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Theorem clwwlkn2 24589
Description: In an undirected simple graph, a closed walk of length 2 represented as word is a word consisting of 2 symbols representing vertices connected by an edge. (Contributed by Alexander van der Vekens, 19-Sep-2018.)
Assertion
Ref Expression
clwwlkn2  |-  ( V USGrph  E  ->  ( W  e.  ( ( V ClWWalksN  E ) `
 2 )  <->  ( ( # `
 W )  =  2  /\  W  e. Word  V  /\  { ( W `
 0 ) ,  ( W `  1
) }  e.  ran  E ) ) )

Proof of Theorem clwwlkn2
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 usgrav 24152 . . 3  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2 2nn0 10824 . . . . 5  |-  2  e.  NN0
3 isclwwlkn 24583 . . . . 5  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  2  e.  NN0 )  ->  ( W  e.  ( ( V ClWWalksN  E ) `  2
)  <->  ( W  e.  ( V ClWWalks  E )  /\  ( # `  W
)  =  2 ) ) )
42, 3mp3an3 1313 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( W  e.  ( ( V ClWWalksN  E ) `  2 )  <->  ( W  e.  ( V ClWWalks  E )  /\  ( # `  W
)  =  2 ) ) )
5 isclwwlk 24582 . . . . 5  |-  ( ( 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 ) ) )
65anbi1d 704 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( ( W  e.  ( V ClWWalks  E )  /\  ( # `  W
)  =  2 )  <-> 
( ( 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
)  =  2 ) ) )
74, 6bitrd 253 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( W  e.  ( ( V ClWWalksN  E ) `  2 )  <->  ( ( 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
)  =  2 ) ) )
81, 7syl 16 . 2  |-  ( V USGrph  E  ->  ( W  e.  ( ( V ClWWalksN  E ) `
 2 )  <->  ( ( 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
)  =  2 ) ) )
9 3anass 977 . . . . 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  /\  ( A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E ) ) )
10 oveq1 6302 . . . . . . . . . . . 12  |-  ( (
# `  W )  =  2  ->  (
( # `  W )  -  1 )  =  ( 2  -  1 ) )
1110oveq2d 6311 . . . . . . . . . . 11  |-  ( (
# `  W )  =  2  ->  (
0..^ ( ( # `  W )  -  1 ) )  =  ( 0..^ ( 2  -  1 ) ) )
1211raleqdv 3069 . . . . . . . . . 10  |-  ( (
# `  W )  =  2  ->  ( A. i  e.  (
0..^ ( ( # `  W )  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  <->  A. i  e.  ( 0..^ ( 2  -  1 ) ) { ( W `  i
) ,  ( W `
 ( i  +  1 ) ) }  e.  ran  E ) )
1312ad2antlr 726 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  ( # `  W )  =  2 )  /\  W  e. Word  V )  ->  ( A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  <->  A. i  e.  ( 0..^ ( 2  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E ) )
14 2m1e1 10662 . . . . . . . . . . . . 13  |-  ( 2  -  1 )  =  1
1514oveq2i 6306 . . . . . . . . . . . 12  |-  ( 0..^ ( 2  -  1 ) )  =  ( 0..^ 1 )
16 fzo01 11877 . . . . . . . . . . . 12  |-  ( 0..^ 1 )  =  {
0 }
1715, 16eqtri 2496 . . . . . . . . . . 11  |-  ( 0..^ ( 2  -  1 ) )  =  {
0 }
1817a1i 11 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  ( # `  W )  =  2 )  /\  W  e. Word  V )  ->  ( 0..^ ( 2  -  1 ) )  =  { 0 } )
1918raleqdv 3069 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  ( # `  W )  =  2 )  /\  W  e. Word  V )  ->  ( A. i  e.  ( 0..^ ( 2  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  <->  A. i  e.  { 0 }  { ( W `
 i ) ,  ( W `  (
i  +  1 ) ) }  e.  ran  E ) )
20 c0ex 9602 . . . . . . . . . 10  |-  0  e.  _V
21 fveq2 5872 . . . . . . . . . . . . 13  |-  ( i  =  0  ->  ( W `  i )  =  ( W ` 
0 ) )
22 oveq1 6302 . . . . . . . . . . . . . . 15  |-  ( i  =  0  ->  (
i  +  1 )  =  ( 0  +  1 ) )
23 0p1e1 10659 . . . . . . . . . . . . . . 15  |-  ( 0  +  1 )  =  1
2422, 23syl6eq 2524 . . . . . . . . . . . . . 14  |-  ( i  =  0  ->  (
i  +  1 )  =  1 )
2524fveq2d 5876 . . . . . . . . . . . . 13  |-  ( i  =  0  ->  ( W `  ( i  +  1 ) )  =  ( W ` 
1 ) )
2621, 25preq12d 4120 . . . . . . . . . . . 12  |-  ( i  =  0  ->  { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  =  { ( W ` 
0 ) ,  ( W `  1 ) } )
2726eleq1d 2536 . . . . . . . . . . 11  |-  ( i  =  0  ->  ( { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( W `  0
) ,  ( W `
 1 ) }  e.  ran  E ) )
2827ralsng 4068 . . . . . . . . . 10  |-  ( 0  e.  _V  ->  ( A. i  e.  { 0 }  { ( W `
 i ) ,  ( W `  (
i  +  1 ) ) }  e.  ran  E  <->  { ( W ` 
0 ) ,  ( W `  1 ) }  e.  ran  E
) )
2920, 28mp1i 12 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  ( # `  W )  =  2 )  /\  W  e. Word  V )  ->  ( A. i  e. 
