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Theorem clwwlkprop 25496
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 25477 . . 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 6525 . 2  |-  ( P  e.  ( V ClWWalks  E )  ->  ( V  e. 
_V  /\  E  e.  _V ) )
3 pm3.22 450 . . . . . 6  |-  ( ( P  e. Word  V  /\  ( V  e.  _V  /\  E  e.  _V )
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  P  e. Word  V ) )
4 df-3an 984 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  P  e. Word  V )  <->  ( ( V  e.  _V  /\  E  e.  _V )  /\  P  e. Word  V ) )
53, 4sylibr 215 . . . . 5  |-  ( ( P  e. Word  V  /\  ( V  e.  _V  /\  E  e.  _V )
)  ->  ( V  e.  _V  /\  E  e. 
_V  /\  P  e. Word  V ) )
65a1d 26 . . . 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 435 . . 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 25492 . . . . . 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 2492 . . . . 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 5881 . . . . . . . . . . 11  |-  ( p  =  P  ->  ( # `
 p )  =  ( # `  P
) )
1110oveq1d 6320 . . . . . . . . . 10  |-  ( p  =  P  ->  (
( # `  p )  -  1 )  =  ( ( # `  P
)  -  1 ) )
1211oveq2d 6321 . . . . . . . . 9  |-  ( p  =  P  ->  (
0..^ ( ( # `  p )  -  1 ) )  =  ( 0..^ ( ( # `  P )  -  1 ) ) )
13 fveq1 5880 . . . . . . . . . . 11  |-  ( p  =  P  ->  (
p `  i )  =  ( P `  i ) )
14 fveq1 5880 . . . . . . . . . . 11  |-  ( p  =  P  ->  (
p `  ( i  +  1 ) )  =  ( P `  ( i  +  1 ) ) )
1513, 14preq12d 4087 . . . . . . . . . 10  |-  ( p  =  P  ->  { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } )
1615eleq1d 2491 . . . . . . . . 9  |-  ( p  =  P  ->  ( { ( p `  i ) ,  ( p `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( P `  i
) ,  ( P `
 ( i  +  1 ) ) }  e.  ran  E ) )
1712, 16raleqbidv 3036 . . . . . . . 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 5881 . . . . . . . . . 10  |-  ( p  =  P  ->  ( lastS  `  p )  =  ( lastS  `  P ) )
19 fveq1 5880 . . . . . . . . . 10  |-  ( p  =  P  ->  (
p `  0 )  =  ( P ` 
0 ) )
2018, 19preq12d 4087 . . . . . . . . 9  |-  ( p  =  P  ->  { ( lastS  `  p ) ,  ( p `  0 ) }  =  { ( lastS  `  P ) ,  ( P `  0 ) } )
2120eleq1d 2491 . . . . . . . 8  |-  ( p  =  P  ->  ( { ( lastS  `  p ) ,  ( p ` 
0 ) }  e.  ran  E  <->  { ( lastS  `  P
) ,  ( P `
 0 ) }  e.  ran  E ) )
2217, 21anbi12d 715 . . . . . . 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 3228 . . . . . 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 466 . . . . . 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 198 . . . . 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 231 . . . 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 82 . . 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 167 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( P  e.  ( V ClWWalks  E )  ->  ( V  e.  _V  /\  E  e.  _V  /\  P  e. Word  V ) ) )
302, 29mpcom 37 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 370    /\ w3a 982    = wceq 1437    e. wcel 1872   A.wral 2771   {crab 2775   _Vcvv 3080   {cpr 4000   ran crn 4854   ` cfv 5601  (class class class)co 6305   0cc0 9546   1c1 9547    + caddc 9549    - cmin 9867  ..^cfzo 11922   #chash 12521  Word cword 12660   lastS clsw 12661   ClWWalks cclwwlk 25474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-map 7485  df-pm 7486  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-card 8381  df-cda 8605  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923  df-hash 12522  df-word 12668  df-clwwlk 25477
This theorem is referenced by:  clwwlkgt0  25497  clwwlknprop  25498  clwwlkn0  25500  clwwisshclww  25533  clwwisshclwwn  25534  erclwwlkeqlen  25538  erclwwlkref  25539  erclwwlksym  25540  erclwwlktr  25541
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