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Theorem wwlkextfun 30510
Description: Lemma 1 for wwlkextbij 30514. (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 5800 . . . . . . 7  |-  ( w  =  t  ->  ( # `
 w )  =  ( # `  t
) )
21eqeq1d 2456 . . . . . 6  |-  ( w  =  t  ->  (
( # `  w )  =  ( N  + 
2 )  <->  ( # `  t
)  =  ( N  +  2 ) ) )
3 oveq1 6208 . . . . . . 7  |-  ( w  =  t  ->  (
w substr  <. 0 ,  ( N  +  1 )
>. )  =  (
t substr  <. 0 ,  ( N  +  1 )
>. ) )
43eqeq1d 2456 . . . . . 6  |-  ( w  =  t  ->  (
( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  <-> 
( t substr  <. 0 ,  ( N  +  1 ) >. )  =  W ) )
5 fveq2 5800 . . . . . . . 8  |-  ( w  =  t  ->  ( lastS  `  w )  =  ( lastS  `  t ) )
65preq2d 4070 . . . . . . 7  |-  ( w  =  t  ->  { ( lastS  `  W ) ,  ( lastS  `  w ) }  =  { ( lastS  `  W ) ,  ( lastS  `  t
) } )
76eleq1d 2523 . . . . . 6  |-  ( w  =  t  ->  ( { ( lastS  `  W ) ,  ( lastS  `  w
) }  e.  ran  E  <->  { ( lastS  `  W ) ,  ( lastS  `  t
) }  e.  ran  E ) )
82, 4, 73anbi123d 1290 . . . . 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 3226 . . . 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 10700 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN0  ->  N  e.  RR )
13 2re 10503 . . . . . . . . . . . . . . . . 17  |-  2  e.  RR
1413a1i 11 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN0  ->  2  e.  RR )
15 nn0ge0 10717 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN0  ->  0  <_  N )
16 2pos 10525 . . . . . . . . . . . . . . . . 17  |-  0  <  2
1716a1i 11 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN0  ->  0  <  2 )
1812, 14, 15, 17addgegt0d 10025 . . . . . . . . . . . . . . 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 4405 . . . . . . . . . . . . . . 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 12251 . . . . . . . . . . . . 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 1009 . . . . . . . . 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 12389 . . . . . 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 1038 . . . . 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 4064 . . . . 5  |-  ( n  =  ( lastS  `  t
)  ->  { ( lastS  `  W ) ,  n }  =  { ( lastS  `  W ) ,  ( lastS  `  t ) } )
3736eleq1d 2523 . . . 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 3226 . . 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 5977 1  |-  ( N  e.  NN0  ->  F : D
--> R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   {crab 2803   (/)c0 3746   {cpr 3988   <.cop 3992   class class class wbr 4401    |-> cmpt 4459   ran crn 4950   -->wf 5523   ` cfv 5527  (class class class)co 6201   RRcr 9393   0cc0 9394   1c1 9395    + caddc 9397    < clt 9530   2c2 10483   NN0cn0 10691   #chash 12221  Word cword 12340   lastS clsw 12341   substr csubstr 12344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-hash 12222  df-word 12348  df-lsw 12349
This theorem is referenced by:  wwlkextinj  30511  wwlkextsur  30512
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