Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wlklniswwlkn2 Structured version   Unicode version

Theorem wlklniswwlkn2 30259
Description: A walk of length n as word corresponds to the sequence of vertices in a walk of length n in an undirected simple graph. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
Assertion
Ref Expression
wlklniswwlkn2  |-  ( V USGrph  E  ->  ( P  e.  ( ( V WWalksN  E
) `  N )  ->  E. f ( f ( V Walks  E ) P  /\  ( # `  f )  =  N ) ) )
Distinct variable groups:    f, E    f, N    P, f    f, V

Proof of Theorem wlklniswwlkn2
StepHypRef Expression
1 wwlknprop 30245 . . 3  |-  ( P  e.  ( ( V WWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) ) )
2 simpl 454 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  V  e.  _V )
32adantr 462 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  ->  V  e.  _V )
4 simpr 458 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  E  e.  _V )
54adantr 462 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  ->  E  e.  _V )
6 simpl 454 . . . . . 6  |-  ( ( N  e.  NN0  /\  P  e. Word  V )  ->  N  e.  NN0 )
76adantl 463 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  ->  N  e.  NN0 )
8 iswwlkn 30243 . . . . 5  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  ( P  e.  ( ( V WWalksN  E ) `  N
)  <->  ( P  e.  ( V WWalks  E )  /\  ( # `  P
)  =  ( N  +  1 ) ) ) )
93, 5, 7, 8syl3anc 1213 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  ->  ( P  e.  ( ( V WWalksN  E
) `  N )  <->  ( P  e.  ( V WWalks  E )  /\  ( # `
 P )  =  ( N  +  1 ) ) ) )
10 lencl 12245 . . . . . . . . . . . . . 14  |-  ( P  e. Word  V  ->  ( # `
 P )  e. 
NN0 )
11 nn0cn 10585 . . . . . . . . . . . . . 14  |-  ( (
# `  P )  e.  NN0  ->  ( # `  P
)  e.  CC )
1210, 11syl 16 . . . . . . . . . . . . 13  |-  ( P  e. Word  V  ->  ( # `
 P )  e.  CC )
1312adantl 463 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  P  e. Word  V )  ->  ( # `  P
)  e.  CC )
14 ax-1cn 9336 . . . . . . . . . . . . 13  |-  1  e.  CC
1514a1i 11 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  P  e. Word  V )  ->  1  e.  CC )
16 nn0cn 10585 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  N  e.  CC )
1716adantr 462 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  P  e. Word  V )  ->  N  e.  CC )
1813, 15, 17subadd2d 9734 . . . . . . . . . . 11  |-  ( ( N  e.  NN0  /\  P  e. Word  V )  ->  ( ( ( # `  P )  -  1 )  =  N  <->  ( N  +  1 )  =  ( # `  P
) ) )
19 eqcom 2443 . . . . . . . . . . 11  |-  ( ( N  +  1 )  =  ( # `  P
)  <->  ( # `  P
)  =  ( N  +  1 ) )
2018, 19syl6rbb 262 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  P  e. Word  V )  ->  ( ( # `  P
)  =  ( N  +  1 )  <->  ( ( # `
 P )  - 
1 )  =  N ) )
2120adantl 463 . . . . . . . . 9  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  ->  ( ( # `
 P )  =  ( N  +  1 )  <->  ( ( # `  P )  -  1 )  =  N ) )
2221biimpcd 224 . . . . . . . 8  |-  ( (
# `  P )  =  ( N  + 
1 )  ->  (
( ( V  e. 
_V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  -> 
( ( # `  P
)  -  1 )  =  N ) )
2322adantl 463 . . . . . . 7  |-  ( ( P  e.  ( V WWalks  E )  /\  ( # `
 P )  =  ( N  +  1 ) )  ->  (
( ( V  e. 
