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Theorem wlklniswwlkn2 30353
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 30339 . . 3  |-  ( P  e.  ( ( V WWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) ) )
2 simpl 457 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  V  e.  _V )
32adantr 465 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  ->  V  e.  _V )
4 simpr 461 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  E  e.  _V )
54adantr 465 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  ->  E  e.  _V )
6 simpl 457 . . . . . 6  |-  ( ( N  e.  NN0  /\  P  e. Word  V )  ->  N  e.  NN0 )
76adantl 466 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  ->  N  e.  NN0 )
8 iswwlkn 30337 . . . . 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 1218 . . . 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 12264 . . . . . . . . . . . . . 14  |-  ( P  e. Word  V  ->  ( # `
 P )  e. 
NN0 )
11 nn0cn 10604 . . . . . . . . . . . . . 14  |-  ( (
# `  P )  e.  NN0  ->  ( # `  P
)  e.  CC )
1210, 11syl 16 . . . . . . . . . . . . 13  |-  ( P  e. Word  V  ->  ( # `
 P )  e.  CC )
1312adantl 466 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  P  e. Word  V )  ->  ( # `  P
)  e.  CC )
14 ax-1cn 9355 . . . . . . . . . . . . 13  |-  1  e.  CC
1514a1i 11 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  P  e. Word  V )  ->  1  e.  CC )
16 nn0cn 10604 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  N  e.  CC )
1716adantr 465 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  P  e. Word  V )  ->  N  e.  CC )
1813, 15, 17subadd2d 9753 . . . . . . . . . . 11  |-  ( ( N  e.  NN0  /\  P  e. Word  V )  ->  ( ( ( # `  P )  -  1 )  =  N  <->  ( N  +  1 )  =  ( # `  P
) ) )
19 eqcom 2445 . . . . . . . . . . 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 466 . . . . . . . . 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 466 . . . . . . 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 30350 . . . . . . . . . . . . 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 465 . . . . . . . . . . 11  |-  ( ( P  e.  ( V WWalks  E )  /\  ( # `
 P )  =  ( N  +  1 ) )  ->  ( V USGrph  E  ->  E. f 
f ( V Walks  E
) P ) )
2827adantl 466 . . . . . . . . . 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 461 . . . . . . . . . . . 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 23442 . . . . . . . . . . . . 13  |-  ( f ( V Walks  E ) P  ->  ( f  e. Word  dom  E  /\  P : ( 0 ... ( # `  f
) ) --> V ) )
32 lencl 12264 . . . . . . . . . . . . . 14  |-  ( f  e. Word  dom  E  ->  (
# `  f )  e.  NN0 )
33 ffn 5574 . . . . . . . . . . . . . . 15  |-  ( P : ( 0 ... ( # `  f
) ) --> V  ->  P  Fn  ( 0 ... ( # `  f
) ) )
34 hashfn 12153 . . . . . . . . . . . . . . 15  |-  ( P  Fn  ( 0 ... ( # `  f
) )  ->  ( # `
 P )  =  ( # `  (
0 ... ( # `  f
) ) ) )
35 hashfz0 12208 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  f )  e.  NN0  ->  ( # `  (
0 ... ( # `  f
) ) )  =  ( ( # `  f
)  +  1 ) )
3635eqeq2d 2454 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  f )  e.  NN0  ->  ( ( # `
 P )  =  ( # `  (
0 ... ( # `  f
) ) )  <->  ( # `  P
)  =  ( (
# `  f )  +  1 ) ) )
37 nn0cn 10604 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  f )  e.  NN0  ->  ( # `  f
)  e.  CC )
3814a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  f )  e.  NN0  ->  1  e.  CC )
3937, 38pncand 9735 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  f )  e.  NN0  ->  ( (
( # `  f )  +  1 )  - 
1 )  =  (
# `  f )
)
4039eqcomd 2448 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  f )  e.  NN0  ->  ( # `  f
)  =  ( ( ( # `  f
)  +  1 )  -  1 ) )
41 oveq1 6113 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  P )  =  ( ( # `  f )  +  1 )  ->  ( ( # `
 P )  - 
1 )  =  ( ( ( # `  f
)  +  1 )  -  1 ) )
4241eqeq2d 2454 . . . . . . . . . . . . . . . . . 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 469 . . . . . . . . . . . . 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 539 . . . . . . . . . . 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 1676 . . . . . . . . 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 2452 . . . . . . . . . 10  |-  ( ( ( # `  P
)  -  1 )  =  N  ->  (
( # `  f )  =  ( ( # `  P )  -  1 )  <->  ( # `  f
)  =  N ) )
5453anbi2d 703 . . . . . . . . 9  |-  ( ( ( # `  P
)  -  1 )  =  N  ->  (
( f ( V Walks 
E ) P  /\  ( # `  f )  =  ( ( # `  P )  -  1 ) )  <->  ( f
( V Walks  E ) P  /\  ( # `  f
)  =  N ) ) )
5554exbidv 1680 . . . . . . . 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 ) ) )
5756expd 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 1369   E.wex 1586    e. wcel 1756   _Vcvv 2987   class class class wbr 4307   dom cdm 4855    Fn wfn 5428   -->wf 5429   ` cfv 5433  (class class class)co 6106   CCcc 9295   0cc0 9297   1c1 9298    + caddc 9300    - cmin 9610   NN0cn0 10594   ...cfz 11452   #chash 12118  Word cword 12236   USGrph cusg 23279   Walks cwalk 23420   WWalks cwwlk 30330   WWalksN cwwlkn 30331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-1o 6935  df-oadd 6939  df-er 7116  df-map 7231  df-pm 7232  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-card 8124  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-nn 10338  df-2 10395  df-n0 10595  df-z 10662  df-uz 10877  df-fz 11453  df-fzo 11564  df-hash 12119  df-word 12244  df-usgra 23281  df-wlk 23430  df-wwlk 30332  df-wwlkn 30333
This theorem is referenced by:  wlklniswwlkn  30354
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