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Theorem efgsdm 15039
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 5524 . . . . 5  |-  ( f  =  F  ->  (
f `  0 )  =  ( F ` 
0 ) )
21eleq1d 2349 . . . 4  |-  ( f  =  F  ->  (
( f `  0
)  e.  D  <->  ( F `  0 )  e.  D ) )
3 fveq2 5525 . . . . . 6  |-  ( f  =  F  ->  ( # `
 f )  =  ( # `  F
) )
43oveq2d 5874 . . . . 5  |-  ( f  =  F  ->  (
1..^ ( # `  f
) )  =  ( 1..^ ( # `  F
) ) )
5 fveq1 5524 . . . . . 6  |-  ( f  =  F  ->  (
f `  i )  =  ( F `  i ) )
6 fveq1 5524 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  ( i  -  1 ) )  =  ( F `  ( i  -  1 ) ) )
76fveq2d 5529 . . . . . . 7  |-  ( f  =  F  ->  ( T `  ( f `  ( i  -  1 ) ) )  =  ( T `  ( F `  ( i  -  1 ) ) ) )
87rneqd 4906 . . . . . 6  |-  ( f  =  F  ->  ran  ( T `  ( f `
 ( i  - 
1 ) ) )  =  ran  ( T `
 ( F `  ( i  -  1 ) ) ) )
95, 8eleq12d 2351 . . . . 5  |-  ( f  =  F  ->  (
( f `  i
)  e.  ran  ( T `  ( f `  ( i  -  1 ) ) )  <->  ( F `  i )  e.  ran  ( T `  ( F `
 ( i  - 
1 ) ) ) ) )
104, 9raleqbidv 2748 . . . 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 691 . . 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 15038 . . . . 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 5394 . . . 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 5524 . . . . . . 7  |-  ( t  =  f  ->  (
t `  0 )  =  ( f ` 
0 ) )
2120eleq1d 2349 . . . . . 6  |-  ( t  =  f  ->  (
( t `  0
)  e.  D  <->  ( f `  0 )  e.  D ) )
22 fveq2 5525 . . . . . . . . 9  |-  ( k  =  i  ->  (
t `  k )  =  ( t `  i ) )
23 oveq1 5865 . . . . . . . . . . . 12  |-  ( k  =  i  ->  (
k  -  1 )  =  ( i  - 
1 ) )
2423fveq2d 5529 . . . . . . . . . . 11  |-  ( k  =  i  ->  (
t `  ( k  -  1 ) )  =  ( t `  ( i  -  1 ) ) )
2524fveq2d 5529 . . . . . . . . . 10  |-  ( k  =  i  ->  ( T `  ( t `  ( k  -  1 ) ) )  =  ( T `  (
t `  ( i  -  1 ) ) ) )
2625rneqd 4906 . . . . . . . . 9  |-  ( k  =  i  ->  ran  ( T `  ( t `
 ( k  - 
1 ) ) )  =  ran  ( T `
 ( t `  ( i  -  1 ) ) ) )
2722, 26eleq12d 2351 . . . . . . . 8  |-  ( k  =  i  ->  (
( t `  k
)  e.  ran  ( T `  ( t `  ( k  -  1 ) ) )  <->  ( t `  i )  e.  ran  ( T `  ( t `
 ( i  - 
1 ) ) ) ) )
2827cbvralv 2764 . . . . . . 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 5525 . . . . . . . . 9  |-  ( t  =  f  ->  ( # `
 t )  =  ( # `  f
) )
3029oveq2d 5874 . . . . . . . 8  |-  ( t  =  f  ->  (
1..^ ( # `  t
) )  =  ( 1..^ ( # `  f
) ) )
31 fveq1 5524 . . . . . . . . 9  |-  ( t  =  f  ->  (
t `  i )  =  ( f `  i ) )
32 fveq1 5524 . . . . . . . . . . 11  |-  ( t  =  f  ->  (
t `  ( i  -  1 ) )  =  ( f `  ( i  -  1 ) ) )
3332fveq2d 5529 . . . . . . . . . 10  |-  ( t  =  f  ->  ( T `  ( t `  ( i  -  1 ) ) )  =  ( T `  (
f `  ( i  -  1 ) ) ) )
3433rneqd 4906 . . . . . . . . 9  |-  ( t  =  f  ->  ran  ( T `  ( t `
 ( i  - 
1 ) ) )  =  ran  ( T `
 ( f `  ( i  -  1 ) ) ) )
3531, 34eleq12d 2351 . . . . . . . 8  |-  ( t  =  f  ->  (
( t `  i
)  e.  ran  ( T `  ( t `  ( i  -  1 ) ) )  <->  ( f `  i )  e.  ran  ( T `  ( f `
 ( i  - 
1 ) ) ) ) )
3630, 35raleqbidv 2748 . . . . . . 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 248 . . . . . 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 691 . . . . 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 2787 . . . 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 2303 . . 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 2925 . 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 938 . 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 243 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547    \ cdif 3149   (/)c0 3455   {csn 3640   <.cop 3643   <.cotp 3644   U_ciun 3905    e. cmpt 4077    _I cid 4304    X. cxp 4687   dom cdm 4689   ran crn 4690   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1oc1o 6472   2oc2o 6473   0cc0 8737   1c1 8738    - cmin 9037   ...cfz 10782  ..^cfzo 10870   #chash 11337  Word cword 11403   splice csplice 11407   <"cs2 11491   ~FG cefg 15015
This theorem is referenced by:  efgsdmi  15041  efgsrel  15043  efgs1  15044  efgs1b  15045  efgsp1  15046  efgsres  15047  efgsfo  15048  efgredlema  15049  efgredlemf  15050  efgredlemd  15053  efgredlemc  15054  efgredlem  15056  efgrelexlemb  15059  efgredeu  15061  efgred2  15062
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-hash 11338  df-word 11409
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