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Theorem wwlkn0s 24478
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 10811 . . 3  |-  0  e.  NN0
2 wwlkn 24455 . . 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 10648 . . . . . 6  |-  ( 0  +  1 )  =  1
54eqeq2i 2485 . . . . 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 24456 . . . . . . . 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 9562 . . . . . . . . . . . . 13  |-  1  =/=  0
11 pm13.181 2779 . . . . . . . . . . . . 13  |-  ( ( ( # `  w
)  =  1  /\  1  =/=  0 )  ->  ( # `  w
)  =/=  0 )
1210, 11mpan2 671 . . . . . . . . . . . 12  |-  ( (
# `  w )  =  1  ->  ( # `
 w )  =/=  0 )
13 vex 3116 . . . . . . . . . . . . . 14  |-  w  e. 
_V
14 hasheq0 12402 . . . . . . . . . . . . . 14  |-  ( w  e.  _V  ->  (
( # `  w )  =  0  <->  w  =  (/) ) )
1513, 14mp1i 12 . . . . . . . . . . . . 13  |-  ( (
# `  w )  =  1  ->  (
( # `  w )  =  0  <->  w  =  (/) ) )
1615necon3bid 2725 . . . . . . . . . . . 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 3932 . . . . . . . . . . . 12  |-  A. i  e.  (/)  { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  E
21 oveq1 6292 . . . . . . . . . . . . . . . 16  |-  ( (
# `  w )  =  1  ->  (
( # `  w )  -  1 )  =  ( 1  -  1 ) )
22 1m1e0 10605 . . . . . . . . . . . . . . . 16  |-  ( 1  -  1 )  =  0
2321, 22syl6eq 2524 . . . . . . . . . . . . . . 15  |-  ( (
# `  w )  =  1  ->  (
( # `  w )  -  1 )  =  0 )
2423oveq2d 6301 . . . . . . . . . . . . . 14  |-  ( (
# `  w )  =  1  ->  (
0..^ ( ( # `  w )  -  1 ) )  =  ( 0..^ 0 ) )
25 fzo0 11818 . . . . . . . . . . . . . 14  |-  ( 0..^ 0 )  =  (/)
2624, 25syl6eq 2524 . . . . . . . . . . . . 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 3072 . . . . . . . . . . . 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 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
) )
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 3103 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  { w  e.  ( V WWalks  E )  |  ( # `  w
)  =  ( 0  +  1 ) }  =  { w  e. Word  V  |  ( # `  w
)  =  1 } )
393, 38eqtrd 2508 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 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   {crab 2818   _Vcvv 3113   (/)c0 3785   {cpr 4029   ran crn 5000   ` cfv 5588  (class class class)co 6285   0cc0 9493   1c1 9494    + caddc 9496    - cmin 9806   NN0cn0 10796  ..^cfzo 11793   #chash 12374  Word cword 12501   WWalks cwwlk 24450   WWalksN cwwlkn 24451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-pm 7424  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-card 8321  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674  df-fzo 11794  df-hash 12375  df-word 12509  df-wwlk 24452  df-wwlkn 24453
This theorem is referenced by:  rusgranumwlkb0  24726
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