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Theorem clwwlknimp 30577
Description: Implications for a set being a closed walk (represented by a word). (Contributed by Alexander van der Vekens, 17-Jun-2018.)
Assertion
Ref Expression
clwwlknimp  |-  ( 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
) )
Distinct variable groups:    i, E    i, V    i, W    i, N

Proof of Theorem clwwlknimp
StepHypRef Expression
1 clwwlknprop 30573 . 2  |-  ( W  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  W  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 W )  =  N ) ) )
2 simp1l 1012 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  W  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  W
)  =  N ) )  ->  V  e.  _V )
3 simp1r 1013 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  W  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  W
)  =  N ) )  ->  E  e.  _V )
4 simp3l 1016 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  W  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  W
)  =  N ) )  ->  N  e.  NN0 )
5 isclwwlkn 30570 . . . . 5  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  ( W  e.  ( ( V ClWWalksN  E ) `  N
)  <->  ( W  e.  ( V ClWWalks  E )  /\  ( # `  W
)  =  N ) ) )
62, 3, 4, 5syl3anc 1219 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  W  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  W
)  =  N ) )  ->  ( W  e.  ( ( V ClWWalksN  E ) `
 N )  <->  ( W  e.  ( V ClWWalks  E )  /\  ( # `  W
)  =  N ) ) )
7 isclwwlk 30569 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( W  e.  ( V ClWWalks  E )  <->  ( W  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E ) ) )
873ad2ant1 1009 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  W  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  W
)  =  N ) )  ->  ( W  e.  ( V ClWWalks  E )  <->  ( W  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  W
)  -  1 ) ) { ( W `
 i ) ,  ( W `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E ) ) )
98anbi1d 704 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  W  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  W
)  =  N ) )  ->  ( ( W  e.  ( V ClWWalks  E )  /\  ( # `  W )  =  N )  <->  ( ( W  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  W
)  -  1 ) ) { ( W `
 i ) ,  ( W `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E )  /\  ( # `  W
)  =  N ) ) )
106, 9bitrd 253 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  W  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  W
)  =  N ) )  ->  ( W  e.  ( ( V ClWWalksN  E ) `
 N )  <->  ( ( W  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  W
)  -  1 ) ) { ( W `
 i ) ,  ( W `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E )  /\  ( # `  W
)  =  N ) ) )
11 simp1 988 . . . . 5  |-  ( ( W  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  W
)  -  1 ) ) { ( W `
 i ) ,  ( W `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E )  ->  W  e. Word  V
)
1211anim1i 568 . . . 4  |-  ( ( ( W  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  W )  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W ) ,  ( W `  0 ) }  e.  ran  E
)  /\  ( # `  W
)  =  N )  ->  ( W  e. Word  V  /\  ( # `  W
)  =  N ) )
13 oveq1 6197 . . . . . . . . 9  |-  ( (
# `  W )  =  N  ->  ( (
# `  W )  -  1 )  =  ( N  -  1 ) )
1413oveq2d 6206 . . . . . . . 8  |-  ( (
# `  W )  =  N  ->  ( 0..^ ( ( # `  W
)  -  1 ) )  =  ( 0..^ ( N  -  1 ) ) )
1514raleqdv 3019 . . . . . . 7  |-  ( (
# `  W )  =  N  ->  ( A. i  e.  ( 0..^ ( ( # `  W
)  -  1 ) ) { ( W `
 i ) ,  ( W `  (
i  +  1 ) ) }  e.  ran  E  <->  A. i  e.  (
0..^ ( N  - 
1 ) ) { ( W `  i
) ,  ( W `
 ( i  +  1 ) ) }  e.  ran  E ) )
1615biimpcd 224 . . . . . 6  |-  ( A. i  e.  ( 0..^ ( ( # `  W
)  -  1 ) ) { ( W `
 i ) ,  ( W `  (
i  +  1 ) ) }  e.  ran  E  ->  ( ( # `  W )  =  N  ->  A. i  e.  ( 0..^ ( N  - 
1 ) ) { ( W `  i
) ,  ( W `
 ( i  +  1 ) ) }  e.  ran  E ) )
17163ad2ant2 1010 . . . . 5  |-  ( ( W  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  W
)  -  1 ) ) { ( W `
 i ) ,  ( W `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E )  ->  ( ( # `  W )  =  N  ->  A. i  e.  ( 0..^ ( N  - 
1 ) ) { ( W `  i
) ,  ( W `
 ( i  +  1 ) ) }  e.  ran  E ) )
1817imp 429 . . . 4  |-  ( ( ( W  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  W )  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W ) ,  ( W `  0 ) }  e.  ran  E
)  /\  ( # `  W
)  =  N )  ->  A. i  e.  ( 0..^ ( N  - 
1 ) ) { ( W `  i
) ,  ( W `
 ( i  +  1 ) ) }  e.  ran  E )
19 simpl3 993 . . . 4  |-  ( ( ( W  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  W )  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W ) ,  ( W `  0 ) }  e.  ran  E
)  /\  ( # `  W
)  =  N )  ->  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E )
2012, 18, 193jca 1168 . . 3  |-  ( ( ( W  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  W )  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W ) ,  ( W `  0 ) }  e.  ran  E
)  /\  ( # `  W
)  =  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
) )
2110, 20syl6bi 228 . 2  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  W  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  W
)  =  N ) )  ->  ( 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 ) ) )
221, 21mpcom 36 1  |-  ( 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
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2795   _Vcvv 3068   {cpr 3977   ran crn 4939   ` cfv 5516  (class class class)co 6190   0cc0 9383   1c1 9384    + caddc 9386    - cmin 9696   NN0cn0 10680  ..^cfzo 11649   #chash 12204  Word cword 12323   lastS clsw 12324   ClWWalks cclwwlk 30551   ClWWalksN cclwwlkn 30552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-map 7316  df-pm 7317  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-n0 10681  df-z 10748  df-uz 10963  df-fz 11539  df-fzo 11650  df-hash 12205  df-word 12331  df-clwwlk 30554  df-clwwlkn 30555
This theorem is referenced by:  clwwlkfo  30597  clwwlknwwlkncl  30600  wwlksubclwwlk  30604  usg2cwwk2dif  30632  extwwlkfablem1  30805  numclwwlkun  30810  extwwlkfab  30821
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