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Theorem efgs1b 15323
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 3430 . . . 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 2496 . . 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 15317 . . . . . . . . 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 972 . . . . . . . 8  |-  ( A  e.  dom  S  ->  A  e.  (Word  W  \  { (/) } ) )
11 eldifsn 3887 . . . . . . . . 9  |-  ( A  e.  (Word  W  \  { (/) } )  <->  ( A  e. Word  W  /\  A  =/=  (/) ) )
12 lennncl 11691 . . . . . . . . 9  |-  ( ( A  e. Word  W  /\  A  =/=  (/) )  ->  ( # `
 A )  e.  NN )
1311, 12sylbi 188 . . . . . . . 8  |-  ( A  e.  (Word  W  \  { (/) } )  -> 
( # `  A )  e.  NN )
1410, 13syl 16 . . . . . . 7  |-  ( A  e.  dom  S  -> 
( # `  A )  e.  NN )
15 elnn1uz2 10508 . . . . . . 7  |-  ( (
# `  A )  e.  NN  <->  ( ( # `  A )  =  1  \/  ( # `  A
)  e.  ( ZZ>= ` 
2 ) ) )
1614, 15sylib 189 . . . . . 6  |-  ( A  e.  dom  S  -> 
( ( # `  A
)  =  1  \/  ( # `  A
)  e.  ( ZZ>= ` 
2 ) ) )
1716ord 367 . . . . 5  |-  ( A  e.  dom  S  -> 
( -.  ( # `  A )  =  1  ->  ( # `  A
)  e.  ( ZZ>= ` 
2 ) ) )
1810eldifad 3292 . . . . . . . . . . . 12  |-  ( A  e.  dom  S  ->  A  e. Word  W )
1918adantr 452 . . . . . . . . . . 11  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  ->  A  e. Word  W )
20 wrdf 11688 . . . . . . . . . . 11  |-  ( A  e. Word  W  ->  A : ( 0..^ (
# `  A )
) --> W )
2119, 20syl 16 . . . . . . . . . 10  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  ->  A : ( 0..^ (
# `  A )
) --> W )
22 1z 10267 . . . . . . . . . . . . . . 15  |-  1  e.  ZZ
23 simpr 448 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( # `  A )  e.  ( ZZ>= `  2
) )
24 df-2 10014 . . . . . . . . . . . . . . . . 17  |-  2  =  ( 1  +  1 )
2524fveq2i 5690 . . . . . . . . . . . . . . . 16  |-  ( ZZ>= ` 
2 )  =  (
ZZ>= `  ( 1  +  1 ) )
2623, 25syl6eleq 2494 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( # `  A )  e.  ( ZZ>= `  (
1  +  1 ) ) )
27 eluzp1m1 10465 . . . . . . . . . . . . . . 15  |-  ( ( 1  e.  ZZ  /\  ( # `  A )  e.  ( ZZ>= `  (
1  +  1 ) ) )  ->  (
( # `  A )  -  1 )  e.  ( ZZ>= `  1 )
)
2822, 26, 27sylancr 645 . . . . . . . . . . . . . 14  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( ( # `  A
)  -  1 )  e.  ( ZZ>= `  1
) )
29 nnuz 10477 . . . . . . . . . . . . . 14  |-  NN  =  ( ZZ>= `  1 )
3028, 29syl6eleqr 2495 . . . . . . . . . . . . 13  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( ( # `  A
)  -  1 )  e.  NN )
31 lbfzo0 11125 . . . . . . . . . . . . 13  |-  ( 0  e.  ( 0..^ ( ( # `  A
)  -  1 ) )  <->  ( ( # `  A )  -  1 )  e.  NN )
3230, 31sylibr 204 . . . . . . . . . . . 12  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
0  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) )
33 fzoend 11142 . . . . . . . . . . . 12  |-  ( 0  e.  ( 0..^ ( ( # `  A
)  -  1 ) )  ->  ( (
( # `  A )  -  1 )  - 
1 )  e.  ( 0..^ ( ( # `  A )  -  1 ) ) )
34 elfzofz 11109 . . . . . . . . . . . 12  |-  ( ( ( ( # `  A
)  -  1 )  -  1 )  e.  ( 0..^ ( (
# `  A )  -  1 ) )  ->  ( ( (
# `  A )  -  1 )  - 
1 )  e.  ( 0 ... ( (
# `  A )  -  1 ) ) )
3532, 33, 343syl 19 . . . . . . . . . . 11  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( ( ( # `  A )  -  1 )  -  1 )  e.  ( 0 ... ( ( # `  A