{ 0 }  {
( W `  i
) ,  ( W `
 ( i  +  1 ) ) }  e.  ran  E  <->  { ( W `  0 ) ,  ( W ` 
1 ) }  e.  ran  E ) )
3013, 19, 293bitrd 279 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  ( # `  W )  =  2 )  /\  W  e. Word  V )  ->  ( A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( W `  0
) ,  ( W `
 1 ) }  e.  ran  E ) )
3130anbi1d 704 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( # `  W )  =  2 )  /\  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 `
 0 ) ,  ( W `  1
) }  e.  ran  E  /\  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E ) ) )
32 lsw 12565 . . . . . . . . . . . . . 14  |-  ( W  e. Word  V  ->  ( lastS  `  W )  =  ( W `  ( (
# `  W )  -  1 ) ) )
3332adantl 466 . . . . . . . . . . . . 13  |-  ( ( ( V USGrph  E  /\  ( # `  W )  =  2 )  /\  W  e. Word  V )  ->  ( lastS  `  W )  =  ( W `  ( ( # `  W
)  -  1 ) ) )
3410ad2antlr 726 . . . . . . . . . . . . . . 15  |-  ( ( ( V USGrph  E  /\  ( # `  W )  =  2 )  /\  W  e. Word  V )  ->  ( ( # `  W
)  -  1 )  =  ( 2  -  1 ) )
3534, 14syl6eq 2524 . . . . . . . . . . . . . 14  |-  ( ( ( V USGrph  E  /\  ( # `  W )  =  2 )  /\  W  e. Word  V )  ->  ( ( # `  W
)  -  1 )  =  1 )
3635fveq2d 5876 . . . . . . . . . . . . 13  |-  ( ( ( V USGrph  E  /\  ( # `  W )  =  2 )  /\  W  e. Word  V )  ->  ( W `  (
( # `  W )  -  1 ) )  =  ( W ` 
1 ) )
3733, 36eqtr2d 2509 . . . . . . . . . . . 12  |-  ( ( ( V USGrph  E  /\  ( # `  W )  =  2 )  /\  W  e. Word  V )  ->  ( W `  1
)  =  ( lastS  `  W
) )
3837preq2d 4119 . . . . . . . . . . 11  |-  ( ( ( V USGrph  E  /\  ( # `  W )  =  2 )  /\  W  e. Word  V )  ->  { ( W ` 
0 ) ,  ( W `  1 ) }  =  { ( W `  0 ) ,  ( lastS  `  W
) } )
39 prcom 4111 . . . . . . . . . . 11  |-  { ( W `  0 ) ,  ( lastS  `  W
) }  =  {
( lastS  `  W ) ,  ( W `  0
) }
4038, 39syl6eq 2524 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  ( # `  W )  =  2 )  /\  W  e. Word  V )  ->  { ( W ` 
0 ) ,  ( W `  1 ) }  =  { ( lastS  `  W ) ,  ( W `  0 ) } )
4140eleq1d 2536 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  ( # `  W )  =  2 )  /\  W  e. Word  V )  ->  ( { ( W `
 0 ) ,  ( W `  1
) }  e.  ran  E  <->  { ( lastS  `  W ) ,  ( W ` 
0 ) }  e.  ran  E ) )
4241biimpd 207 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  ( # `  W )  =  2 )  /\  W  e. Word  V )  ->  ( { ( W `
 0 ) ,  ( W `  1
) }  e.  ran  E  ->  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E ) )
4342pm4.71d 634 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( # `  W )  =  2 )  /\  W  e. Word  V )  ->  ( { ( W `
 0 ) ,  ( W `  1
) }  e.  ran  E  <-> 
( { ( W `
 0 ) ,  ( W `  1
