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Theorem efgs1 15330
Description: A singleton of an irreducible word is an extension sequence. (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
efgs1  |-  ( A  e.  D  ->  <" A ">  e.  dom  S
)
Distinct variable groups:    y, z    t, n, v, w, y, z, m, x    m, M    x, n, M, t, v, w    k, m, t, x, T    k, n, v, w, y, z, W, m, t, x    .~ , m, t, x, y, z    m, I, n, t, v, w, x, y, z    D, m, t
Allowed substitution hints:    A( x, y, z, w, v, t, k, m, n)    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)    I( k)    M( y, z, k)

Proof of Theorem efgs1
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 eldifi 3437 . . . . 5  |-  ( A  e.  ( W  \  U_ x  e.  W  ran  ( T `  x
) )  ->  A  e.  W )
2 efgred.d . . . . 5  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
31, 2eleq2s 2504 . . . 4  |-  ( A  e.  D  ->  A  e.  W )
43s1cld 11719 . . 3  |-  ( A  e.  D  ->  <" A ">  e. Word  W )
5 s1nz 11722 . . . 4  |-  <" A ">  =/=  (/)
6 eldifsn 3895 . . . 4  |-  ( <" A ">  e.  (Word  W  \  { (/)
} )  <->  ( <" A ">  e. Word  W  /\  <" A ">  =/=  (/) ) )
75, 6mpbiran2 886 . . 3  |-  ( <" A ">  e.  (Word  W  \  { (/)
} )  <->  <" A ">  e. Word  W )
84, 7sylibr 204 . 2  |-  ( A  e.  D  ->  <" A ">  e.  (Word  W  \  { (/) } ) )
9 s1fv 11723 . . 3  |-  ( A  e.  D  ->  ( <" A "> `  0 )  =  A )
10 id 20 . . 3  |-  ( A  e.  D  ->  A  e.  D )
119, 10eqeltrd 2486 . 2  |-  ( A  e.  D  ->  ( <" A "> `  0 )  e.  D
)
12 s1len 11721 . . . . . 6  |-  ( # `  <" A "> )  =  1
1312a1i 11 . . . . 5  |-  ( A  e.  D  ->  ( # `
 <" A "> )  =  1
)
1413oveq2d 6064 . . . 4  |-  ( A  e.  D  ->  (
1..^ ( # `  <" A "> )
)  =  ( 1..^ 1 ) )
15 fzo0 11122 . . . 4  |-  ( 1..^ 1 )  =  (/)
1614, 15syl6eq 2460 . . 3  |-  ( A  e.  D  ->  (
1..^ ( # `  <" A "> )
)  =  (/) )
17 rzal 3697 . . 3  |-  ( ( 1..^ ( # `  <" A "> )
)  =  (/)  ->  A. i  e.  ( 1..^ ( # `  <" A "> ) ) ( <" A "> `  i )  e.  ran  ( T `  ( <" A "> `  ( i  -  1 ) ) ) )
1816, 17syl 16 . 2  |-  ( A  e.  D  ->  A. i  e.  ( 1..^ ( # `  <" A "> ) ) ( <" A "> `  i )  e.  ran  ( T `  ( <" A "> `  ( i  -  1 ) ) ) )
19 efgval.w . . 3  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
20 efgval.r . . 3  |-  .~  =  ( ~FG  `  I )
21 efgval2.m . . 3  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
22 efgval2.t . . 3  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
23 efgred.s . . 3  |-  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 ) ) )
2419, 20, 21, 22, 2, 23efgsdm 15325 . 2  |-  ( <" A ">  e.  dom  S  <->  ( <" A ">  e.  (Word  W  \  { (/) } )  /\  ( <" A "> `  0 )  e.  D  /\  A. i  e.  ( 1..^ ( # `  <" A "> )
) ( <" A "> `  i )  e.  ran  ( T `  ( <" A "> `  ( i  - 
1 ) ) ) ) )
258, 11, 18, 24syl3anbrc 1138 1  |-  ( A  e.  D  ->  <" A ">  e.  dom  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2575   A.wral 2674   {crab 2678    \ cdif 3285   (/)c0 3596   {csn 3782   <.cop 3785   <.cotp 3786   U_ciun 4061    e. cmpt 4234    _I cid 4461    X. cxp 4843   dom cdm 4845   ran crn 4846   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050   1oc1o 6684   2oc2o 6685   0cc0 8954   1c1 8955    - cmin 9255   ...cfz 11007  ..^cfzo 11098   #chash 11581  Word cword 11680   <"cs1 11682   splice csplice 11684   <"cs2 11768   ~FG cefg 15301
This theorem is referenced by:  efgsfo  15334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-fzo 11099  df-hash 11582  df-word 11686  df-s1 11688
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