MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isclwwlk Structured version   Unicode version

Theorem isclwwlk 24894
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 24892 . . 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 2527 . 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 5872 . . . . . . . 8  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
43oveq1d 6311 . . . . . . 7  |-  ( w  =  W  ->  (
( # `  w )  -  1 )  =  ( ( # `  W
)  -  1 ) )
54oveq2d 6312 . . . . . 6  |-  ( w  =  W  ->  (
0..^ ( ( # `  w )  -  1 ) )  =  ( 0..^ ( ( # `  W )  -  1 ) ) )
6 fveq1 5871 . . . . . . . 8  |-  ( w  =  W  ->  (
w `  i )  =  ( W `  i ) )
7 fveq1 5871 . . . . . . . 8  |-  ( w  =  W  ->  (
w `  ( i  +  1 ) )  =  ( W `  ( i  +  1 ) ) )
86, 7preq12d 4119 . . . . . . 7  |-  ( w  =  W  ->  { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  =  { ( W `  i ) ,  ( W `  ( i  +  1 ) ) } )
98eleq1d 2526 . . . . . 6  |-  ( w  =  W  ->  ( { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( W `  i
) ,  ( W `
 ( i  +  1 ) ) }  e.  ran  E ) )
105, 9raleqbidv 3068 . . . . 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 5872 . . . . . . 7  |-  ( w  =  W  ->  ( lastS  `  w )  =  ( lastS  `  W ) )
12 fveq1 5871 . . . . . . 7  |-  ( w  =  W  ->  (
w `  0 )  =  ( W ` 
0 ) )
1311, 12preq12d 4119 . . . . . 6  |-  ( w  =  W  ->  { ( lastS  `  w ) ,  ( w `  0 ) }  =  { ( lastS  `  W ) ,  ( W `  0 ) } )
1413eleq1d 2526 . . . . 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 3257 . . 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 977 . . 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 973    = wceq 1395    e. wcel 1819   A.wral 2807   {crab 2811   {cpr 4034   ran crn 5009   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510    + caddc 9512    - cmin 9824  ..^cfzo 11820   #chash 12407  Word cword 12537   lastS clsw 12538   ClWWalks cclwwlk 24874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11821  df-hash 12408  df-word 12545  df-clwwlk 24877
This theorem is referenced by:  clwwlkgt0  24897  clwwlkn2  24901  clwwlknimp  24902  clwlkisclwwlk  24915  clwwlkf  24920  clwwlkext2edg  24928  wwlkext2clwwlk  24929  clwwisshclww  24933  clwlkfclwwlk  24970  extwwlkfablem2  25204  numclwwlkovfel2  25209  numclwwlkovf2ex  25212  numclwwlkovgelim  25215
  Copyright terms: Public domain W3C validator