MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  efgsdm Structured version   Unicode version

Theorem efgsdm 15354
Description: Elementhood in the domain of  S, the set of sequences of extensions starting at an irreducible word. (Contributed by Mario Carneiro, 27-Sep-2015.)
Hypotheses
Ref Expression
efgval.w  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
efgval.r  |-  .~  =  ( ~FG  `  I )
efgval2.m  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
efgval2.t  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
efgred.d  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
efgred.s  |-  S  =  ( m  e.  {
t  e.  (Word  W  \  { (/) } )  |  ( ( t ` 
0 )  e.  D  /\  A. k  e.  ( 1..^ ( # `  t
) ) ( t `
 k )  e. 
ran  ( T `  ( t `  (
k  -  1 ) ) ) ) } 
|->  ( m `  (
( # `  m )  -  1 ) ) )
Assertion
Ref Expression
efgsdm  |-  ( F  e.  dom  S  <->  ( F  e.  (Word  W  \  { (/)
} )  /\  ( F `  0 )  e.  D  /\  A. i  e.  ( 1..^ ( # `  F ) ) ( F `  i )  e.  ran  ( T `
 ( F `  ( i  -  1 ) ) ) ) )
Distinct variable groups:    y, z    i, F    t, n, v, w, y, z    i, m, n, t, v, w, x, M    i, k, T, m, t, x    y,
i, z, W    k, n, v, w, y, z, W, m, t, x    .~ , i, m, t, x, y, z    S, i   
i, I, m, n, t, v, w, x, y, z    D, i, m, t
Allowed substitution hints:    D( x, y, z, w, v, k, n)    .~ ( w, v, k, n)    S( x, y, z, w, v, t, k, m, n)    T( y,
z, w, v, n)    F( x, y, z, w, v, t, k, m, n)    I( k)    M( y, z, k)

Proof of Theorem efgsdm
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 fveq1 5719 . . . . 5  |-  ( f  =  F  ->  (
f `  0 )  =  ( F ` 
0 ) )
21eleq1d 2501 . . . 4  |-  ( f  =  F  ->  (
( f `  0
)  e.  D  <->  ( F `  0 )  e.  D ) )
3 fveq2 5720 . . . . . 6  |-  ( f  =  F  ->  ( # `
 f )  =  ( # `  F
) )
43oveq2d 6089 . . . . 5  |-  ( f  =  F  ->  (
1..^ ( # `  f
) )  =  ( 1..^ ( # `  F
) ) )
5 fveq1 5719 . . . . . 6  |-  ( f  =  F  ->  (
f `  i )  =  ( F `  i ) )
6 fveq1 5719 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  ( i  -  1 ) )  =  ( F `  ( i  -  1 ) ) )
76fveq2d 5724 . . . . . . 7  |-  ( f  =  F  ->  ( T `  ( f `  ( i  -  1 ) ) )  =  ( T `  ( F `  ( i  -  1 ) ) ) )
87rneqd 5089 . . . . . 6  |-  ( f  =  F  ->  ran  ( T `  ( f `
 ( i  - 
1 ) ) )  =  ran  ( T `
 ( F `  ( i  -  1 ) ) ) )
95, 8eleq12d 2503 . . . . 5  |-  ( f  =  F  ->  (
( f `  i
)  e.  ran  ( T `  ( f `  ( i  -  1 ) ) )  <->  ( F `  i )  e.  ran  ( T `  ( F `
 ( i  - 
1 ) ) ) ) )
104, 9raleqbidv 2908 . . . 4  |-  ( f  =  F  ->  ( A. i  e.  (
1..^ ( # `  f
) ) ( f `
 i )  e. 
ran  ( T `  ( f `  (
i  -  1 ) ) )  <->  A. i  e.  ( 1..^ ( # `  F ) ) ( F `  i )  e.  ran  ( T `
 ( F `  ( i  -  1 ) ) ) ) )
112, 10anbi12d 692 . . 3  |-  ( f  =  F  ->  (
( ( f ` 
0 )  e.  D  /\  A. i  e.  ( 1..^ ( # `  f
) ) ( f `
 i )  e. 
ran  ( T `  ( f `  (
i  -  1 ) ) ) )  <->  ( ( F `  0 )  e.  D  /\  A. i  e.  ( 1..^ ( # `  F ) ) ( F `  i )  e.  ran  ( T `
 ( F `  ( i  -  1 ) ) ) ) ) )
12 efgval.w . . . . . 6  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
13 efgval.r . . . . . 6  |-  .~  =  ( ~FG  `  I )
14 efgval2.m . . . . . 6  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
15 efgval2.t . . . . . 6  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
16 efgred.d . . . . . 6  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
17 efgred.s . . . . . 6  |-  S  =  ( m  e.  {
t  e.  (Word  W  \  { (/) } )  |  ( ( t ` 
0 )  e.  D  /\  A. k  e.  ( 1..^ ( # `  t
) ) ( t `
 k )  e. 
