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

Theorem wwlkextprop 30563
Description: Adding additional properties to the set of walks (as words) of a fixed length starting at a fixed vertex. (Contributed by Alexander van der Vekens, 1-Aug-2018.)
Hypotheses
Ref Expression
hashrabrex.x  |-  X  =  ( ( V WWalksN  E
) `  ( N  +  1 ) )
hashrabrex.y  |-  Y  =  { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  P }
Assertion
Ref Expression
wwlkextprop  |-  ( N  e.  NN0  ->  { x  e.  X  |  (
x `  0 )  =  P }  =  {
x  e.  X  |  E. y  e.  Y  ( ( x substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  x ) }  e.  ran  E ) } )
Distinct variable groups:    w, E    w, N    w, P    w, V    y, E    x, N, y, w    y, P    y, X    y, Y
Allowed substitution hints:    P( x)    E( x)    V( x, y)    X( x, w)    Y( x, w)

Proof of Theorem wwlkextprop
StepHypRef Expression
1 eqidd 2444 . . . . 5  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  ( x ` 
0 )  =  P )  ->  ( x substr  <.
0 ,  ( N  +  1 ) >.
)  =  ( x substr  <. 0 ,  ( N  +  1 ) >.
) )
2 hashrabrex.x . . . . . . . . 9  |-  X  =  ( ( V WWalksN  E
) `  ( N  +  1 ) )
32wwlkextproplem1 30560 . . . . . . . 8  |-  ( ( x  e.  X  /\  N  e.  NN0 )  -> 
( ( x substr  <. 0 ,  ( N  + 
1 ) >. ) `  0 )  =  ( x `  0
) )
43ancoms 453 . . . . . . 7  |-  ( ( N  e.  NN0  /\  x  e.  X )  ->  ( ( x substr  <. 0 ,  ( N  + 
1 ) >. ) `  0 )  =  ( x `  0
) )
54adantr 465 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  ( x ` 
0 )  =  P )  ->  ( (
x substr  <. 0 ,  ( N  +  1 )
>. ) `  0 )  =  ( x ` 
0 ) )
6 eqeq2 2452 . . . . . . 7  |-  ( ( x `  0 )  =  P  ->  (
( ( x substr  <. 0 ,  ( N  + 
1 ) >. ) `  0 )  =  ( x `  0
)  <->  ( ( x substr  <. 0 ,  ( N  +  1 ) >.
) `  0 )  =  P ) )
76adantl 466 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  ( x ` 
0 )  =  P )  ->  ( (
( x substr  <. 0 ,  ( N  +  1 ) >. ) `  0
)  =  ( x `
 0 )  <->  ( (
x substr  <. 0 ,  ( N  +  1 )
>. ) `  0 )  =  P ) )
85, 7mpbid 210 . . . . 5  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  ( x ` 
0 )  =  P )  ->  ( (
x substr  <. 0 ,  ( N  +  1 )
>. ) `  0 )  =  P )
92wwlkextproplem2 30561 . . . . . . 7  |-  ( ( x  e.  X  /\  N  e.  NN0 )  ->  { ( lastS  `  ( x substr  <. 0 ,  ( N  +  1 ) >.
) ) ,  ( lastS  `  x ) }  e.  ran  E )
109ancoms 453 . . . . . 6  |-  ( ( N  e.  NN0  /\  x  e.  X )  ->  { ( lastS  `  (
x substr  <. 0 ,  ( N  +  1 )
>. ) ) ,  ( lastS  `  x ) }  e.  ran  E )
1110adantr 465 . . . . 5  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  ( x ` 
0 )  =  P )  ->  { ( lastS  `  ( x substr  <. 0 ,  ( N  + 
1 ) >. )
) ,  ( lastS  `  x
) }  e.  ran  E )
12 simpr 461 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  x  e.  X )  ->  x  e.  X )
1312adantr 465 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  ( x ` 
0 )  =  P )  ->  x  e.  X )
14 simpr 461 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  ( x ` 
0 )  =  P )  ->  ( x `  0 )  =  P )
15 simpll 753 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  ( x ` 
0 )  =  P )  ->  N  e.  NN0 )
16 hashrabrex.y . . . . . . . 8  |-  Y  =  { w  e.  ( ( V WWalksN  E ) `  N )  |  ( w `  0 )  =  P }
172, 16wwlkextproplem3 30562 . . . . . . 7  |-  ( ( x  e.  X  /\  ( x `  0
)  =  P  /\  N  e.  NN0 )  -> 
( x substr  <. 0 ,  ( N  +  1 ) >. )  e.  Y
)
1813, 14, 15, 17syl3anc 1218 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  ( x ` 
0 )  =  P )  ->  ( x substr  <.
0 ,  ( N  +  1 ) >.
