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

Theorem clwwlkprop 25577
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 25558 . . 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 6530 . 2  |-  ( P  e.  ( V ClWWalks  E )  ->  ( V  e. 
_V  /\  E  e.  _V ) )
3 pm3.22 456 . . . . . 6  |-  ( ( P  e. Word  V  /\  ( V  e.  _V  /\  E  e.  _V )
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  P  e. Word  V ) )
4 df-3an 1009 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  P  e. Word  V )  <->  ( ( V  e.  _V  /\  E  e.  _V )  /\  P  e. Word  V ) )
53, 4sylibr 217 . . . . 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 441 . . 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 25573 . . . . . 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 2534 . . . . 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 5879 . . . . . . . . . . 11  |-  ( p  =  P  ->  ( # `
 p )  =  ( # `  P
) )
1110oveq1d 6323 . . . . . . . . . 10  |-  ( p  =  P  ->  (
( # `  p )  -  1 )  =  ( ( # `  P
)  -  1 ) )
1211oveq2d 6324 . . . . . . . . 9  |-  ( p  =  P  ->  (
0..^ ( ( # `  p )  -  1 ) )  =  ( 0..^ ( ( # `  P )  -  1 ) ) )
13 fveq1 5878 . . . . . . . . . . 11  |-  ( p  =  P  ->  (
p `  i )  =  ( P `  i ) )
14 fveq1 5878 . . . . . . . . . . 11  |-  ( p  =  P  ->  (
p `  ( i  +  1 ) )  =  ( P `  ( i  +  1 ) ) )
1513, 14preq12d 4050 . . . . . . . . . 10  |-  ( p  =  P  ->  { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } )
1615eleq1d 2533 . . . . . . . . 9  |-  ( p  =  P  ->  ( { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( P `  i
) ,  ( P `
 ( i  +  1 ) ) }  e.  ran  E ) )
1712, 16raleqbidv 2987 . . . . . . . 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 5879 . . . . . . . . . 10  |-  ( p  =  P  ->  ( lastS  `  p )  =  ( lastS  `  P ) )
19 fveq1 5878 . . . . . . . . . 10  |-  ( p  =  P  ->  (
p `  0 )  =  ( P ` 
0 ) )
2018, 19preq12d 4050 . . . . . . . . 9  |-  ( p  =  P  ->  { ( lastS  `  p ) ,  ( p `  0 ) }  =  { ( lastS  `  P ) ,  ( P `  0 ) } )
2120eleq1d 2533 . . . . . . . 8  |-  ( p  =  P  ->  ( { ( lastS  `  p ) ,  ( p ` 
0 ) }  e.  ran  E  <->  { ( lastS  `  P
) ,  ( P `
 0 ) }  e.  ran  E ) )
2217, 21anbi12d 725 . . . . . . 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 3184 . . . . . 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 pm2.24 112 . . . . . . 7  |-  ( P  e. Word  V  ->  ( -.  P  e. Word  V  -> 
( V  e.  _V  /\  E  e.  _V  /\  P  e. Word  V )
) )
2524adantr 472 . . . . . 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 200 . . . . 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 236 . . . 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 81 . . 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 169 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( P  e.  ( V ClWWalks  E )  ->  ( V  e.  _V  /\  E  e.  _V  /\  P  e. Word  V ) ) )
302, 29mpcom 36 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 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   {crab 2760   _Vcvv 3031   {cpr 3961   ran crn 4840   ` cfv 5589  (class class class)co 6308   0cc0 9557   1c1 9558    + caddc 9560    - cmin 9880  ..^cfzo 11942   #chash 12553  Word cword 12703   lastS clsw 12704   ClWWalks cclwwlk 25555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-clwwlk 25558
This theorem is referenced by:  clwwlkgt0  25578  clwwlknprop  25579  clwwlkn0  25581  clwwisshclww  25614  clwwisshclwwn  25615  erclwwlkeqlen  25619  erclwwlkref  25620  erclwwlksym  25621  erclwwlktr  25622
  Copyright terms: Public domain W3C validator