ran  ( T `  ( t `  (
k  -  1 ) ) ) ) } 
|->  ( m `  (
( # `  m )  -  1 ) ) )
1812, 13, 14, 15, 16, 17efgsf 15353 . . . . 5  |-  S : { t  e.  (Word 
W  \  { (/) } )  |  ( ( t `
 0 )  e.  D  /\  A. k  e.  ( 1..^ ( # `  t ) ) ( t `  k )  e.  ran  ( T `
 ( t `  ( k  -  1 ) ) ) ) } --> W
1918fdmi 5588 . . . 4  |-  dom  S  =  { t  e.  (Word 
W  \  { (/) } )  |  ( ( t `
 0 )  e.  D  /\  A. k  e.  ( 1..^ ( # `  t ) ) ( t `  k )  e.  ran  ( T `
 ( t `  ( k  -  1 ) ) ) ) }
20 fveq1 5719 . . . . . . 7  |-  ( t  =  f  ->  (
t `  0 )  =  ( f ` 
0 ) )
2120eleq1d 2501 . . . . . 6  |-  ( t  =  f  ->  (
( t `  0
)  e.  D  <->  ( f `  0 )  e.  D ) )
22 fveq2 5720 . . . . . . . . 9  |-  ( k  =  i  ->  (
t `  k )  =  ( t `  i ) )
23 oveq1 6080 . . . . . . . . . . . 12  |-  ( k  =  i  ->  (
k  -  1 )  =  ( i  - 
1 ) )
2423fveq2d 5724 . . . . . . . . . . 11  |-  ( k  =  i  ->  (
t `  ( k  -  1 ) )  =  ( t `  ( i  -  1 ) ) )
2524fveq2d 5724 . . . . . . . . . 10  |-  ( k  =  i  ->  ( T `  ( t `  ( k  -  1 ) ) )  =  ( T `  (
t `  ( i  -  1 ) ) ) )
2625rneqd 5089 . . . . . . . . 9  |-  ( k  =  i  ->  ran  ( T `  ( t `
 ( k  - 
1 ) ) )  =  ran  ( T `
 ( t `  ( i  -  1 ) ) ) )
2722, 26eleq12d 2503 . . . . . . . 8  |-  ( k  =  i  ->  (
( t `  k
)  e.  ran  ( T `  ( t `  ( k  -  1 ) ) )  <->  ( t `  i )  e.  ran  ( T `  ( t `
 ( i  - 
1 ) ) ) ) )
2827cbvralv 2924 . . . . . . 7  |-  ( A. k  e.  ( 1..^ ( # `  t
) ) ( t `
 k )  e. 
ran  ( T `  ( t `  (
k  -  1 ) ) )  <->  A. i  e.  ( 1..^ ( # `  t ) ) ( t `  i )  e.  ran  ( T `
 ( t `  ( i  -  1 ) ) ) )
29 fveq2 5720 . . . . . . . . 9  |-  ( t  =  f  ->  ( # `
 t )  =  ( # `  f
) )
3029oveq2d 6089 . . . . . . . 8  |-  ( t  =  f  ->  (
1..^ ( # `  t
) )  =  ( 1..^ ( # `  f
) ) )
31 fveq1 5719 . . . . . . . . 9  |-  ( t  =  f  ->  (
t `  i )  =  ( f `  i ) )
32 fveq1 5719 . . . . . . . . . . 11  |-  ( t  =  f  ->  (
t `  ( i  -  1 ) )  =  ( f `  ( i  -  1 ) ) )
3332fveq2d 5724 . . . . . . . . . 10  |-  ( t  =  f  ->  ( T `  ( t `  ( i  -  1 ) ) )  =  ( T `  (
f `  ( i  -  1 ) ) ) )
3433rneqd 5089 . . . . . . . . 9  |-  ( t  =  f  ->  ran  ( T `  ( t `
 ( i  - 
1 ) ) )  =  ran  ( T `
 ( f `  ( i  -  1 ) ) ) )
3531, 34eleq12d 2503 . . . . . . . 8  |-  ( t  =  f  ->  (
( t `  i
)  e.  ran  ( T `  ( t `  ( i  -  1 ) ) )  <->  ( f `  i )  e.  ran  ( T `  ( f `
 ( i  - 
1 ) ) ) ) )
3630, 35raleqbidv 2908 . . . . . . 7  |-  ( t  =  f  ->  ( A. i  e.  (
1..^ ( # `  t
) ) ( t `
 i )  e. 
