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Theorem wlklniswwlkn1 24998
Description: The sequence of vertices in a walk of length n is a walk as word of length n in an undirected simple graph. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
Assertion
Ref Expression
wlklniswwlkn1  |-  ( V USGrph  E  ->  ( ( F ( V Walks  E ) P  /\  ( # `  F )  =  N )  ->  P  e.  ( ( V WWalksN  E
) `  N )
) )

Proof of Theorem wlklniswwlkn1
StepHypRef Expression
1 wlkbprop 24822 . . . 4  |-  ( F ( V Walks  E ) P  ->  ( ( # `
 F )  e. 
NN0  /\  ( V  e.  _V  /\  E  e. 
_V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
21adantr 463 . . 3  |-  ( ( F ( V Walks  E
) P  /\  ( # `
 F )  =  N )  ->  (
( # `  F )  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
3 wlkiswwlk1 24989 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( F ( V Walks  E ) P  ->  P  e.  ( V WWalks  E ) ) )
43com12 29 . . . . . . . 8  |-  ( F ( V Walks  E ) P  ->  ( V USGrph  E  ->  P  e.  ( V WWalks  E ) ) )
54adantr 463 . . . . . . 7  |-  ( ( F ( V Walks  E
) P  /\  ( # `
 F )  =  N )  ->  ( V USGrph  E  ->  P  e.  ( V WWalks  E ) ) )
65adantl 464 . . . . . 6  |-  ( ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  ( F
( V Walks  E ) P  /\  ( # `  F
)  =  N ) )  ->  ( V USGrph  E  ->  P  e.  ( V WWalks  E ) ) )
76imp 427 . . . . 5  |-  ( ( ( ( ( # `  F )  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  ( F
( V Walks  E ) P  /\  ( # `  F
)  =  N ) )  /\  V USGrph  E
)  ->  P  e.  ( V WWalks  E ) )
8 2mwlk 24820 . . . . . . . . . 10  |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
9 ffn 5668 . . . . . . . . . . . 12  |-  ( P : ( 0 ... ( # `  F
) ) --> V  ->  P  Fn  ( 0 ... ( # `  F
) ) )
10 hashfn 12396 . . . . . . . . . . . 12  |-  ( P  Fn  ( 0 ... ( # `  F
) )  ->  ( # `
 P )  =  ( # `  (
0 ... ( # `  F
) ) ) )
11 hashfz0 12444 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  e.  NN0  ->  ( # `  (
0 ... ( # `  F
) ) )  =  ( ( # `  F
)  +  1 ) )
12113ad2ant1 1016 . . . . . . . . . . . . 13  |-  ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( # `  (
0 ... ( # `  F
) ) )  =  ( ( # `  F
)  +  1 ) )
13 eqeq1 2404 . . . . . . . . . . . . . 14  |-  ( (
# `  ( 0 ... ( # `  F
) ) )  =  ( # `  P
)  ->  ( ( # `
 ( 0 ... ( # `  F
) ) )  =  ( ( # `  F
)  +  1 )  <-> 
( # `  P )  =  ( ( # `  F )  +  1 ) ) )
1413eqcoms 2412 . . . . . . . . . . . . 13  |-  ( (
# `  P )  =  ( # `  (
0 ... ( # `  F
) ) )  -> 
( ( # `  (
0 ... ( # `  F
) ) )  =  ( ( # `  F
)  +  1 )  <-> 
( # `  P )  =  ( ( # `  F )  +  1 ) ) )
1512, 14syl5ib 219 . . . . . . . . . . . 12  |-  ( (
# `  P )  =  ( # `  (
0 ... ( # `  F
) ) )  -> 
( ( ( # `  F )  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( # `  P
)  =  ( (
# `  F )  +  1 ) ) )
169, 10, 153syl 20 . . . . . . . . . . 11  |-  ( P : ( 0 ... ( # `  F
) ) --> V  -> 
( ( ( # `  F )  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( # `  P
)  =  ( (
# `  F )  +  1 ) ) )
1716adantl 464 . . . . . . . . . 10  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( ( (
# `  F )  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  -> 
( # `  P )  =  ( ( # `  F )  +  1 ) ) )
188, 17syl 17 . . . . . . . . 9  |-  ( F ( V Walks  E ) P  ->  ( (
( # `  F )  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  -> 
( # `  P )  =  ( ( # `  F )  +  1 ) ) )
1918adantr 463 . . . . . . . 8  |-  ( ( F ( V Walks  E
) P  /\  ( # `
 F )  =  N )  ->  (
( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( # `  P
)  =  ( (
# `  F )  +  1 ) ) )
20 oveq1 6239 . . . . . . . . . 10  |-  ( (
# `  F )  =  N  ->  ( (
# `  F )  +  1 )  =  ( N  +  1 ) )
2120eqeq2d 2414 . . . . . . . . 9  |-  ( (
# `  F )  =  N  ->  ( (
# `  P )  =  ( ( # `  F )  +  1 )  <->  ( # `  P
)  =  ( N  +  1 ) ) )
2221adantl 464 . . . . . . . 8  |-  ( ( F ( V Walks  E
) P  /\  ( # `
 F )  =  N )  ->  (
( # `  P )  =  ( ( # `  F )  +  1 )  <->  ( # `  P
)  =  ( N  +  1 ) ) )
2319, 22sylibd 214 . . . . . . 7  |-  ( ( F ( V Walks  E
) P  /\  ( # `
 F )  =  N )  ->  (
( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( # `  P
)  =  ( N  +  1 ) ) )
2423impcom 428 . . . . . 6  |-  ( ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  ( F
( V Walks  E ) P  /\  ( # `  F
)  =  N ) )  ->  ( # `  P
)  =  ( N  +  1 ) )
2524adantr 463 . . . . 5  |-  ( ( ( ( ( # `  F )  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  ( F
( V Walks  E ) P  /\  ( # `  F
)  =  N ) )  /\  V USGrph  E
)  ->  ( # `  P
)  =  ( N  +  1 ) )
26 simpl2l 1048 . . . . . . . 8  |-  ( ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  ( F
( V Walks  E ) P  /\  ( # `  F
)  =  N ) )  ->  V  e.  _V )
27 simpl2r 1049 . . . . . . . 8  |-  ( ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  ( F
( V Walks  E ) P  /\  ( # `  F
)  =  N ) )  ->  E  e.  _V )
28 eleq1 2472 . . . . . . . . . . . . 13  |-  ( (
# `  F )  =  N  ->  ( (
# `  F )  e.  NN0  <->  N  e.  NN0 ) )
2928biimpcd 224 . . . . . . . . . . . 12  |-  ( (
# `  F )  e.  NN0  ->  ( ( # `
 F )  =  N  ->  N  e.  NN0 ) )
30293ad2ant1 1016 . . . . . . . . . . 11  |-  ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( ( # `
 F )  =  N  ->  N  e.  NN0 ) )
3130com12 29 . . . . . . . . . 10  |-  ( (
# `  F )  =  N  ->  ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  N  e.  NN0 ) )
3231adantl 464 . . . . . . . . 9  |-  ( ( F ( V Walks  E
) P  /\  ( # `
 F )  =  N )  ->  (
( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  N  e.  NN0 ) )
3332impcom 428 . . . . . . . 8  |-  ( ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  ( F
( V Walks  E ) P  /\  ( # `  F
)  =  N ) )  ->  N  e.  NN0 )
3426, 27, 333jca 1175 . . . . . . 7  |-  ( ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  ( F
( V Walks  E ) P  /\  ( # `  F
)  =  N ) )  ->  ( V  e.  _V  /\  E  e. 
_V  /\  N  e.  NN0 ) )
3534adantr 463 . . . . . 6  |-  ( ( ( ( ( # `  F )  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  ( F
( V Walks  E ) P  /\  ( # `  F
)  =  N ) )  /\  V USGrph  E
)  ->  ( V  e.  _V  /\  E  e. 
_V  /\  N  e.  NN0 ) )
36 iswwlkn 24983 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  ( P  e.  ( ( V WWalksN  E ) `  N
)  <->  ( P  e.  ( V WWalks  E )  /\  ( # `  P
)  =  ( N  +  1 ) ) ) )
3735, 36syl 17 . . . . 5  |-  ( ( ( ( ( # `  F )  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  ( F
( V Walks  E ) P  /\  ( # `  F
)  =  N ) )  /\  V USGrph  E
)  ->  ( P  e.  ( ( V WWalksN  E
) `  N )  <->  ( P  e.  ( V WWalks  E )  /\  ( # `
 P )  =  ( N  +  1 ) ) ) )
387, 25, 37mpbir2and 921 . . . 4  |-  ( ( ( ( ( # `  F )  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  ( F
( V Walks  E ) P  /\  ( # `  F
)  =  N ) )  /\  V USGrph  E
)  ->  P  e.  ( ( V WWalksN  E
) `  N )
)
3938ex 432 . . 3  |-  ( ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  /\  ( F
( V Walks  E ) P  /\  ( # `  F
)  =  N ) )  ->  ( V USGrph  E  ->  P  e.  ( ( V WWalksN  E ) `  N ) ) )
402, 39mpancom 667 . 2  |-  ( ( F ( V Walks  E
) P  /\  ( # `
 F )  =  N )  ->  ( V USGrph  E  ->  P  e.  ( ( V WWalksN  E
) `  N )
) )
4140com12 29 1  |-  ( V USGrph  E  ->  ( ( F ( V Walks  E ) P  /\  ( # `  F )  =  N )  ->  P  e.  ( ( V WWalksN  E
) `  N )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840   _Vcvv 3056   class class class wbr 4392   dom cdm 4940    Fn wfn 5518   -->wf 5519   ` cfv 5523  (class class class)co 6232   0cc0 9440   1c1 9441    + caddc 9443   NN0cn0 10754   ...cfz 11641   #chash 12357  Word cword 12488   USGrph cusg 24629   Walks cwalk 24797   WWalks cwwlk 24976   WWalksN cwwlkn 24977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-1o 7085  df-oadd 7089  df-er 7266  df-map 7377  df-pm 7378  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-card 8270  df-cda 8498  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-nn 10495  df-2 10553  df-n0 10755  df-z 10824  df-uz 11044  df-fz 11642  df-fzo 11766  df-hash 12358  df-word 12496  df-usgra 24632  df-wlk 24807  df-wwlk 24978  df-wwlkn 24979
This theorem is referenced by:  wlklniswwlkn  25000
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