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Theorem wwlkextfun 24391
Description: Lemma 1 for wwlkextbij 24395. (Contributed by Alexander van der Vekens, 7-Aug-2018.)
Hypotheses
Ref Expression
wwlkextbij.d  |-  D  =  { w  e. Word  V  |  ( ( # `  w )  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E ) }
wwlkextbij.r  |-  R  =  { n  e.  V  |  { ( lastS  `  W
) ,  n }  e.  ran  E }
wwlkextbij.f  |-  F  =  ( t  e.  D  |->  ( lastS  `  t )
)
Assertion
Ref Expression
wwlkextfun  |-  ( N  e.  NN0  ->  F : D
--> R )
Distinct variable groups:    t, D    n, E, w    t, N, w    t, R    n, V, t, w    n, W, t, w
Allowed substitution hints:    D( w, n)    R( w, n)    E( t)    F( w, t, n)    N( n)

Proof of Theorem wwlkextfun
StepHypRef Expression
1 fveq2 5857 . . . . . . 7  |-  ( w  =  t  ->  ( # `
 w )  =  ( # `  t
) )
21eqeq1d 2462 . . . . . 6  |-  ( w  =  t  ->  (
( # `  w )  =  ( N  + 
2 )  <->  ( # `  t
)  =  ( N  +  2 ) ) )
3 oveq1 6282 . . . . . . 7  |-  ( w  =  t  ->  (
w substr  <. 0 ,  ( N  +  1 )
>. )  =  (
t substr  <. 0 ,  ( N  +  1 )
>. ) )
43eqeq1d 2462 . . . . . 6  |-  ( w  =  t  ->  (
( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  <-> 
( t substr  <. 0 ,  ( N  +  1 ) >. )  =  W ) )
5 fveq2 5857 . . . . . . . 8  |-  ( w  =  t  ->  ( lastS  `  w )  =  ( lastS  `  t ) )
65preq2d 4106 . . . . . . 7  |-  ( w  =  t  ->  { ( lastS  `  W ) ,  ( lastS  `  w ) }  =  { ( lastS  `  W ) ,  ( lastS  `  t
) } )
76eleq1d 2529 . . . . . 6  |-  ( w  =  t  ->  ( { ( lastS  `  W ) ,  ( lastS  `  w
) }  e.  ran  E  <->  { ( lastS  `  W ) ,  ( lastS  `  t
) }  e.  ran  E ) )
82, 4, 73anbi123d 1294 . . . . 5  |-  ( w  =  t  ->  (
( ( # `  w
)  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E )  <->  ( ( # `  t )  =  ( N  +  2 )  /\  ( t substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  t ) }  e.  ran  E
) ) )
9 wwlkextbij.d . . . . 5  |-  D  =  { w  e. Word  V  |  ( ( # `  w )  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E ) }
108, 9elrab2 3256 . . . 4  |-  ( t  e.  D  <->  ( t  e. Word  V  /\  ( (
# `  t )  =  ( N  + 
2 )  /\  (
t substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  t
) }  e.  ran  E ) ) )
11 simpll 753 . . . . . . . . . . . 12  |-  ( ( ( t  e. Word  V  /\  N  e.  NN0 )  /\  ( # `  t
)  =  ( N  +  2 ) )  ->  t  e. Word  V
)
12 nn0re 10793 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN0  ->  N  e.  RR )
13 2re 10594 . . . . . . . . . . . . . . . . 17  |-  2  e.  RR
1413a1i 11 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN0  ->  2  e.  RR )
15 nn0ge0 10810 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN0  ->  0  <_  N )
16 2pos 10616 . . . . . . . . . . . . . . . . 17  |-  0  <  2
1716a1i 11 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN0  ->  0  <  2 )
1812, 14, 15, 17addgegt0d 10115 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN0  ->  0  < 
( N  +  2 ) )
1918ad2antlr 726 . . . . . . . . . . . . . 14  |-  ( ( ( t  e. Word  V  /\  N  e.  NN0 )  /\  ( # `  t
)  =  ( N  +  2 ) )  ->  0  <  ( N  +  2 ) )
20 breq2 4444 . . . . . . . . . . . . . . 15  |-  ( (
# `  t )  =  ( N  + 
2 )  ->  (
0  <  ( # `  t
)  <->  0  <  ( N  +  2 ) ) )
2120adantl 466 . . . . . . . . . . . . . 14  |-  ( ( ( t  e. Word  V  /\  N  e.  NN0 )  /\  ( # `  t
)  =  ( N  +  2 ) )  ->  ( 0  < 
( # `  t )  <->  0  <  ( N  +  2 ) ) )
2219, 21mpbird 232 . . . . . . . . . . . . 13  |-  ( ( ( t  e. Word  V  /\  N  e.  NN0 )  /\  ( # `  t
)  =  ( N  +  2 ) )  ->  0  <  ( # `
 t ) )
23 hashgt0n0 12390 . . . . . . . . . . . . 13  |-  ( ( t  e. Word  V  /\  0  <  ( # `  t
) )  ->  t  =/=  (/) )
2411, 22, 23syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( t  e. Word  V  /\  N  e.  NN0 )  /\  ( # `  t
