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Theorem wlklniswwlkn2 25104
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 25090 . . 3  |-  ( P  e.  ( ( V WWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) ) )
2 simpl 455 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  V  e.  _V )
32adantr 463 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  ->  V  e.  _V )
4 simpr 459 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  E  e.  _V )
54adantr 463 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  ->  E  e.  _V )
6 simpl 455 . . . . . 6  |-  ( ( N  e.  NN0  /\  P  e. Word  V )  ->  N  e.  NN0 )
76adantl 464 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  ->  N  e.  NN0 )
8 iswwlkn 25088 . . . . 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 1230 . . . 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 12612 . . . . . . . . . . . . . 14  |-  ( P  e. Word  V  ->  ( # `
 P )  e. 
NN0 )
1110nn0cnd 10894 . . . . . . . . . . . . 13  |-  ( P  e. Word  V  ->  ( # `
 P )  e.  CC )
1211adantl 464 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  P  e. Word  V )  ->  ( # `  P
)  e.  CC )
13 1cnd 9641 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  P  e. Word  V )  ->  1  e.  CC )
14 nn0cn 10845 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  N  e.  CC )
1514adantr 463 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  P  e. Word  V )  ->  N  e.  CC )
1612, 13, 15subadd2d 9985 . . . . . . . . . . 11  |-  ( ( N  e.  NN0  /\  P  e. Word  V )  ->  ( ( ( # `  P )  -  1 )  =  N  <->  ( N  +  1 )  =  ( # `  P
) ) )
17 eqcom 2411 . . . . . . . . . . 11  |-  ( ( N  +  1 )  =  ( # `  P
)  <->  ( # `  P
)  =  ( N  +  1 ) )
1816, 17syl6rbb 262 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  P  e. Word  V )  ->  ( ( # `  P
)  =  ( N  +  1 )  <->  ( ( # `
 P )  - 
1 )  =  N ) )
1918adantl 464 . . . . . . . . 9  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  ->  ( ( # `
 P )  =  ( N  +  1 )  <->  ( ( # `  P )  -  1 )  =  N ) )
2019biimpcd 224 . . . . . . . 8  |-  ( (
# `  P )  =  ( N  + 
1 )  ->  (
( ( V  e. 
_V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  P  e. Word  V ) )  -> 
( ( # `  P
)  -  1 )  =  N ) )
2120adantl 464 . . . . . . 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 ) )
2221impcom 428 . . . . . 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 )
23 wlkiswwlk2 25101 . . . . . . . . . . . . 13  |-  ( V USGrph  E  ->  ( P  e.  ( V WWalks  E )  ->  E. f  f ( V Walks  E ) P ) )
2423com12 29 . . . . . . . . . . . 12  |-  ( P  e.  ( V WWalks  E
)  ->  ( V USGrph  E  ->  E. f  f ( V Walks  E ) P ) )
2524adantr 463 . . . . . . . . . . 11  |-  ( ( P  e.  ( V WWalks  E )  /\  ( # `
 P )  =  ( N  +  1 ) )  ->  ( V USGrph  E  ->  E. f 
f ( V Walks  E
) P ) )
2625adantl 464 . . . . . . . . . 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 ) )
2726imp 427 . . . . . . . . 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 )
28 simpr 459 . . . . . . . . . . . 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 )
29 wlklenvm1 24936 . . . . . . . . . . . 12  |-  ( f ( V Walks  E ) P  ->  ( # `  f
)  =  ( (
# `  P )  -  1 ) )
3028, 29jccir 537 . . . . . . . . . . 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 ) ) )
3130ex 432 . . . . . . . . . 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 ) ) ) )
3231eximdv 1731 . . . . . . . . 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 ) ) ) )
3327, 32mpd 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 ) ) )
34 eqeq2 2417 . . . . . . . . . 10  |-  ( ( ( # `  P
)  -  1 )  =  N  ->  (
( # `  f )  =  ( ( # `  P )  -  1 )  <->  ( # `  f
)  =  N ) )
3534anbi2d 702 . . . . . . . . 9  |-  ( ( ( # `  P
)  -  1 )  =  N  ->  (
( f ( V Walks 
E ) P  /\  ( # `  f )  =  ( ( # `  P )  -  1 ) )  <->  ( f
( V Walks  E ) P  /\  ( # `  f
)  =  N ) ) )
3635exbidv 1735 . . . . . . . 8  |-  ( ( ( # `  P
)  -  1 )  =  N  ->  ( E. f ( f ( V Walks  E ) P  /\  ( # `  f
)  =  ( (
# `  P )  -  1 ) )  <->  E. f ( f ( V Walks  E ) P  /\  ( # `  f
)  =  N ) ) )
3733, 36syl5ib 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 ) ) )
3837expd 434 . . . . . 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 ) ) ) )
3922, 38mpcom 34 . . . . 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 ) ) )
4039ex 432 . . . 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 ) ) ) )
419, 40sylbid 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 ) ) ) )
421, 41mpcom 34 . 2  |-  ( P  e.  ( ( V WWalksN  E ) `  N
)  ->  ( V USGrph  E  ->  E. f ( f ( V Walks  E ) P  /\  ( # `  f )  =  N ) ) )
4342com12 29 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 367    = wceq 1405   E.wex 1633    e. wcel 1842   _Vcvv 3058   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   CCcc 9519   1c1 9522    + caddc 9524    - cmin 9840   NN0cn0 10835   #chash 12450  Word cword 12581   USGrph cusg 24734   Walks cwalk 24902   WWalks cwwlk 25081   WWalksN cwwlkn 25082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-fzo 11853  df-hash 12451  df-word 12589  df-usgra 24737  df-wlk 24912  df-wwlk 25083  df-wwlkn 25084
This theorem is referenced by:  wlklniswwlkn  25105
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