalks  e )  |  (
# `  w )  =  ( n  + 
1 ) } )  =  ( n  e. 
NN0  |->  { w  e. Word 
v  |  ( ( w  =/=  (/)  /\  A. i  e.  ( 0..^ ( ( # `  w
)  -  1 ) ) { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  e )  /\  ( # `
 w )  =  ( n  +  1 ) ) } ) )
1716mpt2eq3ia 6236 . . 3  |-  ( v  e.  _V ,  e  e.  _V  |->  ( n  e.  NN0  |->  { w  e.  ( v WWalks  e )  |  ( # `  w
)  =  ( n  +  1 ) } ) )  =  ( v  e.  _V , 
e  e.  _V  |->  ( n  e.  NN0  |->  { w  e. Word  v  |  ( ( w  =/=  (/)  /\  A. i  e.  ( 0..^ ( ( # `  w
)  -  1 ) ) { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  e )  /\  ( # `
 w )  =  ( n  +  1 ) ) } ) )
181, 17eqtri 2478 . 2  |- WWalksN  =  ( v  e.  _V , 
e  e.  _V  |->  ( n  e.  NN0  |->  { w  e. Word  v  |  ( ( w  =/=  (/)  /\  A. i  e.  ( 0..^ ( ( # `  w
)  -  1 ) ) { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  e )  /\  ( # `
 w )  =  ( n  +  1 ) ) } ) )
1918elovmptnn0wrd 30381 1  |-  ( P  e.  ( ( V WWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1757   {cab 2435    =/= wne 2641   A.wral 2792   {crab 2796   _Vcvv 3054   (/)c0 3721   {cpr 3963    |-> cmpt 4434   ran crn 4925   ` cfv 5502  (class class class)co 6176    |-> cmpt2 6178   0cc0 9369   1c1 9370    + caddc 9372    - cmin 9682   NN0cn0 10666  ..^cfzo 11635   #chash 12190  Word cword 12309   WWalks cwwlk 30435   WWalksN cwwlkn 30436
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 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446
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 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-1st 6663  df-2nd 6664  df-recs 6918  df-rdg 6952  df-1o 7006  df-oadd 7010  df-er 7187  df-map 7302  df-pm 7303  df-en 7397  df-dom 7398  df-sdom 7399  df-fin 7400  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-nn 10410  df-n0 10667  df-z 10734  df-uz 10949  df-fz 11525  df-fzo 11636  df-word 12317  df-wwlk 30437  df-wwlkn 30438
This theorem is referenced by:  wwlknimp  30445  wwlkn0  30447  wlklniswwlkn2  30458  wwlkiswwlkn  30460  wwlknred  30479  wwlknext  30480  wwlkextwrd  30484  wwlkextsur  30487  wwlkextbij0  30488  wwlkexthasheq  30490  wwlknndef  30493  wwlkext2clwwlk  30589  wwlkextproplem3  30686  numclwwlk2lem1  30819
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