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Theorem efgs1b 15061
Description: Every extension sequence ending in an irreducible word is trivial. (Contributed by Mario Carneiro, 1-Oct-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
efgs1b  |-  ( A  e.  dom  S  -> 
( ( S `  A )  e.  D  <->  (
# `  A )  =  1 ) )
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 efgs1b
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 eldifn 3312 . . . 4  |-  ( ( S `  A )  e.  ( W  \  U_ x  e.  W  ran  ( T `  x
) )  ->  -.  ( S `  A )  e.  U_ x  e.  W  ran  ( T `
 x ) )
2 efgred.d . . . 4  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
31, 2eleq2s 2388 . . 3  |-  ( ( S `  A )  e.  D  ->  -.  ( S `  A )  e.  U_ x  e.  W  ran  ( T `
 x ) )
4 efgval.w . . . . . . . . . 10  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
5 efgval.r . . . . . . . . . 10  |-  .~  =  ( ~FG  `  I )
6 efgval2.m . . . . . . . . . 10  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
7 efgval2.t . . . . . . . . . 10  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
8 efgred.s . . . . . . . . . 10  |-  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 ) ) )
94, 5, 6, 7, 2, 8efgsdm 15055 . . . . . . . . 9  |-  ( A  e.  dom  S  <->  ( A  e.  (Word  W  \  { (/)
} )  /\  ( A `  0 )  e.  D  /\  A. a  e.  ( 1..^ ( # `  A ) ) ( A `  a )  e.  ran  ( T `
 ( A `  ( a  -  1 ) ) ) ) )
109simp1bi 970 . . . . . . . 8  |-  ( A  e.  dom  S  ->  A  e.  (Word  W  \  { (/) } ) )
11 eldifsn 3762 . . . . . . . . 9  |-  ( A  e.  (Word  W  \  { (/) } )  <->  ( A  e. Word  W  /\  A  =/=  (/) ) )
12 lennncl 11438 . . . . . . . . 9  |-  ( ( A  e. Word  W  /\  A  =/=  (/) )  ->  ( # `
 A )  e.  NN )
1311, 12sylbi 187 . . . . . . . 8  |-  ( A  e.  (Word  W  \  { (/) } )  -> 
( # `  A )  e.  NN )
1410, 13syl 15 . . . . . . 7  |-  ( A  e.  dom  S  -> 
( # `  A )  e.  NN )
15 elnn1uz2 10310 . . . . . . 7  |-  ( (
# `  A )  e.  NN  <->  ( ( # `  A )  =  1  \/  ( # `  A
)  e.  ( ZZ>= ` 
2 ) ) )
1614, 15sylib 188 . . . . . 6  |-  ( A  e.  dom  S  -> 
( ( # `  A
)  =  1  \/  ( # `  A
)  e.  ( ZZ>= ` 
2 ) ) )
1716ord 366 . . . . 5  |-  ( A  e.  dom  S  -> 
( -.  ( # `  A )  =  1  ->  ( # `  A
)  e.  ( ZZ>= ` 
2 ) ) )
18 eldifi 3311 . . . . . . . . . . . . 13  |-  ( A  e.  (Word  W  \  { (/) } )  ->  A  e. Word  W )
1910, 18syl 15 . . . . . . . . . . . 12  |-  ( A  e.  dom  S  ->  A  e. Word  W )
2019adantr 451 . . . . . . . . . . 11  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  ->  A  e. Word  W )
21 wrdf 11435 . . . . . . . . . . 11  |-  ( A  e. Word  W  ->  A : ( 0..^ (
# `  A )
) --> W )
2220, 21syl 15 . . . . . . . . . 10  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  ->  A : ( 0..^ (
# `  A )
) --> W )
23 1z 10069 . . . . . . . . . . . . . . 15  |-  1  e.  ZZ
24 simpr 447 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( # `  A )  e.  ( ZZ>= `  2
) )
25 df-2 9820 . . . . . . . . . . . . . . . . 17  |-  2  =  ( 1  +  1 )
2625fveq2i 5544 . . . . . . . . . . . . . . . 16  |-  ( ZZ>= ` 
2 )  =  (
ZZ>= `  ( 1  +  1 ) )
2724, 26syl6eleq 2386 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( # `  A )  e.  ( ZZ>= `  (
1  +  1 ) ) )
28 eluzp1m1 10267 . . . . . . . . . . . . . . 15  |-  ( ( 1  e.  ZZ  /\  ( # `  A )  e.  ( ZZ>= `  (