)  =  ( N  +  2 ) )  ->  t  =/=  (/) )
2511, 24jca 532 . . . . . . . . . . 11  |-  ( ( ( t  e. Word  V  /\  N  e.  NN0 )  /\  ( # `  t
)  =  ( N  +  2 ) )  ->  ( t  e. Word  V  /\  t  =/=  (/) ) )
2625expcom 435 . . . . . . . . . 10  |-  ( (
# `  t )  =  ( N  + 
2 )  ->  (
( t  e. Word  V  /\  N  e.  NN0 )  ->  ( t  e. Word  V  /\  t  =/=  (/) ) ) )
27263ad2ant1 1012 . . . . . . . . 9  |-  ( ( ( # `  t
)  =  ( N  +  2 )  /\  ( t substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  t
) }  e.  ran  E )  ->  ( (
t  e. Word  V  /\  N  e.  NN0 )  -> 
( t  e. Word  V  /\  t  =/=  (/) ) ) )
2827expd 436 . . . . . . . 8  |-  ( ( ( # `  t
)  =  ( N  +  2 )  /\  ( t substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  t
) }  e.  ran  E )  ->  ( t  e. Word  V  ->  ( N  e.  NN0  ->  ( t  e. Word  V  /\  t  =/=  (/) ) ) ) )
2928impcom 430 . . . . . . 7  |-  ( ( t  e. Word  V  /\  ( ( # `  t
)  =  ( N  +  2 )  /\  ( t substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  t
) }  e.  ran  E ) )  ->  ( N  e.  NN0  ->  (
t  e. Word  V  /\  t  =/=  (/) ) ) )
3029impcom 430 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( t  e. Word  V  /\  ( ( # `  t
)  =  ( N  +  2 )  /\  ( t substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  t
) }  e.  ran  E ) ) )  -> 
( t  e. Word  V  /\  t  =/=  (/) ) )
31 lswcl 12541 . . . . . 6  |-  ( ( t  e. Word  V  /\  t  =/=  (/) )  ->  ( lastS  `  t )  e.  V
)
3230, 31syl 16 . . . . 5  |-  ( ( N  e.  NN0  /\  ( t  e. Word  V  /\  ( ( # `  t
)  =  ( N  +  2 )  /\  ( t substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  t
) }  e.  ran  E ) ) )  -> 
( lastS  `  t )  e.  V )
33 simprr3 1041 . . . . 5  |-  ( ( N  e.  NN0  /\  ( t  e. Word  V  /\  ( ( # `  t
)  =  ( N  +  2 )  /\  ( t substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  t
) }  e.  ran  E ) ) )  ->  { ( lastS  `  W ) ,  ( lastS  `  t
) }  e.  ran  E )
3432, 33jca 532 . . . 4  |-  ( ( N  e.  NN0  /\  ( t  e. Word  V  /\  ( ( # `  t
)  =  ( N  +  2 )  /\  ( t substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  t
) }  e.  ran  E ) ) )  -> 
( ( lastS  `  t
)  e.  V  /\  { ( lastS  `  W ) ,  ( lastS  `  t ) }  e.  ran  E
) )
3510, 34sylan2b 475 . . 3  |-  ( ( N  e.  NN0  /\  t  e.  D )  ->  ( ( lastS  `  t
)  e.  V  /\  { ( lastS  `  W ) ,  ( lastS  `  t ) }  e.  ran  E
) )
36 preq2 4100 . . . . 5  |-  ( n  =  ( lastS  `  t
)  ->  { ( lastS  `  W ) ,  n }  =  { ( lastS  `  W ) ,  ( lastS  `  t ) } )
3736eleq1d 2529 . . . 4  |-  ( n  =  ( lastS  `  t
)  ->  ( {
( lastS  `  W ) ,  n }  e.  ran  E  <->  { ( lastS  `  W ) ,  ( lastS  `  t
) }  e.  ran  E ) )
38 wwlkextbij.r . . . 4  |-  R  =  { n  e.  V  |  { ( lastS  `  W
) ,  n }  e.  ran  E }
3937, 38elrab2 3256 . . 3  |-  ( ( lastS  `  t )  e.  R  <->  ( ( lastS  `  t )  e.  V  /\  { ( lastS  `  W ) ,  ( lastS  `  t ) }  e.  ran  E ) )
4035, 39sylibr 212 . 2  |-  ( ( N  e.  NN0  /\  t  e.  D )  ->  ( lastS  `  t )  e.  R )
41 wwlkextbij.f . 2  |-  F  =  ( t  e.  D  |->  ( lastS  `  t )
)
4240, 41fmptd 6036 1  |-  ( N  e.  NN0  ->  F : D
--> R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   {crab 2811   (/)c0 3778   {cpr 4022   <.cop 4026   class class class wbr 4440    |-> cmpt 4498   ran crn 4993   -->wf 5575   ` cfv 5579  (class class class)co 6275   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    < clt 9617   2c2 10574   NN0cn0 10784   #chash 12360  Word cword 12487   lastS clsw 12488   substr csubstr 12491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-hash 12361  df-word 12495  df-lsw 12496
This theorem is referenced by:  wwlkextinj  24392  wwlkextsur  24393
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