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Theorem efgsdm 16351
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 5801 . . . . 5  |-  ( f  =  F  ->  (
f `  0 )  =  ( F ` 
0 ) )
21eleq1d 2523 . . . 4  |-  ( f  =  F  ->  (
( f `  0
)  e.  D  <->  ( F `  0 )  e.  D ) )
3 fveq2 5802 . . . . . 6  |-  ( f  =  F  ->  ( # `
 f )  =  ( # `  F
) )
43oveq2d 6219 . . . . 5  |-  ( f  =  F  ->  (
1..^ ( # `  f
) )  =  ( 1..^ ( # `  F
) ) )
5 fveq1 5801 . . . . . 6  |-  ( f  =  F  ->  (
f `  i )  =  ( F `  i ) )
6 fveq1 5801 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  ( i  -  1 ) )  =  ( F `  ( i  -  1 ) ) )
76fveq2d 5806 . . . . . . 7  |-  ( f  =  F  ->  ( T `  ( f `  ( i  -  1 ) ) )  =  ( T `  ( F `  ( i  -  1 ) ) ) )
87rneqd 5178 . . . . . 6  |-  ( f  =  F  ->  ran  ( T `  ( f `
 ( i  - 
1 ) ) )  =  ran  ( T `
 ( F `  ( i  -  1 ) ) ) )
95, 8eleq12d 2536 . . . . 5  |-  ( f  =  F  ->  (
( f `  i
)  e.  ran  ( T `  ( f `  ( i  -  1 ) ) )  <->  ( F `  i )  e.  ran  ( T `  ( F `
 ( i  - 
1 ) ) ) ) )
104, 9raleqbidv 3037 . . . 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 710 . . 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 16350 . . . . 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 5675 . . . 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 5801 . . . . . . 7  |-  ( t  =  f  ->  (
t `  0 )  =  ( f ` 
0 ) )
2120eleq1d 2523 . . . . . 6  |-  ( t  =  f  ->  (
( t `  0
)  e.  D  <->  ( f `  0 )  e.  D ) )
22 fveq2 5802 . . . . . . . . 9  |-  ( k  =  i  ->  (
t `  k )  =  ( t `  i ) )
23 oveq1 6210 . . . . . . . . . . . 12  |-  ( k  =  i  ->  (
k  -  1 )  =  ( i  - 
1 ) )
2423fveq2d 5806 . . . . . . . . . . 11  |-  ( k  =  i  ->  (
t `  ( k  -  1 ) )  =  ( t `  ( i  -  1 ) ) )
2524fveq2d 5806 . . . . . . . . . 10  |-  ( k  =  i  ->  ( T `  ( t `  ( k  -  1 ) ) )  =  ( T `  (
t `  ( i  -  1 ) ) ) )
2625rneqd 5178 . . . . . . . . 9  |-  ( k  =  i  ->  ran  ( T `  ( t `
 ( k  - 
1 ) ) )  =  ran  ( T `
 ( t `  ( i  -  1 ) ) ) )
2722, 26eleq12d 2536 . . . . . . . 8  |-  ( k  =  i  ->  (
( t `  k
)  e.  ran  ( T `  ( t `  ( k  -  1 ) ) )  <->  ( t `  i )  e.  ran  ( T `  ( t `
 ( i  - 
1 ) ) ) ) )
2827cbvralv 3053 . . . . . . 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 5802 . . . . . . . . 9  |-  ( t  =  f  ->  ( # `
 t )  =  ( # `  f
) )
3029oveq2d 6219 . . . . . . . 8  |-  ( t  =  f  ->  (
1..^ ( # `  t
) )  =  ( 1..^ ( # `  f
) ) )
31 fveq1 5801 . . . . . . . . 9  |-  ( t  =  f  ->  (
t `  i )  =  ( f `  i ) )
32 fveq1 5801 . . . . . . . . . . 11  |-  ( t  =  f  ->  (
t `  ( i  -  1 ) )  =  ( f `  ( i  -  1 ) ) )
3332fveq2d 5806 . . . . . . . . . 10  |-  ( t  =  f  ->  ( T `  ( t `  ( i  -  1 ) ) )  =  ( T `  (
f `  ( i  -  1 ) ) ) )
3433rneqd 5178 . . . . . . . . 9  |-  ( t  =  f  ->  ran  ( T `  ( t `
 ( i  - 
1 ) ) )  =  ran  ( T `
 ( f `  ( i  -  1 ) ) ) )
3531, 34eleq12d 2536 . . . . . . . 8  |-  ( t  =  f  ->  (
( t `  i
)  e.  ran  ( T `  ( t `  ( i  -  1 ) ) )  <->  ( f `  i )  e.  ran  ( T `  ( f `
 ( i  - 
1 ) ) ) ) )
3630, 35raleqbidv 3037 . . . . . . 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 257 . . . . . 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 710 . . . . 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 3077 . . . 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 2483 . . 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 3226 . 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 969 . 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 252 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 setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   {crab 2803    \ cdif 3436   (/)c0 3748   {csn 3988   <.cop 3994   <.cotp 3996   U_ciun 4282    |-> cmpt 4461    _I cid 4742    X. cxp 4949   dom cdm 4951   ran crn 4952   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   1oc1o 7026   2oc2o 7027   0cc0 9396   1c1 9397    - cmin 9709   ...cfz 11557  ..^cfzo 11668   #chash 12223  Word cword 12342   splice csplice 12347   <"cs2 12589   ~FG cefg 16327
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 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
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 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-fzo 11669  df-hash 12224  df-word 12350
This theorem is referenced by:  efgsdmi  16353  efgsrel  16355  efgs1  16356  efgs1b  16357  efgsp1  16358  efgsres  16359  efgsfo  16360  efgredlema  16361  efgredlemf  16362  efgredlemd  16365  efgredlemc  16366  efgredlem  16368  efgrelexlemb  16371  efgredeu  16373  efgred2  16374
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