) }  e.  ran  E  /\  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E ) ) )
4431, 43bitr4d 256 . . . . . 6  |-  ( ( ( V USGrph  E  /\  ( # `  W )  =  2 )  /\  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 ` 
0 ) ,  ( W `  1 ) }  e.  ran  E
) )
4544pm5.32da 641 . . . . 5  |-  ( ( V USGrph  E  /\  ( # `
 W )  =  2 )  ->  (
( 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  /\  { ( W `
 0 ) ,  ( W `  1
) }  e.  ran  E ) ) )
469, 45syl5bb 257 . . . 4  |-  ( ( V USGrph  E  /\  ( # `
 W )  =  2 )  ->  (
( 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  /\  { ( W `
 0 ) ,  ( W `  1
) }  e.  ran  E ) ) )
4746ex 434 . . 3  |-  ( V USGrph  E  ->  ( ( # `  W )  =  2  ->  ( ( 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  /\  { ( W ` 
0 ) ,  ( W `  1 ) }  e.  ran  E
) ) ) )
4847pm5.32rd 640 . 2  |-  ( V USGrph  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
)  =  2 )  <-> 
( ( W  e. Word  V  /\  { ( W `
 0 ) ,  ( W `  1
) }  e.  ran  E )  /\  ( # `  W )  =  2 ) ) )
49 3anass 977 . . . 4  |-  ( ( ( # `  W
)  =  2  /\  W  e. Word  V  /\  { ( W `  0
) ,  ( W `
 1 ) }  e.  ran  E )  <-> 
( ( # `  W
)  =  2  /\  ( W  e. Word  V  /\  { ( W ` 
0 ) ,  ( W `  1 ) }  e.  ran  E
) ) )
50 ancom 450 . . . 4  |-  ( ( ( # `  W
)  =  2  /\  ( W  e. Word  V  /\  { ( W ` 
0 ) ,  ( W `  1 ) }  e.  ran  E
) )  <->  ( ( W  e. Word  V  /\  {
( W `  0
) ,  ( W `
 1 ) }  e.  ran  E )  /\  ( # `  W
)  =  2 ) )
5149, 50bitr2i 250 . . 3  |-  ( ( ( W  e. Word  V  /\  { ( W ` 
0 ) ,  ( W `  1 ) }  e.  ran  E
)  /\  ( # `  W
)  =  2 )  <-> 
( ( # `  W
)  =  2  /\  W  e. Word  V  /\  { ( W `  0
) ,  ( W `
 1 ) }  e.  ran  E ) )
5251a1i 11 . 2  |-  ( V USGrph  E  ->  ( ( ( W  e. Word  V  /\  { ( W `  0
) ,  ( W `
 1 ) }  e.  ran  E )  /\  ( # `  W
)  =  2 )  <-> 
( ( # `  W
)  =  2  /\  W  e. Word  V  /\  { ( W `  0
) ,  ( W `
 1 ) }  e.  ran  E ) ) )
538, 48, 523bitrd 279 1  |-  ( V USGrph  E  ->  ( W  e.  ( ( V ClWWalksN  E ) `
 2 )  <->  ( ( # `
 W )  =  2  /\  W  e. Word  V  /\  { ( W `
 0 ) ,  ( W `  1
) }  e.  ran  E ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118   {csn 4033   {cpr 4035   class class class wbr 4453   ran crn 5006   ` cfv 5594  (class class class)co 6295   0cc0 9504   1c1 9505    + caddc 9507    - cmin 9817   2c2 10597   NN0cn0 10807  ..^cfzo 11804   #chash 12385  Word cword 12515   lastS clsw 12516   USGrph cusg 24144   ClWWalks cclwwlk 24562   ClWWalksN cclwwlkn 24563
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-word 12523  df-lsw 12524  df-usgra 24147  df-clwwlk 24565  df-clwwlkn 24566
This theorem is referenced by:  numclwwlkovf2  24899
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