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Theorem wlklniswwlkn1 30338
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 23438 . . . 4  |-  ( F ( V Walks  E ) P  ->  ( ( # `
 F )  e. 
NN0  /\  ( V  e.  _V  /\  E  e. 
_V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
21adantr 465 . . 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 30329 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( F ( V Walks  E ) P  ->  P  e.  ( V WWalks  E ) ) )
43com12 31 . . . . . . . 8  |-  ( F ( V Walks  E ) P  ->  ( V USGrph  E  ->  P  e.  ( V WWalks  E ) ) )
54adantr 465 . . . . . . 7  |-  ( ( F ( V Walks  E
) P  /\  ( # `
 F )  =  N )  ->  ( V USGrph  E  ->  P  e.  ( V WWalks  E ) ) )
65adantl 466 . . . . . 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 429 . . . . 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 23432 . . . . . . . . . 10  |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
9 ffn 5564 . . . . . . . . . . . 12  |-  ( P : ( 0 ... ( # `  F
) ) --> V  ->  P  Fn  ( 0 ... ( # `  F
) ) )
10 hashfn 12143 . . . . . . . . . . . 12  |-  ( P  Fn  ( 0 ... ( # `  F
) )  ->  ( # `
 P )  =  ( # `  (
0 ... ( # `  F
) ) ) )
11 hashfz0 12198 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  e.  NN0  ->  ( # `  (
0 ... ( # `  F
) ) )  =  ( ( # `  F
)  +  1 ) )
12113ad2ant1 1009 . . . . . . . . . . . . 13  |-  ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( # `  (
0 ... ( # `  F
) ) )  =  ( ( # `  F
)  +  1 ) )
13 eqeq1 2449 . . . . . . . . . . . . . 14  |-  ( (
# `  ( 0 ... ( # `  F
) ) )  =  ( # `  P
)  ->  ( ( # `
 ( 0 ... ( # `  F
) ) )  =  ( ( # `  F
)  +  1 )  <-> 
( # `  P )  =  ( ( # `  F )  +  1 ) ) )
1413eqcoms 2446 . . . . . . . . . . . . 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 466 . . . . . . . . . 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 16 . . . . . . . . 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 465 . . . . . . . 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 6103 . . . . . . . . . 10  |-  ( (
# `  F )  =  N  ->  ( (
# `  F )  +  1 )  =  ( N  +  1 ) )
2120eqeq2d 2454 . . . . . . . . 9  |-  ( (
# `  F )  =  N  ->  ( (
# `  P )  =  ( ( # `  F )  +  1 )  <->  ( # `  P
)  =  ( N  +  1 ) ) )
2221adantl 466 . . . . . . . 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 430 . . . . . 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 465 . . . . 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 1041 . . . . . . . 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 1042 . . . . . . . 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 2503 . . . . . . . . . . . . 13  |-  ( (
# `  F )  =  N  ->  ( (
# `  F )  e.  NN0  <->  N  e.  NN0 ) )
2928biimpcd 224 . . . . . . . . . . . 12  |-  ( (
# `  F )  e.  NN0  ->  ( ( # `
 F )  =  N  ->  N  e.  NN0 ) )
30293ad2ant1 1009 . . . . . . . . . . 11  |-  ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( ( # `
 F )  =  N  ->  N  e.  NN0 ) )
3130com12 31 . . . . . . . . . 10  |-  ( (
# `  F )  =  N  ->  ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  N  e.  NN0 ) )
3231adantl 466 . . . . . . . . 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 430 . . . . . . . 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 1168 . . . . . . 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 465 . . . . . 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 30323 . . . . . 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 16 . . . . 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 913 . . . 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 434 . . 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 669 . 2  |-  ( ( F ( V Walks  E
) P  /\  ( # `
 F )  =  N )  ->  ( V USGrph  E  ->  P  e.  ( ( V WWalksN  E
) `  N )
) )
4140com12 31 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2977   class class class wbr 4297   dom cdm 4845    Fn wfn 5418   -->wf 5419   ` cfv 5423  (class class class)co 6096   0cc0 9287   1c1 9288    + caddc 9290   NN0cn0 10584   ...cfz 11442   #chash 12108  Word cword 12226   USGrph cusg 23269   Walks cwalk 23410   WWalks cwwlk 30316   WWalksN cwwlkn 30317
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 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  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 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-fzo 11554  df-hash 12109  df-word 12234  df-usgra 23271  df-wlk 23420  df-wwlk 30318  df-wwlkn 30319
This theorem is referenced by:  wlklniswwlkn  30340
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