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Theorem clwwlkprop 25174
Description: Properties of a closed walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Mar-2018.)
Assertion
Ref Expression
clwwlkprop  |-  ( P  e.  ( V ClWWalks  E )  ->  ( V  e. 
_V  /\  E  e.  _V  /\  P  e. Word  V
) )

Proof of Theorem clwwlkprop
Dummy variables  e 
i  v  w  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-clwwlk 25155 . . 3  |- ClWWalks  =  ( v  e.  _V , 
e  e.  _V  |->  { 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 ) } )
21elmpt2cl 6497 . 2  |-  ( P  e.  ( V ClWWalks  E )  ->  ( V  e. 
_V  /\  E  e.  _V ) )
3 pm3.22 447 . . . . . 6  |-  ( ( P  e. Word  V  /\  ( V  e.  _V  /\  E  e.  _V )
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  P  e. Word  V ) )
4 df-3an 976 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  P  e. Word  V )  <->  ( ( V  e.  _V  /\  E  e.  _V )  /\  P  e. Word  V ) )
53, 4sylibr 212 . . . . 5  |-  ( ( P  e. Word  V  /\  ( V  e.  _V  /\  E  e.  _V )
)  ->  ( V  e.  _V  /\  E  e. 
_V  /\  P  e. Word  V ) )
65a1d 25 . . . 4  |-  ( ( P  e. Word  V  /\  ( V  e.  _V  /\  E  e.  _V )
)  ->  ( P  e.  ( V ClWWalks  E )  ->  ( V  e.  _V  /\  E  e.  _V  /\  P  e. Word  V )
) )
76ex 432 . . 3  |-  ( P  e. Word  V  ->  (
( V  e.  _V  /\  E  e.  _V )  ->  ( P  e.  ( V ClWWalks  E )  ->  ( V  e.  _V  /\  E  e.  _V  /\  P  e. Word  V ) ) ) )
8 clwwlk 25170 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V ClWWalks  E )  =  { p  e. Word  V  |  ( A. i  e.  ( 0..^ ( (
# `  p )  -  1 ) ) { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  p
) ,  ( p `
 0 ) }  e.  ran  E ) } )
98eleq2d 2472 . . . . 5  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( P  e.  ( V ClWWalks  E )  <->  P  e.  { p  e. Word  V  | 
( A. i  e.  ( 0..^ ( (
# `  p )  -  1 ) ) { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  p
) ,  ( p `
 0 ) }  e.  ran  E ) } ) )
10 fveq2 5848 . . . . . . . . . . 11  |-  ( p  =  P  ->  ( # `
 p )  =  ( # `  P
) )
1110oveq1d 6292 . . . . . . . . . 10  |-  ( p  =  P  ->  (
( # `  p )  -  1 )  =  ( ( # `  P
)  -  1 ) )
1211oveq2d 6293 . . . . . . . . 9  |-  ( p  =  P  ->  (
0..^ ( ( # `  p )  -  1 ) )  =  ( 0..^ ( ( # `  P )  -  1 ) ) )
13 fveq1 5847 . . . . . . . . . . 11  |-  ( p  =  P  ->  (
p `  i )  =  ( P `  i ) )
14 fveq1 5847 . . . . . . . . . . 11  |-  ( p  =  P  ->  (
p `  ( i  +  1 ) )  =  ( P `  ( i  +  1 ) ) )
1513, 14preq12d 4058 . . . . . . . . . 10  |-  ( p  =  P  ->  { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } )
1615eleq1d 2471 . . . . . . . . 9  |-  ( p  =  P  ->  ( { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( P `  i
) ,  ( P `
 ( i  +  1 ) ) }  e.  ran  E ) )
1712, 16raleqbidv 3017 . . . . . . . 8  |-  ( p  =  P  ->  ( A. i  e.  (
0..^ ( ( # `  p )  -  1 ) ) { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  <->  A. i  e.  ( 0..^ ( ( # `  P )  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E ) )
18 fveq2 5848 . . . . . . . . . 10  |-  ( p  =  P  ->  ( lastS  `  p )  =  ( lastS  `  P ) )
