Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wwlkn0s Structured version   Unicode version

Theorem wwlkn0s 30488
Description: The set of all walks as words of length 0 is the set of all words of length 1 over the verices. (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 10706 . . 3  |-  0  e.  NN0
2 wwlkn 30465 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y  /\  0  e.  NN0 )  -> 
( ( V WWalksN  E
) `  0 )  =  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( 0  +  1 ) } )
31, 2mp3an3 1304 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( V WWalksN  E
) `  0 )  =  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( 0  +  1 ) } )
4 0p1e1 10545 . . . . . 6  |-  ( 0  +  1 )  =  1
54eqeq2i 2472 . . . . 5  |-  ( (
# `  w )  =  ( 0  +  1 )  <->  ( # `  w
)  =  1 )
65anbi2i 694 . . . 4  |-  ( ( w  e.  ( V WWalks  E )  /\  ( # `
 w )  =  ( 0  +  1 ) )  <->  ( w  e.  ( V WWalks  E )  /\  ( # `  w
)  =  1 ) )
7 iswwlk 30466 . . . . . . . 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 465 . . . . . . 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 989 . . . . . . . 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 9463 . . . . . . . . . . . . 13  |-  1  =/=  0
11 pm13.181 2764 . . . . . . . . . . . . 13  |-  ( ( ( # `  w
)  =  1  /\  1  =/=  0 )  ->  ( # `  w
)  =/=  0 )
1210, 11mpan2 671 . . . . . . . . . . . 12  |-  ( (
# `  w )  =  1  ->  ( # `
 w )  =/=  0 )
13 vex 3081 . . . . . . . . . . . . . 14  |-  w  e. 
_V
14 hasheq0 12249 . . . . . . . . . . . . . 14  |-  ( w  e.  _V  ->  (
( # `  w )  =  0  <->  w  =  (/) ) )
1513, 14mp1i 12 . . . . . . . . . . . . 13  |-  ( (
# `  w )  =  1  ->  (
( # `  w )  =  0  <->  w  =  (/) ) )
1615necon3bid 2710 . . . . . . . . . . . 12  |-  ( (
# `  w )  =  1  ->  (
( # `  w )  =/=  0  <->  w  =/=  (/) ) )
1712, 16mpbid 210 . . . . . . . . . . 11  |-  ( (
# `  w )  =  1  ->  w  =/=  (/) )
1817ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( # `
 w )  =  1 )  /\  w  e. Word  V )  ->  w  =/=  (/) )
19 simpr 461 . . . . . . . . . 10  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( # `
 w )  =  1 )  /\  w  e. Word  V )  ->  w  e. Word  V )
20 ral0 3893 . . . . . . . . . . . 12  |-  A. i  e.  (/)  { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  E
21 oveq1 6208 . . . . . . . . . . . . . . . 16  |-  ( (
# `  w )  =  1  ->  (
( # `  w )  -  1 )  =  ( 1  -  1 ) )
22 1m1e0 10502 . . . . . . . . . . . . . . . 16  |-  ( 1  -  1 )  =  0
2321, 22syl6eq 2511 . . . . . . . . . . . . . . 15  |-  ( (
# `  w )  =  1  ->  (
( # `  w )  -  1 )  =  0 )
2423oveq2d 6217 . . . . . . . . . . . . . 14  |-  ( (
# `  w )  =  1  ->  (
0..^ ( ( # `  w )  -  1 ) )  =  ( 0..^ 0 ) )
25 fzo0 11691 . . . . . . . . . . . . . 14  |-  ( 0..^ 0 )  =  (/)
2624, 25syl6eq 2511 . . . . . . . . . . . . 13  |-  ( (
# `  w )  =  1  ->  (
0..^ ( ( # `  w )  -  1 ) )  =  (/) )
27 biidd 237 . . . . . . . . . . . . 13  |-  ( (
# `  w )  =  1  ->  ( { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  ran  E ) )
2826, 27raleqbidv 3037 . . . . . . . . . . . 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 ) )
2920, 28mpbiri 233 . . . . . . . . . . 11  |-  ( (
# `  w )  =  1  ->  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)
3029ad2antlr 726 . . . . . . . . . 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
)
3118, 19, 303jca 1168 . . . . . . . . 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
) )
3231ex 434 . . . . . . . 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 ) ) )
339, 32impbid2 204 . . . . . . 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 ) )
348, 33bitrd 253 . . . . . 6  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( # `  w
)  =  1 )  ->  ( w  e.  ( V WWalks  E )  <-> 
w  e. Word  V )
)
3534ex 434 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( # `  w
)  =  1  -> 
( w  e.  ( V WWalks  E )  <->  w  e. Word  V ) ) )
3635pm5.32rd 640 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( w  e.  ( V WWalks  E )  /\  ( # `  w
)  =  1 )  <-> 
( w  e. Word  V  /\  ( # `  w
)  =  1 ) ) )
376, 36syl5bb 257 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( w  e.  ( V WWalks  E )  /\  ( # `  w
)  =  ( 0  +  1 ) )  <-> 
( w  e. Word  V  /\  ( # `  w
)  =  1 ) ) )
3837rabbidva2 3068 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( 0  +  1 ) }  =  { w  e. Word  V  |  ( # `  w
)  =  1 } )
393, 38eqtrd 2495 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 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   {crab 2803   _Vcvv 3078   (/)c0 3746   {cpr 3988   ran crn 4950   ` cfv 5527  (class class class)co 6201   0cc0 9394   1c1 9395    + caddc 9397    - cmin 9707   NN0cn0 10691  ..^cfzo 11666   #chash 12221  Word cword 12340   WWalks cwwlk 30460   WWalksN cwwlkn 30461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-pm 7328  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-hash 12222  df-word 12348  df-wwlk 30462  df-wwlkn 30463
This theorem is referenced by:  rusgranumwlkb0  30720
  Copyright terms: Public domain W3C validator