)  e.  Y )
19 eqeq2 2452 . . . . . . . 8  |-  ( y  =  ( x substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( ( x substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  <->  ( x substr  <. 0 ,  ( N  + 
1 ) >. )  =  ( x substr  <. 0 ,  ( N  + 
1 ) >. )
) )
20 fveq1 5690 . . . . . . . . 9  |-  ( y  =  ( x substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( y `  0
)  =  ( ( x substr  <. 0 ,  ( N  +  1 )
>. ) `  0 ) )
2120eqeq1d 2451 . . . . . . . 8  |-  ( y  =  ( x substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( ( y ` 
0 )  =  P  <-> 
( ( x substr  <. 0 ,  ( N  + 
1 ) >. ) `  0 )  =  P ) )
22 fveq2 5691 . . . . . . . . . 10  |-  ( y  =  ( x substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( lastS  `  y )  =  ( lastS  `  ( x substr  <. 0 ,  ( N  +  1 ) >.
) ) )
2322preq1d 3960 . . . . . . . . 9  |-  ( y  =  ( x substr  <. 0 ,  ( N  + 
1 ) >. )  ->  { ( lastS  `  y
) ,  ( lastS  `  x
) }  =  {
( lastS  `  ( x substr  <. 0 ,  ( N  + 
1 ) >. )
) ,  ( lastS  `  x
) } )
2423eleq1d 2509 . . . . . . . 8  |-  ( y  =  ( x substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( { ( lastS  `  y
) ,  ( lastS  `  x
) }  e.  ran  E  <->  { ( lastS  `  ( x substr  <. 0 ,  ( N  +  1 ) >.
) ) ,  ( lastS  `  x ) }  e.  ran  E ) )
2519, 21, 243anbi123d 1289 . . . . . . 7  |-  ( y  =  ( x substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( ( ( x substr  <. 0 ,  ( N  +  1 ) >.
)  =  y  /\  ( y `  0
)  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  x ) }  e.  ran  E
)  <->  ( ( x substr  <. 0 ,  ( N  +  1 ) >.
)  =  ( x substr  <. 0 ,  ( N  +  1 ) >.
)  /\  ( (
x substr  <. 0 ,  ( N  +  1 )
>. ) `  0 )  =  P  /\  {
( lastS  `  ( x substr  <. 0 ,  ( N  + 
1 ) >. )
) ,  ( lastS  `  x
) }  e.  ran  E ) ) )
2625adantl 466 . . . . . 6  |-  ( ( ( ( N  e. 
NN0  /\  x  e.  X )  /\  (
x `  0 )  =  P )  /\  y  =  ( x substr  <. 0 ,  ( N  + 
1 ) >. )
)  ->  ( (
( x substr  <. 0 ,  ( N  +  1 ) >. )  =  y  /\  ( y ` 
0 )  =  P  /\  { ( lastS  `  y
) ,  ( lastS  `  x
) }  e.  ran  E )  <->  ( ( x substr  <. 0 ,  ( N  +  1 ) >.
)  =  ( x substr  <. 0 ,  ( N  +  1 ) >.
)  /\  ( (
x substr  <. 0 ,  ( N  +  1 )
>. ) `  0 )  =  P  /\  {
( lastS  `  ( x substr  <. 0 ,  ( N  + 
1 ) >. )
) ,  ( lastS  `  x
) }  e.  ran  E ) ) )
2718, 26rspcedv 3077 . . . . 5  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  ( x ` 
0 )  =  P )  ->  ( (
( x substr  <. 0 ,  ( N  +  1 ) >. )  =  ( x substr  <. 0 ,  ( N  +  1 )
>. )  /\  (
( x substr  <. 0 ,  ( N  +  1 ) >. ) `  0
)  =  P  /\  { ( lastS  `  ( x substr  <.
0 ,  ( N  +  1 ) >.
