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Theorem wwlkn0s 24832
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 10831 . . 3  |-  0  e.  NN0
2 wwlkn 24809 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y  /\  0  e.  NN0 )  -> 
( ( V WWalksN  E
) `  0 )  =  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( 0  +  1 ) } )
31, 2mp3an3 1313 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( V WWalksN  E
) `  0 )  =  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( 0  +  1 ) } )
4 0p1e1 10668 . . . . . 6  |-  ( 0  +  1 )  =  1
54eqeq2i 2475 . . . . 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 24810 . . . . . . . 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 997 . . . . . . . 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 9578 . . . . . . . . . . . . 13  |-  1  =/=  0
11 pm13.181 2769 . . . . . . . . . . . . 13  |-  ( ( ( # `  w
)  =  1  /\  1  =/=  0 )  ->  ( # `  w
)  =/=  0 )
1210, 11mpan2 671 . . . . . . . . . . . 12  |-  ( (
# `  w )  =  1  ->  ( # `
 w )  =/=  0 )
13 vex 3112 . . . . . . . . . . . . . 14  |-  w  e. 
_V
14 hasheq0 12436 . . . . . . . . . . . . . 14  |-  ( w  e.  _V  ->  (
( # `  w )  =  0  <->  w  =  (/) ) )
1513, 14mp1i 12 . . . . . . . . . . . . 13  |-  ( (
# `  w )  =  1  ->  (
( # `  w )  =  0  <->  w  =  (/) ) )
1615necon3bid 2715 . . . . . . . . . . . 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 3937 . . . . . . . . . . . 12  |-  A. i  e.  (/)  { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  E
21 oveq1 6303 . . . . . . . . . . . . . . . 16  |-  ( (
# `  w )  =  1  ->  (
( # `  w )  -  1 )  =  ( 1  -  1 ) )
22 1m1e0 10625 . . . . . . . . . . . . . . . 16  |-  ( 1  -  1 )  =  0
2321, 22syl6eq 2514 . . . . . . . . . . . . . . 15  |-  ( (
# `  w )  =  1  ->  (
( # `  w )  -  1 )  =  0 )
2423oveq2d 6312 . . . . . . . . . . . . . 14  |-  ( (
# `  w )  =  1  ->  (
0..^ ( ( # `  w )  -  1 ) )  =  ( 0..^ 0 ) )
25 fzo0 11848 . . . . . . . . . . . . . 14  |-  ( 0..^ 0 )  =  (/)
2624, 25syl6eq 2514 . . . . . . . . . . . . 13  |-  ( (
# `  w )  =  1  ->  (
0..^ ( ( # `  w )  -  1 ) )  =  (/) )
2726raleqdv 3060 . . . . . . . . . . . 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 233 . . . . . . . . . . 11  |-  ( (
# `  w )  =  1  ->  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E
)
2928ad2antlr 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
)
3018, 19, 293jca 1176 . . . . . . . . 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 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 ) ) )
329, 31impbid2 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 ) )
338, 32bitrd 253 . . . . . 6  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( # `  w
)  =  1 )  ->  ( w  e.  ( V WWalks  E )  <-> 
w  e. Word  V )
)
3433ex 434 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( # `  w
)  =  1  -> 
( w  e.  ( V WWalks  E )  <->  w  e. Word  V ) ) )
3534pm5.32rd 640 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( w  e.  ( V WWalks  E )  /\  ( # `  w
)  =  1 )  <-> 
( w  e. Word  V  /\  ( # `  w
)  =  1 ) ) )
366, 35syl5bb 257 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( w  e.  ( V WWalks  E )  /\  ( # `  w
)  =  ( 0  +  1 ) )  <-> 
( w  e. Word  V  /\  ( # `  w
)  =  1 ) ) )
3736rabbidva2 3099 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( 0  +  1 ) }  =  { w  e. Word  V  |  ( # `  w
)  =  1 } )
383, 37eqtrd 2498 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   {crab 2811   _Vcvv 3109   (/)c0 3793   {cpr 4034   ran crn 5009   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510    + caddc 9512    - cmin 9824   NN0cn0 10816  ..^cfzo 11821   #chash 12408  Word cword 12538   WWalks cwwlk 24804   WWalksN cwwlkn 24805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-wwlk 24806  df-wwlkn 24807
This theorem is referenced by:  rusgranumwlkb0  25080
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