)  -  1 ) ) )
36 eluzelz 10452 . . . . . . . . . . . . 13  |-  ( (
# `  A )  e.  ( ZZ>= `  2 )  ->  ( # `  A
)  e.  ZZ )
3736adantl 453 . . . . . . . . . . . 12  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( # `  A )  e.  ZZ )
38 fzoval 11096 . . . . . . . . . . . 12  |-  ( (
# `  A )  e.  ZZ  ->  ( 0..^ ( # `  A
) )  =  ( 0 ... ( (
# `  A )  -  1 ) ) )
3937, 38syl 16 . . . . . . . . . . 11  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( 0..^ ( # `  A ) )  =  ( 0 ... (
( # `  A )  -  1 ) ) )
4035, 39eleqtrrd 2481 . . . . . . . . . 10  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( ( ( # `  A )  -  1 )  -  1 )  e.  ( 0..^ (
# `  A )
) )
4121, 40ffvelrnd 5830 . . . . . . . . 9  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( A `  (
( ( # `  A
)  -  1 )  -  1 ) )  e.  W )
42 uz2m1nn 10506 . . . . . . . . . 10  |-  ( (
# `  A )  e.  ( ZZ>= `  2 )  ->  ( ( # `  A
)  -  1 )  e.  NN )
434, 5, 6, 7, 2, 8efgsdmi 15319 . . . . . . . . . 10  |-  ( ( A  e.  dom  S  /\  ( ( # `  A
)  -  1 )  e.  NN )  -> 
( S `  A
)  e.  ran  ( T `  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) ) ) )
4442, 43sylan2 461 . . . . . . . . 9  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( S `  A
)  e.  ran  ( T `  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) ) ) )
45 fveq2 5687 . . . . . . . . . . . 12  |-  ( a  =  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) )  ->  ( T `  a )  =  ( T `  ( A `
 ( ( (
# `  A )  -  1 )  - 
1 ) ) ) )
4645rneqd 5056 . . . . . . . . . . 11  |-  ( a  =  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) )  ->  ran  ( T `
 a )  =  ran  ( T `  ( A `  ( ( ( # `  A
)  -  1 )  -  1 ) ) ) )
4746eleq2d 2471 . . . . . . . . . 10  |-  ( a  =  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) )  ->  ( ( S `  A )  e.  ran  ( T `  a )  <->  ( S `  A )  e.  ran  ( T `  ( A `
 ( ( (
# `  A )  -  1 )  - 
1 ) ) ) ) )
4847rspcev 3012 . . . . . . . . 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 ) )
4941, 44, 48syl2anc 643 . . . . . . . 8  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  ->  E. a  e.  W  ( S `  A )  e.  ran  ( T `
 a ) )
50 eliun 4057 . . . . . . . 8  |-  ( ( S `  A )  e.  U_ a  e.  W  ran  ( T `
 a )  <->  E. a  e.  W  ( S `  A )  e.  ran  ( T `  a ) )
5149, 50sylibr 204 . . . . . . 7  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( S `  A
)  e.  U_ a  e.  W  ran  ( T `
 a ) )
52 fveq2 5687 . . . . . . . . 9  |-  ( a  =  x  ->  ( T `  a )  =  ( T `  x ) )
5352rneqd 5056 . . . . . . . 8  |-  ( a  =  x  ->  ran  ( T `  a )  =  ran  ( T `
 x ) )
5453cbviunv 4090 . . . . . . 7  |-  U_ a  e.  W  ran  ( T `
 a )  = 
U_ x  e.  W  ran  ( T `  x
)
5551, 54syl6eleq 2494 . . . . . 6  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( S `  A
)  e.  U_ x  e.  W  ran  ( T `
 x ) )
5655ex 424 . . . . 5  |-  ( A  e.  dom  S  -> 
( ( # `  A
)  e.  ( ZZ>= ` 
2 )  ->  ( S `  A )  e.  U_ x  e.  W  ran  ( T `  x
) ) )
5717, 56syld 42 . . . 4  |-  ( A  e.  dom  S  -> 
( -.  ( # `  A )  =  1  ->  ( S `  A )  e.  U_ x  e.  W  ran  ( T `  x ) ) )
5857con1d 118 . . 3  |-  ( A  e.  dom  S  -> 
( -.  ( S `
 A )  e. 