ran  ( T `  ( t `  (
i  -  1 ) ) )  <->  A. i  e.  ( 1..^ ( # `  f ) ) ( f `  i )  e.  ran  ( T `
 ( f `  ( i  -  1 ) ) ) ) )
3728, 36syl5bb 249 . . . . . 6  |-  ( t  =  f  ->  ( A. k  e.  (
1..^ ( # `  t
) ) ( t `
 k )  e. 
ran  ( T `  ( t `  (
k  -  1 ) ) )  <->  A. i  e.  ( 1..^ ( # `  f ) ) ( f `  i )  e.  ran  ( T `
 ( f `  ( i  -  1 ) ) ) ) )
3821, 37anbi12d 692 . . . . 5  |-  ( t  =  f  ->  (
( ( t ` 
0 )  e.  D  /\  A. k  e.  ( 1..^ ( # `  t
) ) ( t `
 k )  e. 
ran  ( T `  ( t `  (
k  -  1 ) ) ) )  <->  ( (
f `  0 )  e.  D  /\  A. i  e.  ( 1..^ ( # `  f ) ) ( f `  i )  e.  ran  ( T `
 ( f `  ( i  -  1 ) ) ) ) ) )
3938cbvrabv 2947 . . . 4  |-  { t  e.  (Word  W  \  { (/) } )  |  ( ( t ` 
0 )  e.  D  /\  A. k  e.  ( 1..^ ( # `  t
) ) ( t `
 k )  e. 
ran  ( T `  ( t `  (
k  -  1 ) ) ) ) }  =  { f  e.  (Word  W  \  { (/)
} )  |  ( ( f `  0
)  e.  D  /\  A. i  e.  ( 1..^ ( # `  f
) ) ( f `
 i )  e. 
ran  ( T `  ( f `  (
i  -  1 ) ) ) ) }
4019, 39eqtri 2455 . . 3  |-  dom  S  =  { f  e.  (Word 
W  \  { (/) } )  |  ( ( f `
 0 )  e.  D  /\  A. i  e.  ( 1..^ ( # `  f ) ) ( f `  i )  e.  ran  ( T `
 ( f `  ( i  -  1 ) ) ) ) }
4111, 40elrab2 3086 . 2  |-  ( F  e.  dom  S  <->  ( F  e.  (Word  W  \  { (/)
} )  /\  (
( F `  0
)  e.  D  /\  A. i  e.  ( 1..^ ( # `  F
) ) ( F `
 i )  e. 
ran  ( T `  ( F `  ( i  -  1 ) ) ) ) ) )
42 3anass 940 . 2  |-  ( ( F  e.  (Word  W  \  { (/) } )  /\  ( F `  0 )  e.  D  /\  A. i  e.  ( 1..^ ( # `  F
) ) ( F `
 i )  e. 
ran  ( T `  ( F `  ( i  -  1 ) ) ) )  <->  ( F  e.  (Word  W  \  { (/)
} )  /\  (
( F `  0
)  e.  D  /\  A. i  e.  ( 1..^ ( # `  F
) ) ( F `
 i )  e. 
ran  ( T `  ( F `  ( i  -  1 ) ) ) ) ) )
4341, 42bitr4i 244 1  |-  ( F  e.  dom  S  <->  ( F  e.  (Word  W  \  { (/)
} )  /\  ( F `  0 )  e.  D  /\  A. i  e.  ( 1..^ ( # `  F ) ) ( F `  i )  e.  ran  ( T `
 ( F `  ( i  -  1 ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701    \ cdif 3309   (/)c0 3620   {csn 3806   <.cop 3809   <.cotp 3810   U_ciun 4085    e. cmpt 4258    _I cid 4485    X. cxp 4868   dom cdm 4870   ran crn 4871   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1oc1o 6709   2oc2o 6710   0cc0 8982   1c1 8983    - cmin 9283   ...cfz 11035  ..^cfzo 11127   #chash 11610  Word cword 11709   splice csplice 11713   <"cs2 11797   ~FG cefg 15330
This theorem is referenced by:  efgsdmi  15356  efgsrel  15358  efgs1  15359  efgs1b  15360  efgsp1  15361  efgsres  15362  efgsfo  15363  efgredlema  15364  efgredlemf  15365  efgredlemd  15368  efgredlemc  15369  efgredlem  15371  efgrelexlemb  15374  efgredeu  15376  efgred2  15377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-hash 11611  df-word 11715
  Copyright terms: Public domain W3C validator