) ) ,  ( lastS  `  x ) }  e.  ran  E )  ->  E. y  e.  Y  ( (
x substr  <. 0 ,  ( N  +  1 )
>. )  =  y  /\  ( y `  0
)  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  x ) }  e.  ran  E
) ) )
281, 8, 11, 27mp3and 1317 . . . 4  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  ( x ` 
0 )  =  P )  ->  E. y  e.  Y  ( (
x substr  <. 0 ,  ( N  +  1 )
>. )  =  y  /\  ( y `  0
)  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  x ) }  e.  ran  E
) )
2928ex 434 . . 3  |-  ( ( N  e.  NN0  /\  x  e.  X )  ->  ( ( x ` 
0 )  =  P  ->  E. y  e.  Y  ( ( x substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  x ) }  e.  ran  E ) ) )
3020eqcoms 2446 . . . . . . . . 9  |-  ( ( x substr  <. 0 ,  ( N  +  1 )
>. )  =  y  ->  ( y `  0
)  =  ( ( x substr  <. 0 ,  ( N  +  1 )
>. ) `  0 ) )
3130eqeq1d 2451 . . . . . . . 8  |-  ( ( x substr  <. 0 ,  ( N  +  1 )
>. )  =  y  ->  ( ( y ` 
0 )  =  P  <-> 
( ( x substr  <. 0 ,  ( N  + 
1 ) >. ) `  0 )  =  P ) )
323eqcomd 2448 . . . . . . . . . . 11  |-  ( ( x  e.  X  /\  N  e.  NN0 )  -> 
( x `  0
)  =  ( ( x substr  <. 0 ,  ( N  +  1 )
>. ) `  0 ) )
3332ancoms 453 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  x  e.  X )  ->  ( x `  0
)  =  ( ( x substr  <. 0 ,  ( N  +  1 )
>. ) `  0 ) )
3433adantr 465 . . . . . . . . 9  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  y  e.  Y
)  ->  ( x `  0 )  =  ( ( x substr  <. 0 ,  ( N  + 
1 ) >. ) `  0 ) )
35 eqeq2 2452 . . . . . . . . . 10  |-  ( P  =  ( ( x substr  <. 0 ,  ( N  +  1 ) >.
) `  0 )  ->  ( ( x ` 
0 )  =  P  <-> 
( x `  0
)  =  ( ( x substr  <. 0 ,  ( N  +  1 )
>. ) `  0 ) ) )
3635eqcoms 2446 . . . . . . . . 9  |-  ( ( ( x substr  <. 0 ,  ( N  + 
1 ) >. ) `  0 )  =  P  ->  ( (
x `  0 )  =  P  <->  ( x ` 
0 )  =  ( ( x substr  <. 0 ,  ( N  + 
1 ) >. ) `  0 ) ) )
3734, 36syl5ibr 221 . . . . . . . 8  |-  ( ( ( x substr  <. 0 ,  ( N  + 
1 ) >. ) `  0 )  =  P  ->  ( (
( N  e.  NN0  /\  x  e.  X )  /\  y  e.  Y
)  ->  ( x `  0 )  =  P ) )
3831, 37syl6bi 228 . . . . . . 7  |-  ( ( x substr  <. 0 ,  ( N  +  1 )
>. )  =  y  ->  ( ( y ` 
0 )  =  P  ->  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  y  e.  Y
)  ->  ( x `  0 )  =  P ) ) )
3938imp 429 . . . . . 6  |-  ( ( ( x substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P )  ->  (
( ( N  e. 
NN0  /\  x  e.  X )  /\  y  e.  Y )  ->  (
x `  0 )  =  P ) )
40393adant3 1008 . . . . 5  |-  ( ( ( x substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  x ) }  e.  ran  E )  ->  (
( ( N  e. 
NN0  /\  x  e.  X )  /\  y  e.  Y )  ->  (
x `  0 )  =  P ) )
4140com12 31 . . . 4  |-  ( ( ( N  e.  NN0  /\  x  e.  X )  /\  y  e.  Y
)  ->  ( (
( x substr  <. 0 ,  ( N  +  1 ) >. )  =  y  /\  ( y ` 
0 )  =  P  /\  { ( lastS  `  y
) ,  ( lastS  `  x
) }  e.  ran  E )  ->  ( x `  0 )  =  P ) )
4241rexlimdva 2841 . . 3  |-  ( ( N  e.  NN0  /\  x  e.  X )  ->  ( E. y  e.  Y  ( ( x substr  <. 0 ,  ( N  +  1 ) >.
)  =  y  /\  ( y `  0
)  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  x ) }  e.  ran  E
)  ->  ( x `  0 )  =  P ) )
4329, 42impbid 191 . 2  |-  ( ( N  e.  NN0  /\  x  e.  X )  ->  ( ( x ` 
0 )  =  P  <->  E. y  e.  Y  ( ( x substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  x ) }  e.  ran  E ) ) )
4443rabbidva 2963 1  |-  ( N  e.  NN0  ->  { x  e.  X  |  (
x `  0 )  =  P }  =  {
x  e.  X  |  E. y  e.  Y  ( ( x substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  x ) }  e.  ran  E ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2716   {crab 2719   {cpr 3879   <.cop 3883   ran crn 4841   ` cfv 5418  (class class class)co 6091   0cc0 9282   1c1 9283    + caddc 9285   NN0cn0 10579   lastS clsw 12222   substr csubstr 12225   WWalksN cwwlkn 30312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-hash 12104  df-word 12229  df-lsw 12230  df-substr 12233  df-wwlk 30313  df-wwlkn 30314
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator