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Theorem wwlkn0s 25425
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 10886 . . 3  |-  0  e.  NN0
2 wwlkn 25402 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y  /\  0  e.  NN0 )  -> 
( ( V WWalksN  E
) `  0 )  =  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( 0  +  1 ) } )
31, 2mp3an3 1350 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( V WWalksN  E
) `  0 )  =  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( 0  +  1 ) } )
4 0p1e1 10723 . . . . . 6  |-  ( 0  +  1 )  =  1
54eqeq2i 2441 . . . . 5  |-  ( (
# `  w )  =  ( 0  +  1 )  <->  ( # `  w
)  =  1 )
65anbi2i 699 . . . 4  |-  ( ( w  e.  ( V WWalks  E )  /\  ( # `
 w )  =  ( 0  +  1 ) )  <->  ( w  e.  ( V WWalks  E )  /\  ( # `  w
)  =  1 ) )
7 iswwlk 25403 . . . . . . . 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 467 . . . . . . 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 1007 . . . . . . . 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 9610 . . . . . . . . . . . . 13  |-  1  =/=  0
11 pm13.181 2737 . . . . . . . . . . . . 13  |-  ( ( ( # `  w
)  =  1  /\  1  =/=  0 )  ->  ( # `  w
)  =/=  0 )
1210, 11mpan2 676 . . . . . . . . . . . 12  |-  ( (
# `  w )  =  1  ->  ( # `
 w )  =/=  0 )
13 vex 3085 . . . . . . . . . . . . . 14  |-  w  e. 
_V
14 hasheq0 12545 . . . . . . . . . . . . . 14  |-  ( w  e.  _V  ->  (
( # `  w )  =  0  <->  w  =  (/) ) )
1513, 14mp1i 13 . . . . . . . . . . . . 13  |-  ( (
# `  w )  =  1  ->  (
( # `  w )  =  0  <->  w  =  (/) ) )
1615necon3bid 2683 . . . . . . . . . . . 12  |-  ( (
# `  w )  =  1  ->  (
( # `  w )  =/=  0  <->  w  =/=  (/) ) )
1712, 16mpbid 214 . . . . . . . . . . 11  |-  ( (
# `  w )  =  1  ->  w  =/=  (/) )
1817ad2antlr 732 . . . . . . . . . 10  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( # `
 w )  =  1 )  /\  w  e. Word  V )  ->  w  =/=  (/) )
19 simpr 463 . . . . . . . . . 10  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( # `
 w )  =  1 )  /\  w  e. Word  V )  ->  w  e. Word  V )
20 ral0 3903 . . . . . . . . . . . 12  |-  A. i  e.  (/)  { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  E
21 oveq1 6310 . . . . . . . . . . . . . . . 16  |-  ( (
# `  w )  =  1  ->  (
( # `  w )  -  1 )  =  ( 1  -  1 ) )
22 1m1e0 10680 . . . . . . . . . . . . . . . 16  |-  ( 1  -  1 )  =  0
2321, 22syl6eq 2480 . . . . . . . . . . . . . . 15  |-  ( (
# `  w )  =  1  ->  (
( # `  w )  -  1 )  =  0 )
2423oveq2d 6319 . . . . . . . . . . . . . 14  |-  ( (
# `  w )  =  1  ->  (
0..^ ( ( # `  w )  -  1 ) )  =  ( 0..^ 0 ) )
25 fzo0 11944 . . . . . . . . . . . . . 14  |-  ( 0..^ 0 )  =  (/)
2624, 25syl6eq 2480 . . . . . . . . . . . . 13  |-  ( (
# `  w )  =  1  ->  (
0..^ ( ( # `  w )  -  1 ) )  =  (/) )
2726raleqdv 3032 . . . . . . . . . . . 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 237 . . . . . . . . . . 11  |-  ( (
# `  w )  =  1  ->  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)
2928ad2antlr 732 . . . . . . . . . 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 1186 . . . . . . . . 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 436 . . . . . . . 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 208 . . . . . . 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 257 . . . . . 6  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( # `  w
)  =  1 )  ->  ( w  e.  ( V WWalks  E )  <-> 
w  e. Word  V )
)
3433ex 436 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( # `  w
)  =  1  -> 
( w  e.  ( V WWalks  E )  <->  w  e. Word  V ) ) )
3534pm5.32rd 645 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( w  e.  ( V WWalks  E )  /\  ( # `  w
)  =  1 )  <-> 
( w  e. Word  V  /\  ( # `  w
)  =  1 ) ) )
366, 35syl5bb 261 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( w  e.  ( V WWalks  E )  /\  ( # `  w
)  =  ( 0  +  1 ) )  <-> 
( w  e. Word  V  /\  ( # `  w
)  =  1 ) ) )
3736rabbidva2 3071 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( 0  +  1 ) }  =  { w  e. Word  V  |  ( # `  w
)  =  1 } )
383, 37eqtrd 2464 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 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869    =/= wne 2619   A.wral 2776   {crab 2780   _Vcvv 3082   (/)c0 3762   {cpr 3999   ran crn 4852   ` cfv 5599  (class class class)co 6303   0cc0 9541   1c1 9542    + caddc 9544    - cmin 9862   NN0cn0 10871  ..^cfzo 11917   #chash 12516  Word cword 12654   WWalks cwwlk 25397   WWalksN cwwlkn 25398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-oadd 7192  df-er 7369  df-map 7480  df-pm 7481  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-card 8376  df-cda 8600  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-nn 10612  df-2 10670  df-n0 10872  df-z 10940  df-uz 11162  df-fz 11787  df-fzo 11918  df-hash 12517  df-word 12662  df-wwlk 25399  df-wwlkn 25400
This theorem is referenced by:  rusgranumwlkb0  25673
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