U_ x  e.  W  ran  ( T `  x
)  ->  ( # `  A
)  =  1 ) )
593, 58syl5 30 . 2  |-  ( A  e.  dom  S  -> 
( ( S `  A )  e.  D  ->  ( # `  A
)  =  1 ) )
609simp2bi 973 . . . 4  |-  ( A  e.  dom  S  -> 
( A `  0
)  e.  D )
61 oveq1 6047 . . . . . . 7  |-  ( (
# `  A )  =  1  ->  (
( # `  A )  -  1 )  =  ( 1  -  1 ) )
62 1m1e0 10024 . . . . . . 7  |-  ( 1  -  1 )  =  0
6361, 62syl6eq 2452 . . . . . 6  |-  ( (
# `  A )  =  1  ->  (
( # `  A )  -  1 )  =  0 )
6463fveq2d 5691 . . . . 5  |-  ( (
# `  A )  =  1  ->  ( A `  ( ( # `
 A )  - 
1 ) )  =  ( A `  0
) )
6564eleq1d 2470 . . . 4  |-  ( (
# `  A )  =  1  ->  (
( A `  (
( # `  A )  -  1 ) )  e.  D  <->  ( A `  0 )  e.  D ) )
6660, 65syl5ibrcom 214 . . 3  |-  ( A  e.  dom  S  -> 
( ( # `  A
)  =  1  -> 
( A `  (
( # `  A )  -  1 ) )  e.  D ) )
674, 5, 6, 7, 2, 8efgsval 15318 . . . 4  |-  ( A  e.  dom  S  -> 
( S `  A
)  =  ( A `
 ( ( # `  A )  -  1 ) ) )
6867eleq1d 2470 . . 3  |-  ( A  e.  dom  S  -> 
( ( S `  A )  e.  D  <->  ( A `  ( (
# `  A )  -  1 ) )  e.  D ) )
6966, 68sylibrd 226 . 2  |-  ( A  e.  dom  S  -> 
( ( # `  A
)  =  1  -> 
( S `  A
)  e.  D ) )
7059, 69impbid 184 1  |-  ( A  e.  dom  S  -> 
( ( S `  A )  e.  D  <->  (
# `  A )  =  1 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   {crab 2670    \ cdif 3277   (/)c0 3588   {csn 3774   <.cop 3777   <.cotp 3778   U_ciun 4053    e. cmpt 4226    _I cid 4453    X. cxp 4835   dom cdm 4837   ran crn 4838   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1oc1o 6676   2oc2o 6677   0cc0 8946   1c1 8947    + caddc 8949    - cmin 9247   NNcn 9956   2c2 10005   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999  ..^cfzo 11090   #chash 11573  Word cword 11672   splice csplice 11676   <"cs2 11760   ~FG cefg 15293
This theorem is referenced by:  efgredlema  15327  efgredeu  15339
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-hash 11574  df-word 11678
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