1  +  1 ) ) )  ->  (
( # `  A )  -  1 )  e.  ( ZZ>= `  1 )
)
2923, 27, 28sylancr 644 . . . . . . . . . . . . . 14  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( ( # `  A
)  -  1 )  e.  ( ZZ>= `  1
) )
30 nnuz 10279 . . . . . . . . . . . . . 14  |-  NN  =  ( ZZ>= `  1 )
3129, 30syl6eleqr 2387 . . . . . . . . . . . . 13  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( ( # `  A
)  -  1 )  e.  NN )
32 lbfzo0 10919 . . . . . . . . . . . . 13  |-  ( 0  e.  ( 0..^ ( ( # `  A
)  -  1 ) )  <->  ( ( # `  A )  -  1 )  e.  NN )
3331, 32sylibr 203 . . . . . . . . . . . 12  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
0  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) )
34 fzoend 10930 . . . . . . . . . . . 12  |-  ( 0  e.  ( 0..^ ( ( # `  A
)  -  1 ) )  ->  ( (
( # `  A )  -  1 )  - 
1 )  e.  ( 0..^ ( ( # `  A )  -  1 ) ) )
35 elfzofz 10905 . . . . . . . . . . . 12  |-  ( ( ( ( # `  A
)  -  1 )  -  1 )  e.  ( 0..^ ( (
# `  A )  -  1 ) )  ->  ( ( (
# `  A )  -  1 )  - 
1 )  e.  ( 0 ... ( (
# `  A )  -  1 ) ) )
3633, 34, 353syl 18 . . . . . . . . . . 11  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( ( ( # `  A )  -  1 )  -  1 )  e.  ( 0 ... ( ( # `  A
)  -  1 ) ) )
37 eluzelz 10254 . . . . . . . . . . . . 13  |-  ( (
# `  A )  e.  ( ZZ>= `  2 )  ->  ( # `  A
)  e.  ZZ )
3837adantl 452 . . . . . . . . . . . 12  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( # `  A )  e.  ZZ )
39 fzoval 10892 . . . . . . . . . . . 12  |-  ( (
# `  A )  e.  ZZ  ->  ( 0..^ ( # `  A
) )  =  ( 0 ... ( (
# `  A )  -  1 ) ) )
4038, 39syl 15 . . . . . . . . . . 11  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( 0..^ ( # `  A ) )  =  ( 0 ... (
( # `  A )  -  1 ) ) )
4136, 40eleqtrrd 2373 . . . . . . . . . 10  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( ( ( # `  A )  -  1 )  -  1 )  e.  ( 0..^ (
# `  A )
) )
42 ffvelrn 5679 . . . . . . . . . 10  |-  ( ( A : ( 0..^ ( # `  A
) ) --> W  /\  ( ( ( # `  A )  -  1 )  -  1 )  e.  ( 0..^ (
# `  A )
) )  ->  ( A `  ( (
( # `  A )  -  1 )  - 
1 ) )  e.  W )
4322, 41, 42syl2anc 642 . . . . . . . . 9  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( A `  (
( ( # `  A
)  -  1 )  -  1 ) )  e.  W )
44 uz2m1nn 10308 . . . . . . . . . 10  |-  ( (
# `  A )  e.  ( ZZ>= `  2 )  ->  ( ( # `  A
)  -  1 )  e.  NN )
454, 5, 6, 7, 2, 8efgsdmi 15057 . . . . . . . . . 10  |-  ( ( A  e.  dom  S  /\  ( ( # `  A
)  -  1 )  e.  NN )  -> 
( S `  A
)  e.  ran  ( T `  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) ) ) )
4644, 45sylan2 460 . . . . . . . . 9  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( S `  A
)  e.  ran  ( T `  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) ) ) )
47 fveq2 5541 . . . . . . . . . . . 12  |-  ( a  =  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) )  ->  ( T `  a )  =  ( T `  ( A `
 ( ( (
# `  A )  -  1 )  - 
1 ) ) ) )
4847rneqd 4922 . . . . . . . . . . 11  |-  ( a  =  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) )  ->  ran  ( T `
 a )  =  ran  ( T `  ( A `  ( ( ( # `  A
)  -  1 )  -  1 ) ) ) )
4948eleq2d 2363 . . . . . . . . . 10  |-  ( a  =  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) )  ->  ( ( S `  A )  e.  ran  ( T `  a )  <->  ( S `  A )  e.  ran  ( T `  ( A `
 ( ( (
