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Theorem clwwlknimp 24978
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 24974 . 2  |-  ( W  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  W  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 W )  =  N ) ) )
2 simp1l 1018 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  W  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  W
)  =  N ) )  ->  V  e.  _V )
3 simp1r 1019 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  W  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  W
)  =  N ) )  ->  E  e.  _V )
4 simp3l 1022 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  W  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  W
)  =  N ) )  ->  N  e.  NN0 )
5 isclwwlkn 24971 . . . . 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 1226 . . . 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 24970 . . . . . 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 1015 . . . . 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 702 . . . 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 994 . . . . 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 566 . . . 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 6277 . . . . . . . . 9  |-  ( (
# `  W )  =  N  ->  ( (
# `  W )  -  1 )  =  ( N  -  1 ) )
1413oveq2d 6286 . . . . . . . 8  |-  ( (
# `  W )  =  N  ->  ( 0..^ ( ( # `  W
)  -  1 ) )  =  ( 0..^ ( N  -  1 ) ) )
1514raleqdv 3057 . . . . . . 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 1016 . . . . 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 427 . . . 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 999 . . . 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 1174 . . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106   {cpr 4018   ran crn 4989   ` cfv 5570  (class class class)co 6270   0cc0 9481   1c1 9482    + caddc 9484    - cmin 9796   NN0cn0 10791  ..^cfzo 11799   #chash 12387  Word cword 12518   lastS clsw 12519   ClWWalks cclwwlk 24950   ClWWalksN cclwwlkn 24951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12388  df-word 12526  df-clwwlk 24953  df-clwwlkn 24954
This theorem is referenced by:  clwwlkfo  24999  clwwlknwwlkncl  25002  wwlksubclwwlk  25006  usg2cwwk2dif  25022  extwwlkfablem1  25276  numclwwlkun  25281  extwwlkfab  25292
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