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Theorem isclwwlk 30580
Description: Properties of a word to represent a closed walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Mar-2018.)
Assertion
Ref Expression
isclwwlk  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( 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 ) ) )
Distinct variable groups:    i, E    i, V    i, W
Allowed substitution hints:    X( i)    Y( i)

Proof of Theorem isclwwlk
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 clwwlk 30578 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( 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 ) } )
21eleq2d 2524 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( W  e.  ( V ClWWalks  E )  <->  W  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 ) } ) )
3 fveq2 5800 . . . . . . . 8  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
43oveq1d 6216 . . . . . . 7  |-  ( w  =  W  ->  (
( # `  w )  -  1 )  =  ( ( # `  W
)  -  1 ) )
54oveq2d 6217 . . . . . 6  |-  ( w  =  W  ->  (
0..^ ( ( # `  w )  -  1 ) )  =  ( 0..^ ( ( # `  W )  -  1 ) ) )
6 fveq1 5799 . . . . . . . 8  |-  ( w  =  W  ->  (
w `  i )  =  ( W `  i ) )
7 fveq1 5799 . . . . . . . 8  |-  ( w  =  W  ->  (
w `  ( i  +  1 ) )  =  ( W `  ( i  +  1 ) ) )
86, 7preq12d 4071 . . . . . . 7  |-  ( w  =  W  ->  { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  =  { ( W `  i ) ,  ( W `  ( i  +  1 ) ) } )
98eleq1d 2523 . . . . . 6  |-  ( w  =  W  ->  ( { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( W `  i
) ,  ( W `
 ( i  +  1 ) ) }  e.  ran  E ) )
105, 9raleqbidv 3037 . . . . 5  |-  ( w  =  W  ->  ( A. i  e.  (
0..^ ( ( # `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  A. i  e.  ( 0..^ ( ( # `  W )  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E ) )
11 fveq2 5800 . . . . . . 7  |-  ( w  =  W  ->  ( lastS  `  w )  =  ( lastS  `  W ) )
12 fveq1 5799 . . . . . . 7  |-  ( w  =  W  ->  (
w `  0 )  =  ( W ` 
0 ) )
1311, 12preq12d 4071 . . . . . 6  |-  ( w  =  W  ->  { ( lastS  `  w ) ,  ( w `  0 ) }  =  { ( lastS  `  W ) ,  ( W `  0 ) } )
1413eleq1d 2523 . . . . 5  |-  ( w  =  W  ->  ( { ( lastS  `  w ) ,  ( w ` 
0 ) }  e.  ran  E  <->  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E ) )
1510, 14anbi12d 710 . . . 4  |-  ( w  =  W  ->  (
( A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  w
) ,  ( w `
 0 ) }  e.  ran  E )  <-> 
( A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E ) ) )
1615elrab 3224 . . 3  |-  ( W  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 ) }  <->  ( 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 ) ) )
17 3anass 969 . . 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  e. Word  V  /\  ( A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E ) ) )
1816, 17bitr4i 252 . 2  |-  ( W  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 ) }  <->  ( 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 ) )
192, 18syl6bb 261 1  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( 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 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   {crab 2803   {cpr 3988   ran crn 4950   ` cfv 5527  (class class class)co 6201   0cc0 9394   1c1 9395    + caddc 9397    - cmin 9707  ..^cfzo 11666   #chash 12221  Word cword 12340   lastS clsw 12341   ClWWalks cclwwlk 30562
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 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-pm 7328  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-word 12348  df-clwwlk 30565
This theorem is referenced by:  clwwlkgt0  30583  clwwlkn2  30587  clwwlknimp  30588  clwlkisclwwlk  30600  clwwlkf  30605  clwwlkext2edg  30613  wwlkext2clwwlk  30614  clwwisshclww  30620  clwlkfclwwlk  30666  extwwlkfablem2  30820  numclwwlkovfel2  30825  numclwwlkovf2ex  30828  numclwwlkovgelim  30831
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