19 fveq1 5847 . . . . . . . . . 10  |-  ( p  =  P  ->  (
p `  0 )  =  ( P ` 
0 ) )
2018, 19preq12d 4058 . . . . . . . . 9  |-  ( p  =  P  ->  { ( lastS  `  p ) ,  ( p `  0 ) }  =  { ( lastS  `  P ) ,  ( P `  0 ) } )
2120eleq1d 2471 . . . . . . . 8  |-  ( p  =  P  ->  ( { ( lastS  `  p ) ,  ( p ` 
0 ) }  e.  ran  E  <->  { ( lastS  `  P
) ,  ( P `
 0 ) }  e.  ran  E ) )
2217, 21anbi12d 709 . . . . . . 7  |-  ( p  =  P  ->  (
( A. i  e.  ( 0..^ ( (
# `  p )  -  1 ) ) { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  p
) ,  ( p `
 0 ) }  e.  ran  E )  <-> 
( A. i  e.  ( 0..^ ( (
# `  P )  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P
) ,  ( P `
 0 ) }  e.  ran  E ) ) )
2322elrab 3206 . . . . . 6  |-  ( P  e.  { p  e. Word  V  |  ( A. i  e.  ( 0..^ ( ( # `  p
)  -  1 ) ) { ( p `
 i ) ,  ( p `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  p
) ,  ( p `
 0 ) }  e.  ran  E ) }  <->  ( P  e. Word  V  /\  ( A. i  e.  ( 0..^ ( (
# `  P )  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P
) ,  ( P `
 0 ) }  e.  ran  E ) ) )
24 pm2.24 109 . . . . . . 7  |-  ( P  e. Word  V  ->  ( -.  P  e. Word  V  -> 
( V  e.  _V  /\  E  e.  _V  /\  P  e. Word  V )
) )
2524adantr 463 . . . . . 6  |-  ( ( P  e. Word  V  /\  ( A. i  e.  ( 0..^ ( ( # `  P )  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P ) ,  ( P `  0 ) }  e.  ran  E
) )  ->  ( -.  P  e. Word  V  -> 
( V  e.  _V  /\  E  e.  _V  /\  P  e. Word  V )
) )
2623, 25sylbi 195 . . . . 5  |-  ( P  e.  { p  e. Word  V  |  ( A. i  e.  ( 0..^ ( ( # `  p
)  -  1 ) ) { ( p `
 i ) ,  ( p `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  p
) ,  ( p `
 0 ) }  e.  ran  E ) }  ->  ( -.  P  e. Word  V  ->  ( V  e.  _V  /\  E  e.  _V  /\  P  e. Word  V ) ) )
279, 26syl6bi 228 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( P  e.  ( V ClWWalks  E )  ->  ( -.  P  e. Word  V  -> 
( V  e.  _V  /\  E  e.  _V  /\  P  e. Word  V )
) ) )
2827com3r 79 . . 3  |-  ( -.  P  e. Word  V  -> 
( ( V  e. 
_V  /\  E  e.  _V )  ->  ( P  e.  ( V ClWWalks  E )  ->  ( V  e. 
_V  /\  E  e.  _V  /\  P  e. Word  V
) ) ) )
297, 28pm2.61i 164 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( P  e.  ( V ClWWalks  E )  ->  ( V  e.  _V  /\  E  e.  _V  /\  P  e. Word  V ) ) )
302, 29mpcom 34 1  |-  ( P  e.  ( V ClWWalks  E )  ->  ( V  e. 
_V  /\  E  e.  _V  /\  P  e. Word  V
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2753   {crab 2757   _Vcvv 3058   {cpr 3973   ran crn 4823   ` cfv 5568  (class class class)co 6277   0cc0 9521   1c1 9522    + caddc 9524    - cmin 9840  ..^cfzo 11852   #chash 12450  Word cword 12581   lastS clsw 12582   ClWWalks cclwwlk 25152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-fzo 11853  df-hash 12451  df-word 12589  df-clwwlk 25155
This theorem is referenced by:  clwwlkgt0  25175  clwwlknprop  25176  clwwlkn0  25178  clwwisshclww  25211  clwwisshclwwn  25212  erclwwlkeqlen  25216  erclwwlkref  25217  erclwwlksym  25218  erclwwlktr  25219
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