_V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  -> 
( ( # `  P
)  -  1 )  =  N ) )
2423impcom 430 . . . . . 6  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  /\  ( P  e.  ( V WWalks  E )  /\  ( # `
 P )  =  ( N  +  1 ) ) )  -> 
( ( # `  P
)  -  1 )  =  N )
25 wlkiswwlk2 30256 . . . . . . . . . . . . 13  |-  ( V USGrph  E  ->  ( P  e.  ( V WWalks  E )  ->  E. f  f ( V Walks  E ) P ) )
2625com12 31 . . . . . . . . . . . 12  |-  ( P  e.  ( V WWalks  E
)  ->  ( V USGrph  E  ->  E. f  f ( V Walks  E ) P ) )
2726adantr 462 . . . . . . . . . . 11  |-  ( ( P  e.  ( V WWalks  E )  /\  ( # `
 P )  =  ( N  +  1 ) )  ->  ( V USGrph  E  ->  E. f 
f ( V Walks  E
) P ) )
2827adantl 463 . . . . . . . . . 10  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  /\  ( P  e.  ( V WWalks  E )  /\  ( # `
 P )  =  ( N  +  1 ) ) )  -> 
( V USGrph  E  ->  E. f  f ( V Walks 
E ) P ) )
2928imp 429 . . . . . . . . 9  |-  ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  /\  ( P  e.  ( V WWalks  E )  /\  ( # `
 P )  =  ( N  +  1 ) ) )  /\  V USGrph  E )  ->  E. f 
f ( V Walks  E
) P )
30 simpr 458 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  /\  ( P  e.  ( V WWalks  E
)  /\  ( # `  P
)  =  ( N  +  1 ) ) )  /\  V USGrph  E
)  /\  f ( V Walks  E ) P )  ->  f ( V Walks 
E ) P )
31 2mwlk 23362 . . . . . . . . . . . . 13  |-  ( f ( V Walks  E ) P  ->  ( f  e. Word  dom  E  /\  P : ( 0 ... ( # `  f
) ) --> V ) )
32 lencl 12245 . . . . . . . . . . . . . 14  |-  ( f  e. Word  dom  E  ->  (
# `  f )  e.  NN0 )
33 ffn 5556 . . . . . . . . . . . . . . 15  |-  ( P : ( 0 ... ( # `  f
) ) --> V  ->  P  Fn  ( 0 ... ( # `  f
) ) )
34 hashfn 12134 . . . . . . . . . . . . . . 15  |-  ( P  Fn  ( 0 ... ( # `  f
) )  ->  ( # `
 P )  =  ( # `  (
0 ... ( # `  f
) ) ) )
35 hashfz0 12189 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  f )  e.  NN0  ->  ( # `  (
0 ... ( # `  f
) ) )  =  ( ( # `  f
)  +  1 ) )
3635eqeq2d 2452 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  f )  e.  NN0  ->  ( ( # `
 P )  =  ( # `  (
0 ... ( # `  f
) ) )  <->  ( # `  P
)  =  ( (
# `  f )  +  1 ) ) )
37 nn0cn 10585 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  f )  e.  NN0  ->  ( # `  f
)  e.  CC )
3814a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  f )  e.  NN0  ->  1  e.  CC )
3937, 38pncand 9716 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  f )  e.  NN0  ->  ( (
( # `  f )  +  1 )  - 
1 )  =  (
# `  f )
)
4039eqcomd 2446 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  f )  e.  NN0  ->  ( # `  f
)  =  ( ( ( # `  f
)  +  1 )  -  1 ) )
41 oveq1 6097 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  P )  =  ( ( # `  f )  +  1 )  ->  ( ( # `
 P )  - 
1 )  =  ( ( ( # `  f
)  +  1 )  -  1 ) )
4241eqeq2d 2452 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  P )  =  ( ( # `  f )  +  1 )  ->  ( ( # `
 f )  =  ( ( # `  P
)  -  1 )  <-> 
( # `  f )  =  ( ( (
# `  f )  +  1 )  - 
1 ) ) )
4340, 42syl5ibrcom 222 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  f )  e.  NN0  ->  ( ( # `
 P )  =  ( ( # `  f
)  +  1 )  ->  ( # `  f
)  =  ( (
# `  P )  -  1 ) ) )
4436, 43sylbid 215 . . . . . . . . . . . . . . . 16  |-  ( (
# `  f )  e.  NN0  ->  ( ( # `
 P )  =  ( # `  (
0 ... ( # `  f
) ) )  -> 
( # `  f )  =  ( ( # `  P )  -  1 ) ) )
4544com12 31 . . . . . . . . . . . . . . 15  |-  ( (
# `  P )  =  ( # `  (
0 ... ( # `  f
) ) )  -> 
( ( # `  f
)  e.  NN0  ->  (
# `  f )  =  ( ( # `  P )  -  1 ) ) )
4633, 34, 453syl 20 . . . . . . . . . . . . . 14  |-  ( P : ( 0 ... ( # `  f
) ) --> V  -> 
( ( # `  f
)  e.  NN0  ->  (
# `  f )  =  ( ( # `  P )  -  1 ) ) )
4732, 46mpan9 466 . . . . . . . . . . . . 13  |-  ( ( f  e. Word  dom  E  /\  P : ( 0 ... ( # `  f
) ) --> V )  ->  ( # `  f
)  =  ( (
