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Theorem efgs1 15254
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 3385 . . . . 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 2458 . . . 4  |-  ( A  e.  D  ->  A  e.  W )
43s1cld 11643 . . 3  |-  ( A  e.  D  ->  <" A ">  e. Word  W )
5 s1nz 11646 . . . 4  |-  <" A ">  =/=  (/)
6 eldifsn 3842 . . . 4  |-  ( <" A ">  e.  (Word  W  \  { (/)
} )  <->  ( <" A ">  e. Word  W  /\  <" A ">  =/=  (/) ) )
75, 6mpbiran2 885 . . 3  |-  ( <" A ">  e.  (Word  W  \  { (/)
} )  <->  <" A ">  e. Word  W )
84, 7sylibr 203 . 2  |-  ( A  e.  D  ->  <" A ">  e.  (Word  W  \  { (/) } ) )
9 s1fv 11647 . . 3  |-  ( A  e.  D  ->  ( <" A "> `  0 )  =  A )
10 id 19 . . 3  |-  ( A  e.  D  ->  A  e.  D )
119, 10eqeltrd 2440 . 2  |-  ( A  e.  D  ->  ( <" A "> `  0 )  e.  D
)
12 s1len 11645 . . . . . 6  |-  ( # `  <" A "> )  =  1
1312a1i 10 . . . . 5  |-  ( A  e.  D  ->  ( # `
 <" A "> )  =  1
)
1413oveq2d 5997 . . . 4  |-  ( A  e.  D  ->  (
1..^ ( # `  <" A "> )
)  =  ( 1..^ 1 ) )
15 fzo0 11049 . . . 4  |-  ( 1..^ 1 )  =  (/)
1614, 15syl6eq 2414 . . 3  |-  ( A  e.  D  ->  (
1..^ ( # `  <" A "> )
)  =  (/) )
17 rzal 3644 . . 3  |-  ( ( 1..^ ( # `  <" A "> )
)  =  (/)  ->  A. i  e.  ( 1..^ ( # `  <" A "> ) ) ( <" A "> `  i )  e.  ran  ( T `  ( <" A "> `  ( i  -  1 ) ) ) )
1816, 17syl 15 . 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 15249 . 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 1137 1  |-  ( A  e.  D  ->  <" A ">  e.  dom  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715    =/= wne 2529   A.wral 2628   {crab 2632    \ cdif 3235   (/)c0 3543   {csn 3729   <.cop 3732   <.cotp 3733   U_ciun 4007    e. cmpt 4179    _I cid 4407    X. cxp 4790   dom cdm 4792   ran crn 4793   ` cfv 5358  (class class class)co 5981    e. cmpt2 5983   1oc1o 6614   2oc2o 6615   0cc0 8884   1c1 8885    - cmin 9184   ...cfz 10935  ..^cfzo 11025   #chash 11505  Word cword 11604   <"cs1 11606   splice csplice 11608   <"cs2 11692   ~FG cefg 15225
This theorem is referenced by:  efgsfo  15258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-card 7719  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-n0 10115  df-z 10176  df-uz 10382  df-fz 10936  df-fzo 11026  df-hash 11506  df-word 11610  df-s1 11612
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