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Theorem wlklniswwlkn2 24532
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 24518 . . 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 24516 . . . . 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 1228 . . . 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 12542 . . . . . . . . . . . . . 14  |-  ( P  e. Word  V  ->  ( # `
 P )  e. 
NN0 )
1110nn0cnd 10866 . . . . . . . . . . . . 13  |-  ( P  e. Word  V  ->  ( # `
 P )  e.  CC )
1211adantl 466 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  P  e. Word  V )  ->  ( # `  P
)  e.  CC )
13 1cnd 9624 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  P  e. Word  V )  ->  1  e.  CC )
14 nn0cn 10817 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  N  e.  CC )
1514adantr 465 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  P  e. Word  V )  ->  N  e.  CC )
1612, 13, 15subadd2d 9961 . . . . . . . . . . 11  |-  ( ( N  e.  NN0  /\  P  e. Word  V )  ->  ( ( ( # `  P )  -  1 )  =  N  <->  ( N  +  1 )  =  ( # `  P
) ) )
17 eqcom 2476 . . . . . . . . . . 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 466 . . . . . . . . 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 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 ) )
2221impcom 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 )
23 wlkiswwlk2 24529 . . . . . . . . . . . . 13  |-  ( V USGrph  E  ->  ( P  e.  ( V WWalks  E )  ->  E. f  f ( V Walks  E ) P ) )
2423com12 31 . . . . . . . . . . . 12  |-  ( P  e.  ( V WWalks  E
)  ->  ( V USGrph  E  ->  E. f  f ( V Walks  E ) P ) )
2524adantr 465 . . . . . . . . . . 11  |-  ( ( P  e.  ( V WWalks  E )  /\  ( # `
 P )  =  ( N  +  1 ) )  ->  ( V USGrph  E  ->  E. f 
f ( V Walks  E
) P ) )
2625adantl 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 ) )
2726imp 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 )
28 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 )
29 wlklenvm1 24364 . . . . . . . . . . . 12  |-  ( f ( V Walks  E ) P  ->  ( # `  f
)  =  ( (
# `  P )  -  1 ) )
3028, 29jccir 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 ) ) )
3130ex 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 ) ) ) )
3231eximdv 1686 . . . . . . . . 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 2482 . . . . . . . . . 10  |-  ( ( ( # `  P
)  -  1 )  =  N  ->  (
( # `  f )  =  ( ( # `  P )  -  1 )  <->  ( # `  f
)  =  N ) )
3534anbi2d 703 . . . . . . . . 9  |-  ( ( ( # `  P
)  -  1 )  =  N  ->  (
( f ( V Walks 
E ) P  /\  ( # `  f )  =  ( ( # `  P )  -  1 ) )  <->  ( f
( V Walks  E ) P  /\  ( # `  f
)  =  N ) ) )
3635exbidv 1690 . . . . . . . 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 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 ) ) ) )
3922, 38mpcom 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 ) ) )
4039ex 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 ) ) ) )
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 36 . 2  |-  ( P  e.  ( ( V WWalksN  E ) `  N
)  ->  ( V USGrph  E  ->  E. f ( f ( V Walks  E ) P  /\  ( # `  f )  =  N ) ) )
4342com12 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 1379   E.wex 1596    e. wcel 1767   _Vcvv 3118   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   CCcc 9502   1c1 9505    + caddc 9507    - cmin 9817   NN0cn0 10807   #chash 12385  Word cword 12514   USGrph cusg 24162   Walks cwalk 24330   WWalks cwwlk 24509   WWalksN cwwlkn 24510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-hash 12386  df-word 12522  df-usgra 24165  df-wlk 24340  df-wwlk 24511  df-wwlkn 24512
This theorem is referenced by:  wlklniswwlkn  24533
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