MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wwlkn0s Structured version   Visualization version   Unicode version

Theorem wwlkn0s 25512
Description: The set of all walks as words of length 0 is the set of all words of length 1 over the vertices. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
Assertion
Ref Expression
wwlkn0s  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( V WWalksN  E
) `  0 )  =  { w  e. Word  V  |  ( # `  w
)  =  1 } )
Distinct variable groups:    w, E    w, V    w, X    w, Y

Proof of Theorem wwlkn0s
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 0nn0 10908 . . 3  |-  0  e.  NN0
2 wwlkn 25489 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y  /\  0  e.  NN0 )  -> 
( ( V WWalksN  E
) `  0 )  =  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( 0  +  1 ) } )
31, 2mp3an3 1379 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( V WWalksN  E
) `  0 )  =  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( 0  +  1 ) } )
4 0p1e1 10743 . . . . . 6  |-  ( 0  +  1 )  =  1
54eqeq2i 2483 . . . . 5  |-  ( (
# `  w )  =  ( 0  +  1 )  <->  ( # `  w
)  =  1 )
65anbi2i 708 . . . 4  |-  ( ( w  e.  ( V WWalks  E )  /\  ( # `
 w )  =  ( 0  +  1 ) )  <->  ( w  e.  ( V WWalks  E )  /\  ( # `  w
)  =  1 ) )
7 iswwlk 25490 . . . . . . . 8  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( w  e.  ( V WWalks  E )  <->  ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
) ) )
87adantr 472 . . . . . . 7  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( # `  w
)  =  1 )  ->  ( w  e.  ( V WWalks  E )  <-> 
( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
) ) )
9 simp2 1031 . . . . . . . 8  |-  ( ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  ->  w  e. Word  V )
10 ax-1ne0 9626 . . . . . . . . . . . . 13  |-  1  =/=  0
11 pm13.181 2725 . . . . . . . . . . . . 13  |-  ( ( ( # `  w
)  =  1  /\  1  =/=  0 )  ->  ( # `  w
)  =/=  0 )
1210, 11mpan2 685 . . . . . . . . . . . 12  |-  ( (
# `  w )  =  1  ->  ( # `
 w )  =/=  0 )
13 vex 3034 . . . . . . . . . . . . . 14  |-  w  e. 
_V
14 hasheq0 12582 . . . . . . . . . . . . . 14  |-  ( w  e.  _V  ->  (
( # `  w )  =  0  <->  w  =  (/) ) )
1513, 14mp1i 13 . . . . . . . . . . . . 13  |-  ( (
# `  w )  =  1  ->  (
( # `  w )  =  0  <->  w  =  (/) ) )
1615necon3bid 2687 . . . . . . . . . . . 12  |-  ( (
# `  w )  =  1  ->  (
( # `  w )  =/=  0  <->  w  =/=  (/) ) )
1712, 16mpbid 215 . . . . . . . . . . 11  |-  ( (
# `  w )  =  1  ->  w  =/=  (/) )
1817ad2antlr 741 . . . . . . . . . 10  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( # `
 w )  =  1 )  /\  w  e. Word  V )  ->  w  =/=  (/) )
19 simpr 468 . . . . . . . . . 10  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( # `
 w )  =  1 )  /\  w  e. Word  V )  ->  w  e. Word  V )
20 ral0 3865 . . . . . . . . . . . 12  |-  A. i  e.  (/)  { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  E
21 oveq1 6315 . . . . . . . . . . . . . . . 16  |-  ( (
# `  w )  =  1  ->  (
( # `  w )  -  1 )  =  ( 1  -  1 ) )
22 1m1e0 10700 . . . . . . . . . . . . . . . 16  |-  ( 1  -  1 )  =  0
2321, 22syl6eq 2521 . . . . . . . . . . . . . . 15  |-  ( (
# `  w )  =  1  ->  (
( # `  w )  -  1 )  =  0 )
2423oveq2d 6324 . . . . . . . . . . . . . 14  |-  ( (
# `  w )  =  1  ->  (
0..^ ( ( # `  w )  -  1 ) )  =  ( 0..^ 0 ) )
25 fzo0 11969 . . . . . . . . . . . . . 14  |-  ( 0..^ 0 )  =  (/)
2624, 25syl6eq 2521 . . . . . . . . . . . . 13  |-  ( (
# `  w )  =  1  ->  (
0..^ ( ( # `  w )  -  1 ) )  =  (/) )
2726raleqdv 2979 . . . . . . . . . . . 12  |-  ( (
# `  w )  =  1  ->  ( A. i  e.  (
0..^ ( ( # `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  A. i  e.  (/)  { ( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  ran  E ) )
2820, 27mpbiri 241 . . . . . . . . . . 11  |-  ( (
# `  w )  =  1  ->  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)
2928ad2antlr 741 . . . . . . . . . 10  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( # `
 w )  =  1 )  /\  w  e. Word  V )  ->  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)
3018, 19, 293jca 1210 . . . . . . . . 9  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( # `
 w )  =  1 )  /\  w  e. Word  V )  ->  (
w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
) )
3130ex 441 . . . . . . . 8  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( # `  w
)  =  1 )  ->  ( w  e. Word  V  ->  ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E ) ) )
329, 31impbid2 209 . . . . . . 7  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( # `  w
)  =  1 )  ->  ( ( w  =/=  (/)  /\  w  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)  <->  w  e. Word  V ) )
338, 32bitrd 261 . . . . . 6  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( # `  w
)  =  1 )  ->  ( w  e.  ( V WWalks  E )  <-> 
w  e. Word  V )
)
3433ex 441 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( # `  w
)  =  1  -> 
( w  e.  ( V WWalks  E )  <->  w  e. Word  V ) ) )
3534pm5.32rd 652 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( w  e.  ( V WWalks  E )  /\  ( # `  w
)  =  1 )  <-> 
( w  e. Word  V  /\  ( # `  w
)  =  1 ) ) )
366, 35syl5bb 265 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( w  e.  ( V WWalks  E )  /\  ( # `  w
)  =  ( 0  +  1 ) )  <-> 
( w  e. Word  V  /\  ( # `  w
)  =  1 ) ) )
3736rabbidva2 3020 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( 0  +  1 ) }  =  { w  e. Word  V  |  ( # `  w
)  =  1 } )
383, 37eqtrd 2505 1  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( V WWalksN  E
) `  0 )  =  { w  e. Word  V  |  ( # `  w
)  =  1 } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   {crab 2760   _Vcvv 3031   (/)c0 3722   {cpr 3961   ran crn 4840   ` cfv 5589  (class class class)co 6308   0cc0 9557   1c1 9558    + caddc 9560    - cmin 9880   NN0cn0 10893  ..^cfzo 11942   #chash 12553  Word cword 12703   WWalks cwwlk 25484   WWalksN cwwlkn 25485
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-wwlk 25486  df-wwlkn 25487
This theorem is referenced by:  rusgranumwlkb0  25760
  Copyright terms: Public domain W3C validator