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Theorem efgsdm 15055
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 5540 . . . . 5  |-  ( f  =  F  ->  (
f `  0 )  =  ( F ` 
0 ) )
21eleq1d 2362 . . . 4  |-  ( f  =  F  ->  (
( f `  0
)  e.  D  <->  ( F `  0 )  e.  D ) )
3 fveq2 5541 . . . . . 6  |-  ( f  =  F  ->  ( # `
 f )  =  ( # `  F
) )
43oveq2d 5890 . . . . 5  |-  ( f  =  F  ->  (
1..^ ( # `  f
) )  =  ( 1..^ ( # `  F
) ) )
5 fveq1 5540 . . . . . 6  |-  ( f  =  F  ->  (
f `  i )  =  ( F `  i ) )
6 fveq1 5540 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  ( i  -  1 ) )  =  ( F `  ( i  -  1 ) ) )
76fveq2d 5545 . . . . . . 7  |-  ( f  =  F  ->  ( T `  ( f `  ( i  -  1 ) ) )  =  ( T `  ( F `  ( i  -  1 ) ) ) )
87rneqd 4922 . . . . . 6  |-  ( f  =  F  ->  ran  ( T `  ( f `
 ( i  - 
1 ) ) )  =  ran  ( T `
 ( F `  ( i  -  1 ) ) ) )
95, 8eleq12d 2364 . . . . 5  |-  ( f  =  F  ->  (
( f `  i
)  e.  ran  ( T `  ( f `  ( i  -  1 ) ) )  <->  ( F `  i )  e.  ran  ( T `  ( F `
 ( i  - 
1 ) ) ) ) )
104, 9raleqbidv 2761 . . . 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 15054 . . . . 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 5410 . . . 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 5540 . . . . . . 7  |-  ( t  =  f  ->  (
t `  0 )  =  ( f ` 
0 ) )
2120eleq1d 2362 . . . . . 6  |-  ( t  =  f  ->  (
( t `  0
)  e.  D  <->  ( f `  0 )  e.  D ) )
22 fveq2 5541 . . . . . . . . 9  |-  ( k  =  i  ->  (
t `  k )  =  ( t `  i ) )
23 oveq1 5881 . . . . . . . . . . . 12  |-  ( k  =  i  ->  (
k  -  1 )  =  ( i  - 
1 ) )
2423fveq2d 5545 . . . . . . . . . . 11  |-  ( k  =  i  ->  (
t `  ( k  -  1 ) )  =  ( t `  ( i  -  1 ) ) )
2524fveq2d 5545 . . . . . . . . . 10  |-  ( k  =  i  ->  ( T `  ( t `  ( k  -  1 ) ) )  =  ( T `  (
t `  ( i  -  1 ) ) ) )
2625rneqd 4922 . . . . . . . . 9  |-  ( k  =  i  ->  ran  ( T `  ( t `
 ( k  - 
1 ) ) )  =  ran  ( T `
 ( t `  ( i  -  1 ) ) ) )
2722, 26eleq12d 2364 . . . . . . . 8  |-  ( k  =  i  ->  (
( t `  k
)  e.  ran  ( T `  ( t `  ( k  -  1 ) ) )  <->  ( t `  i )  e.  ran  ( T `  ( t `
 ( i  - 
1 ) ) ) ) )
2827cbvralv 2777 . . . . . . 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 5541 . . . . . . . . 9  |-  ( t  =  f  ->  ( # `
 t )  =  ( # `  f
) )
3029oveq2d 5890 . . . . . . . 8  |-  ( t  =  f  ->  (
1..^ ( # `  t
) )  =  ( 1..^ ( # `  f
) ) )
31 fveq1 5540 . . . . . . . . 9  |-  ( t  =  f  ->  (
t `  i )  =  ( f `  i ) )
32 fveq1 5540 . . . . . . . . . . 11  |-  ( t  =  f  ->  (
t `  ( i  -  1 ) )  =  ( f `  ( i  -  1 ) ) )
3332fveq2d 5545 . . . . . . . . . 10  |-  ( t  =  f  ->  ( T `  ( t `  ( i  -  1 ) ) )  =  ( T `  (
f `  ( i  -  1 ) ) ) )
3433rneqd 4922 . . . . . . . . 9  |-  ( t  =  f  ->  ran  ( T `  ( t `
 ( i  - 
1 ) ) )  =  ran  ( T `
 ( f `  ( i  -  1 ) ) ) )
3531, 34eleq12d 2364 . . . . . . . 8  |-  ( t  =  f  ->  (
( t `  i
)  e.  ran  ( T `  ( t `  ( i  -  1 ) ) )  <->  ( f `  i )  e.  ran  ( T `  ( f `
 ( i  - 
1 ) ) ) ) )
3630, 35raleqbidv 2761 . . . . . . 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 2800 . . . 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 2316 . . 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 2938 . 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 1632    e. wcel 1696   A.wral 2556   {crab 2560    \ cdif 3162   (/)c0 3468   {csn 3653   <.cop 3656   <.cotp 3657   U_ciun 3921    e. cmpt 4093    _I cid 4320    X. cxp 4703   dom cdm 4705   ran crn 4706   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1oc1o 6488   2oc2o 6489   0cc0 8753   1c1 8754    - cmin 9053   ...cfz 10798  ..^cfzo 10886   #chash 11353  Word cword 11419   splice csplice 11423   <"cs2 11507   ~FG cefg 15031
This theorem is referenced by:  efgsdmi  15057  efgsrel  15059  efgs1  15060  efgs1b  15061  efgsp1  15062  efgsres  15063  efgsfo  15064  efgredlema  15065  efgredlemf  15066  efgredlemd  15069  efgredlemc  15070  efgredlem  15072  efgrelexlemb  15075  efgredeu  15077  efgred2  15078
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-hash 11354  df-word 11425
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