MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  efgs1b Unicode version

Theorem efgs1b 15045
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 3299 . . . 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 2375 . . 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 15039 . . . . . . . . 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 3749 . . . . . . . . 9  |-  ( A  e.  (Word  W  \  { (/) } )  <->  ( A  e. Word  W  /\  A  =/=  (/) ) )
12 lennncl 11422 . . . . . . . . 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 10294 . . . . . . 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 3298 . . . . . . . . . . . . 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 11419 . . . . . . . . . . 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 10053 . . . . . . . . . . . . . . 15  |-  1  e.  ZZ
24 simpr 447 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( # `  A )  e.  ( ZZ>= `  2
) )
25 df-2 9804 . . . . . . . . . . . . . . . . 17  |-  2  =  ( 1  +  1 )
2625fveq2i 5528 . . . . . . . . . . . . . . . 16  |-  ( ZZ>= ` 
2 )  =  (
ZZ>= `  ( 1  +  1 ) )
2724, 26syl6eleq 2373 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( # `  A )  e.  ( ZZ>= `  (
1  +  1 ) ) )
28 eluzp1m1 10251 . . . . . . . . . . . . . . 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 10263 . . . . . . . . . . . . . 14  |-  NN  =  ( ZZ>= `  1 )
3129, 30syl6eleqr 2374 . . . . . . . . . . . . 13  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( ( # `  A
)  -  1 )  e.  NN )
32 lbfzo0 10903 . . . . . . . . . . . . 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 10914 . . . . . . . . . . . 12  |-  ( 0  e.  ( 0..^ ( ( # `  A
)  -  1 ) )  ->  ( (
( # `  A )  -  1 )  - 
1 )  e.  ( 0..^ ( ( # `  A )  -  1 ) ) )
35 elfzofz 10889 . . . . . . . . . . . 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 10238 . . . . . . . . . . . . 13  |-  ( (
# `  A )  e.  ( ZZ>= `  2 )  ->  ( # `  A
)  e.  ZZ )
3837adantl 452 . . . . . . . . . . . 12  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( # `  A )  e.  ZZ )
39 fzoval 10876 . . . . . . . . . . . 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 2360 . . . . . . . . . 10  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( ( ( # `  A )  -  1 )  -  1 )  e.  ( 0..^ (
# `  A )
) )
42 ffvelrn 5663 . . . . . . . . . 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 10292 . . . . . . . . . 10  |-  ( (
# `  A )  e.  ( ZZ>= `  2 )  ->  ( ( # `  A
)  -  1 )  e.  NN )
454, 5, 6, 7, 2, 8efgsdmi 15041 . . . . . . . . . 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 5525 . . . . . . . . . . . 12  |-  ( a  =  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) )  ->  ( T `  a )  =  ( T `  ( A `
 ( ( (
# `  A )  -  1 )  - 
1 ) ) ) )
4847rneqd 4906 . . . . . . . . . . 11  |-  ( a  =  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) )  ->  ran  ( T `
 a )  =  ran  ( T `  ( A `  ( ( ( # `  A
)  -  1 )  -  1 ) ) ) )
4948eleq2d 2350 . . . . . . . . . 10  |-  ( a  =  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) )  ->  ( ( S `  A )  e.  ran  ( T `  a )  <->  ( S `  A )  e.  ran  ( T `  ( A `
 ( ( (
# `  A )  -  1 )  - 
1 ) ) ) ) )
5049rspcev 2884 . . . . . . . . 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 3909 . . . . . . . 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 5525 . . . . . . . . 9  |-  ( a  =  x  ->  ( T `  a )  =  ( T `  x ) )
5554rneqd 4906 . . . . . . . 8  |-  ( a  =  x  ->  ran  ( T `  a )  =  ran  ( T `
 x ) )
5655cbviunv 3941 . . . . . . 7  |-  U_ a  e.  W  ran  ( T `
 a )  = 
U_ x  e.  W  ran  ( T `  x
)
5753, 56syl6eleq 2373 . . . . . 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 5865 . . . . . . 7  |-  ( (
# `  A )  =  1  ->  (
( # `  A )  -  1 )  =  ( 1  -  1 ) )
64 1m1e0 9814 . . . . . . 7  |-  ( 1  -  1 )  =  0
6563, 64syl6eq 2331 . . . . . 6  |-  ( (
# `  A )  =  1  ->  (
( # `  A )  -  1 )  =  0 )
6665fveq2d 5529 . . . . 5  |-  ( (
# `  A )  =  1  ->  ( A `  ( ( # `
 A )  - 
1 ) )  =  ( A `  0
) )
6766eleq1d 2349 . . . 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 15040 . . . 4  |-  ( A  e.  dom  S  -> 
( S `  A
)  =  ( A `
 ( ( # `  A )  -  1 ) ) )
7069eleq1d 2349 . . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547    \ cdif 3149   (/)c0 3455   {csn 3640   <.cop 3643   <.cotp 3644   U_ciun 3905    e. cmpt 4077    _I cid 4304    X. cxp 4687   dom cdm 4689   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1oc1o 6472   2oc2o 6473   0cc0 8737   1c1 8738    + caddc 8740    - cmin 9037   NNcn 9746   2c2 9795   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782  ..^cfzo 10870   #chash 11337  Word cword 11403   splice csplice 11407   <"cs2 11491   ~FG cefg 15015
This theorem is referenced by:  efgredlema  15049  efgredeu  15061
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-hash 11338  df-word 11409
  Copyright terms: Public domain W3C validator