# `  P )  -  1 ) )
4831, 47syl 16 . . . . . . . . . . . 12  |-  ( f ( V Walks  E ) P  ->  ( # `  f
)  =  ( (
# `  P )  -  1 ) )
4930, 48jccir 536 . . . . . . . . . . 11  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  /\  ( P  e.  ( V WWalks  E
)  /\  ( # `  P
)  =  ( N  +  1 ) ) )  /\  V USGrph  E
)  /\  f ( V Walks  E ) P )  ->  ( f ( V Walks  E ) P  /\  ( # `  f
)  =  ( (
# `  P )  -  1 ) ) )
5049ex 434 . . . . . . . . . 10  |-  ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  /\  ( P  e.  ( V WWalks  E )  /\  ( # `
 P )  =  ( N  +  1 ) ) )  /\  V USGrph  E )  ->  (
f ( V Walks  E
) P  ->  (
f ( V Walks  E
) P  /\  ( # `
 f )  =  ( ( # `  P
)  -  1 ) ) ) )
5150eximdv 1681 . . . . . . . . 9  |-  ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  /\  ( P  e.  ( V WWalks  E )  /\  ( # `
 P )  =  ( N  +  1 ) ) )  /\  V USGrph  E )  ->  ( E. f  f ( V Walks  E ) P  ->  E. f ( f ( V Walks  E ) P  /\  ( # `  f
)  =  ( (
# `  P )  -  1 ) ) ) )
5229, 51mpd 15 . . . . . . . 8  |-  ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  /\  ( P  e.  ( V WWalks  E )  /\  ( # `
 P )  =  ( N  +  1 ) ) )  /\  V USGrph  E )  ->  E. f
( f ( V Walks 
E ) P  /\  ( # `  f )  =  ( ( # `  P )  -  1 ) ) )
53 eqeq2 2450 . . . . . . . . . 10  |-  ( ( ( # `  P
)  -  1 )  =  N  ->  (
( # `  f )  =  ( ( # `  P )  -  1 )  <->  ( # `  f
)  =  N ) )
5453anbi2d 698 . . . . . . . . 9  |-  ( ( ( # `  P
)  -  1 )  =  N  ->  (
( f ( V Walks 
E ) P  /\  ( # `  f )  =  ( ( # `  P )  -  1 ) )  <->  ( f
( V Walks  E ) P  /\  ( # `  f
)  =  N ) ) )
5554exbidv 1685 . . . . . . . 8  |-  ( ( ( # `  P
)  -  1 )  =  N  ->  ( E. f ( f ( V Walks  E ) P  /\  ( # `  f
)  =  ( (
# `  P )  -  1 ) )  <->  E. f ( f ( V Walks  E ) P  /\  ( # `  f
)  =  N ) ) )
5652, 55syl5ib 219 . . . . . . 7  |-  ( ( ( # `  P
)  -  1 )  =  N  ->  (
( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  /\  ( P  e.  ( V WWalks  E
)  /\  ( # `  P
)  =  ( N  +  1 ) ) )  /\  V USGrph  E
)  ->  E. f
( f ( V Walks 
E ) P  /\  ( # `  f )  =  N ) ) )
5756exp3a 436 . . . . . 6  |-  ( ( ( # `  P
)  -  1 )  =  N  ->  (
( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  /\  ( P  e.  ( V WWalks  E )  /\  ( # `
 P )  =  ( N  +  1 ) ) )  -> 
( V USGrph  E  ->  E. f ( f ( V Walks  E ) P  /\  ( # `  f
)  =  N ) ) ) )
5824, 57mpcom 36 . . . . 5  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  /\  ( P  e.  ( V WWalks  E )  /\  ( # `
 P )  =  ( N  +  1 ) ) )  -> 
( V USGrph  E  ->  E. f ( f ( V Walks  E ) P  /\  ( # `  f
)  =  N ) ) )
5958ex 434 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  ->  ( ( P  e.  ( V WWalks  E )  /\  ( # `  P )  =  ( N  +  1 ) )  ->  ( V USGrph  E  ->  E. f ( f ( V Walks  E ) P  /\  ( # `  f )  =  N ) ) ) )
609, 59sylbid 215 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  ->  ( P  e.  ( ( V WWalksN  E
) `  N )  ->  ( V USGrph  E  ->  E. f ( f ( V Walks  E ) P  /\  ( # `  f
)  =  N ) ) ) )
611, 60mpcom 36 . 2  |-  ( P  e.  ( ( V WWalksN  E ) `  N
)  ->  ( V USGrph  E  ->  E. f ( f ( V Walks  E ) P  /\  ( # `  f )  =  N ) ) )
6261com12 31 1  |-  ( V USGrph  E  ->  ( P  e.  ( ( V WWalksN  E
) `  N )  ->  E. f ( f ( V Walks  E ) P  /\  ( # `  f )  =  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364   E.wex 1591    e. wcel 1761   _Vcvv 2970   class class class wbr 4289   dom cdm 4836    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279    + caddc 9281    - cmin 9591   NN0cn0 10575   ...cfz 11433   #chash 12099  Word cword 12217   USGrph cusg 23199   Walks cwalk 23340   WWalks cwwlk 30236   WWalksN cwwlkn 30237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-hash 12100  df-word 12225  df-usgra 23201  df-wlk 23350  df-wwlk 30238  df-wwlkn 30239
This theorem is referenced by:  wlklniswwlkn  30260
  Copyright terms: Public domain W3C validator