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Theorem usg2cwwk2dif 24593
Description: If a word represents a closed walk of length at least 2 in a undirected simple graph, the first two symbols of the word must be different. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
Assertion
Ref Expression
usg2cwwk2dif  |-  ( ( V USGrph  E  /\  N  e.  ( ZZ>= `  2 )  /\  W  e.  (
( V ClWWalksN  E ) `  N ) )  -> 
( W `  1
)  =/=  ( W `
 0 ) )

Proof of Theorem usg2cwwk2dif
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 clwwlknimp 24549 . . . 4  |-  ( W  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( W  e. Word  V  /\  ( # `
 W )  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W ) ,  ( W `  0 ) }  e.  ran  E
) )
2 simpr 461 . . . . . 6  |-  ( ( ( ( ( W  e. Word  V  /\  ( # `
 W )  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W ) ,  ( W `  0 ) }  e.  ran  E
)  /\  N  e.  ( ZZ>= `  2 )
)  /\  V USGrph  E )  ->  V USGrph  E )
3 uz2m1nn 11157 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( N  -  1 )  e.  NN )
4 lbfzo0 11831 . . . . . . . . . . . 12  |-  ( 0  e.  ( 0..^ ( N  -  1 ) )  <->  ( N  - 
1 )  e.  NN )
53, 4sylibr 212 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  2
)  ->  0  e.  ( 0..^ ( N  - 
1 ) ) )
6 fveq2 5866 . . . . . . . . . . . . . 14  |-  ( i  =  0  ->  ( W `  i )  =  ( W ` 
0 ) )
76adantl 466 . . . . . . . . . . . . 13  |-  ( ( N  e.  ( ZZ>= ` 
2 )  /\  i  =  0 )  -> 
( W `  i
)  =  ( W `
 0 ) )
8 oveq1 6292 . . . . . . . . . . . . . . . 16  |-  ( i  =  0  ->  (
i  +  1 )  =  ( 0  +  1 ) )
98adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  ( ZZ>= ` 
2 )  /\  i  =  0 )  -> 
( i  +  1 )  =  ( 0  +  1 ) )
10 0p1e1 10648 . . . . . . . . . . . . . . 15  |-  ( 0  +  1 )  =  1
119, 10syl6eq 2524 . . . . . . . . . . . . . 14  |-  ( ( N  e.  ( ZZ>= ` 
2 )  /\  i  =  0 )  -> 
( i  +  1 )  =  1 )
1211fveq2d 5870 . . . . . . . . . . . . 13  |-  ( ( N  e.  ( ZZ>= ` 
2 )  /\  i  =  0 )  -> 
( W `  (
i  +  1 ) )  =  ( W `
 1 ) )
137, 12preq12d 4114 . . . . . . . . . . . 12  |-  ( ( N  e.  ( ZZ>= ` 
2 )  /\  i  =  0 )  ->  { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  =  { ( W `  0 ) ,  ( W ` 
1 ) } )
1413eleq1d 2536 . . . . . . . . . . 11  |-  ( ( N  e.  ( ZZ>= ` 
2 )  /\  i  =  0 )  -> 
( { ( W `
 i ) ,  ( W `  (
i  +  1 ) ) }  e.  ran  E  <->  { ( W ` 
0 ) ,  ( W `  1 ) }  e.  ran  E
) )
155, 14rspcdv 3217 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( A. i  e.  ( 0..^ ( N  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  ->  { ( W `  0 ) ,  ( W ` 
1 ) }  e.  ran  E ) )
1615com12 31 . . . . . . . . 9  |-  ( A. i  e.  ( 0..^ ( N  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  ->  ( N  e.  ( ZZ>= `  2 )  ->  { ( W ` 
0 ) ,  ( W `  1 ) }  e.  ran  E
) )
17163ad2ant2 1018 . . . . . . . 8  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E )  ->  ( N  e.  ( ZZ>= `  2 )  ->  { ( W ` 
0 ) ,  ( W `  1 ) }  e.  ran  E
) )
1817imp 429 . . . . . . 7  |-  ( ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E )  /\  N  e.  (
ZZ>= `  2 ) )  ->  { ( W `
 0 ) ,  ( W `  1
) }  e.  ran  E )
1918adantr 465 . . . . . 6  |-  ( ( ( ( ( W  e. Word  V  /\  ( # `
 W )  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W ) ,  ( W `  0 ) }  e.  ran  E
)  /\  N  e.  ( ZZ>= `  2 )
)  /\  V USGrph  E )  ->  { ( W `
 0 ) ,  ( W `  1
) }  e.  ran  E )
20 usgraedgrn 24154 . . . . . . 7  |-  ( ( V USGrph  E  /\  { ( W `  0 ) ,  ( W ` 
1 ) }  e.  ran  E )  ->  ( W `  0 )  =/=  ( W `  1
) )
2120necomd 2738 . . . . . 6  |-  ( ( V USGrph  E  /\  { ( W `  0 ) ,  ( W ` 
1 ) }  e.  ran  E )  ->  ( W `  1 )  =/=  ( W `  0
) )
222, 19, 21syl2anc 661 . . . . 5  |-  ( ( ( ( ( W  e. Word  V  /\  ( # `
 W )  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W ) ,  ( W `  0 ) }  e.  ran  E
)  /\  N  e.  ( ZZ>= `  2 )
)  /\  V USGrph  E )  ->  ( W ` 
1 )  =/=  ( W `  0 )
)
2322exp31 604 . . . 4  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E )  ->  ( N  e.  ( ZZ>= `  2 )  ->  ( V USGrph  E  -> 
( W `  1
)  =/=  ( W `
 0 ) ) ) )
241, 23syl 16 . . 3  |-  ( W  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( N  e.  ( ZZ>= `  2 )  ->  ( V USGrph  E  -> 
( W `  1
)  =/=  ( W `
 0 ) ) ) )
2524com13 80 . 2  |-  ( V USGrph  E  ->  ( N  e.  ( ZZ>= `  2 )  ->  ( W  e.  ( ( V ClWWalksN  E ) `  N )  ->  ( W `  1 )  =/=  ( W `  0
) ) ) )
26253imp 1190 1  |-  ( ( V USGrph  E  /\  N  e.  ( ZZ>= `  2 )  /\  W  e.  (
( V ClWWalksN  E ) `  N ) )  -> 
( W `  1
)  =/=  ( W `
 0 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   {cpr 4029   class class class wbr 4447   ran crn 5000   ` cfv 5588  (class class class)co 6285   0cc0 9493   1c1 9494    + caddc 9496    - cmin 9806   NNcn 10537   2c2 10586   ZZ>=cuz 11083  ..^cfzo 11793   #chash 12374  Word cword 12501   lastS clsw 12502   USGrph cusg 24103   ClWWalksN cclwwlkn 24522
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-pm 7424  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-card 8321  df-cda 8549  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674  df-fzo 11794  df-hash 12375  df-word 12509  df-usgra 24106  df-clwwlk 24524  df-clwwlkn 24525
This theorem is referenced by:  usg2cwwkdifex  24594
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