# `  A )  -  1 )  - 
1 ) ) ) ) )
5049rspcev 2897 . . . . . . . . 9  |-  ( ( ( A `  (
( ( # `  A
)  -  1 )  -  1 ) )  e.  W  /\  ( S `  A )  e.  ran  ( T `  ( A `  ( ( ( # `  A
)  -  1 )  -  1 ) ) ) )  ->  E. a  e.  W  ( S `  A )  e.  ran  ( T `  a ) )
5143, 46, 50syl2anc 642 . . . . . . . 8  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  ->  E. a  e.  W  ( S `  A )  e.  ran  ( T `
 a ) )
52 eliun 3925 . . . . . . . 8  |-  ( ( S `  A )  e.  U_ a  e.  W  ran  ( T `
 a )  <->  E. a  e.  W  ( S `  A )  e.  ran  ( T `  a ) )
5351, 52sylibr 203 . . . . . . 7  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( S `  A
)  e.  U_ a  e.  W  ran  ( T `
 a ) )
54 fveq2 5541 . . . . . . . . 9  |-  ( a  =  x  ->  ( T `  a )  =  ( T `  x ) )
5554rneqd 4922 . . . . . . . 8  |-  ( a  =  x  ->  ran  ( T `  a )  =  ran  ( T `
 x ) )
5655cbviunv 3957 . . . . . . 7  |-  U_ a  e.  W  ran  ( T `
 a )  = 
U_ x  e.  W  ran  ( T `  x
)
5753, 56syl6eleq 2386 . . . . . 6  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( S `  A
)  e.  U_ x  e.  W  ran  ( T `
 x ) )
5857ex 423 . . . . 5  |-  ( A  e.  dom  S  -> 
( ( # `  A
)  e.  ( ZZ>= ` 
2 )  ->  ( S `  A )  e.  U_ x  e.  W  ran  ( T `  x
) ) )
5917, 58syld 40 . . . 4  |-  ( A  e.  dom  S  -> 
( -.  ( # `  A )  =  1  ->  ( S `  A )  e.  U_ x  e.  W  ran  ( T `  x ) ) )
6059con1d 116 . . 3  |-  ( A  e.  dom  S  -> 
( -.  ( S `
 A )  e. 
U_ x  e.  W  ran  ( T `  x
)  ->  ( # `  A
)  =  1 ) )
613, 60syl5 28 . 2  |-  ( A  e.  dom  S  -> 
( ( S `  A )  e.  D  ->  ( # `  A
)  =  1 ) )
629simp2bi 971 . . . 4  |-  ( A  e.  dom  S  -> 
( A `  0
)  e.  D )
63 oveq1 5881 . . . . . . 7  |-  ( (
# `  A )  =  1  ->  (
( # `  A )  -  1 )  =  ( 1  -  1 ) )
64 1m1e0 9830 . . . . . . 7  |-  ( 1  -  1 )  =  0
6563, 64syl6eq 2344 . . . . . 6  |-  ( (
# `  A )  =  1  ->  (
( # `  A )  -  1 )  =  0 )
6665fveq2d 5545 . . . . 5  |-  ( (
# `  A )  =  1  ->  ( A `  ( ( # `
 A )  - 
1 ) )  =  ( A `  0
) )
6766eleq1d 2362 . . . 4  |-  ( (
# `  A )  =  1  ->  (
( A `  (
( # `  A )  -  1 ) )  e.  D  <->  ( A `  0 )  e.  D ) )
6862, 67syl5ibrcom 213 . . 3  |-  ( A  e.  dom  S  -> 
( ( # `  A
)  =  1  -> 
( A `  (
( # `  A )  -  1 ) )  e.  D ) )
694, 5, 6, 7, 2, 8efgsval 15056 . . . 4  |-  ( A  e.  dom  S  -> 
( S `  A
)  =  ( A `
 ( ( # `  A )  -  1 ) ) )
7069eleq1d 2362 . . 3  |-  ( A  e.  dom  S  -> 
( ( S `  A )  e.  D  <->  ( A `  ( (
# `  A )  -  1 ) )  e.  D ) )
7168, 70sylibrd 225 . 2  |-  ( A  e.  dom  S  -> 
( ( # `  A
)  =  1  -> 
( S `  A
)  e.  D ) )
7261, 71impbid 183 1  |-  ( A  e.  dom  S  -> 
( ( S `  A )  e.  D  <->  (
# `  A )  =  1 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {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   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1oc1o 6488   2oc2o 6489   0cc0 8753   1c1 8754    + caddc 8756    - cmin 9053   NNcn 9762   2c2 9811   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798  ..^cfzo 10886   #chash 11353  Word cword 11419   splice csplice 11423   <"cs2 11507   ~FG cefg 15031
This theorem is referenced by:  efgredlema  15065  efgredeu  15077
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-2 9820  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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