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Theorem efgs1b 17379
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 3555 . . . 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 2546 . . 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 17373 . . . . . . . . 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 1022 . . . . . . . 8  |-  ( A  e.  dom  S  ->  A  e.  (Word  W  \  { (/) } ) )
11 eldifsn 4096 . . . . . . . . 9  |-  ( A  e.  (Word  W  \  { (/) } )  <->  ( A  e. Word  W  /\  A  =/=  (/) ) )
12 lennncl 12685 . . . . . . . . 9  |-  ( ( A  e. Word  W  /\  A  =/=  (/) )  ->  ( # `
 A )  e.  NN )
1311, 12sylbi 199 . . . . . . . 8  |-  ( A  e.  (Word  W  \  { (/) } )  -> 
( # `  A )  e.  NN )
1410, 13syl 17 . . . . . . 7  |-  ( A  e.  dom  S  -> 
( # `  A )  e.  NN )
15 elnn1uz2 11232 . . . . . . 7  |-  ( (
# `  A )  e.  NN  <->  ( ( # `  A )  =  1  \/  ( # `  A
)  e.  ( ZZ>= ` 
2 ) ) )
1614, 15sylib 200 . . . . . 6  |-  ( A  e.  dom  S  -> 
( ( # `  A
)  =  1  \/  ( # `  A
)  e.  ( ZZ>= ` 
2 ) ) )
1716ord 379 . . . . 5  |-  ( A  e.  dom  S  -> 
( -.  ( # `  A )  =  1  ->  ( # `  A
)  e.  ( ZZ>= ` 
2 ) ) )
1810eldifad 3415 . . . . . . . . . . . 12  |-  ( A  e.  dom  S  ->  A  e. Word  W )
1918adantr 467 . . . . . . . . . . 11  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  ->  A  e. Word  W )
20 wrdf 12673 . . . . . . . . . . 11  |-  ( A  e. Word  W  ->  A : ( 0..^ (
# `  A )
) --> W )
2119, 20syl 17 . . . . . . . . . 10  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  ->  A : ( 0..^ (
# `  A )
) --> W )
22 1z 10964 . . . . . . . . . . . . . . 15  |-  1  e.  ZZ
23 simpr 463 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( # `  A )  e.  ( ZZ>= `  2
) )
24 df-2 10665 . . . . . . . . . . . . . . . . 17  |-  2  =  ( 1  +  1 )
2524fveq2i 5866 . . . . . . . . . . . . . . . 16  |-  ( ZZ>= ` 
2 )  =  (
ZZ>= `  ( 1  +  1 ) )
2623, 25syl6eleq 2538 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( # `  A )  e.  ( ZZ>= `  (
1  +  1 ) ) )
27 eluzp1m1 11179 . . . . . . . . . . . . . . 15  |-  ( ( 1  e.  ZZ  /\  ( # `  A )  e.  ( ZZ>= `  (
1  +  1 ) ) )  ->  (
( # `  A )  -  1 )  e.  ( ZZ>= `  1 )
)
2822, 26, 27sylancr 668 . . . . . . . . . . . . . 14  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( ( # `  A
)  -  1 )  e.  ( ZZ>= `  1
) )
29 nnuz 11191 . . . . . . . . . . . . . 14  |-  NN  =  ( ZZ>= `  1 )
3028, 29syl6eleqr 2539 . . . . . . . . . . . . 13  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( ( # `  A
)  -  1 )  e.  NN )
31 lbfzo0 11952 . . . . . . . . . . . . 13  |-  ( 0  e.  ( 0..^ ( ( # `  A
)  -  1 ) )  <->  ( ( # `  A )  -  1 )  e.  NN )
3230, 31sylibr 216 . . . . . . . . . . . 12  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
0  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) )
33 fzoend 11999 . . . . . . . . . . . 12  |-  ( 0  e.  ( 0..^ ( ( # `  A
)  -  1 ) )  ->  ( (
( # `  A )  -  1 )  - 
1 )  e.  ( 0..^ ( ( # `  A )  -  1 ) ) )
34 elfzofz 11932 . . . . . . . . . . . 12  |-  ( ( ( ( # `  A
)  -  1 )  -  1 )  e.  ( 0..^ ( (
# `  A )  -  1 ) )  ->  ( ( (
# `  A )  -  1 )  - 
1 )  e.  ( 0 ... ( (
# `  A )  -  1 ) ) )
3532, 33, 343syl 18 . . . . . . . . . . 11  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( ( ( # `  A )  -  1 )  -  1 )  e.  ( 0 ... ( ( # `  A
)  -  1 ) ) )
36 eluzelz 11165 . . . . . . . . . . . . 13  |-  ( (
# `  A )  e.  ( ZZ>= `  2 )  ->  ( # `  A
)  e.  ZZ )
3736adantl 468 . . . . . . . . . . . 12  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( # `  A )  e.  ZZ )
38 fzoval 11918 . . . . . . . . . . . 12  |-  ( (
# `  A )  e.  ZZ  ->  ( 0..^ ( # `  A
) )  =  ( 0 ... ( (
# `  A )  -  1 ) ) )
3937, 38syl 17 . . . . . . . . . . 11  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( 0..^ ( # `  A ) )  =  ( 0 ... (
( # `  A )  -  1 ) ) )
4035, 39eleqtrrd 2531 . . . . . . . . . 10  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( ( ( # `  A )  -  1 )  -  1 )  e.  ( 0..^ (
# `  A )
) )
4121, 40ffvelrnd 6021 . . . . . . . . 9  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( A `  (
( ( # `  A
)  -  1 )  -  1 ) )  e.  W )
42 uz2m1nn 11230 . . . . . . . . . 10  |-  ( (
# `  A )  e.  ( ZZ>= `  2 )  ->  ( ( # `  A
)  -  1 )  e.  NN )
434, 5, 6, 7, 2, 8efgsdmi 17375 . . . . . . . . . 10  |-  ( ( A  e.  dom  S  /\  ( ( # `  A
)  -  1 )  e.  NN )  -> 
( S `  A
)  e.  ran  ( T `  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) ) ) )
4442, 43sylan2 477 . . . . . . . . 9  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( S `  A
)  e.  ran  ( T `  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) ) ) )
45 fveq2 5863 . . . . . . . . . . . 12  |-  ( a  =  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) )  ->  ( T `  a )  =  ( T `  ( A `
 ( ( (
# `  A )  -  1 )  - 
1 ) ) ) )
4645rneqd 5061 . . . . . . . . . . 11  |-  ( a  =  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) )  ->  ran  ( T `
 a )  =  ran  ( T `  ( A `  ( ( ( # `  A
)  -  1 )  -  1 ) ) ) )
4746eleq2d 2513 . . . . . . . . . 10  |-  ( a  =  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) )  ->  ( ( S `  A )  e.  ran  ( T `  a )  <->  ( S `  A )  e.  ran  ( T `  ( A `
 ( ( (
# `  A )  -  1 )  - 
1 ) ) ) ) )
4847rspcev 3149 . . . . . . . . 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 666 . . . . . . . 8  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  ->  E. a  e.  W  ( S `  A )  e.  ran  ( T `
 a ) )
50 eliun 4282 . . . . . . . 8  |-  ( ( S `  A )  e.  U_ a  e.  W  ran  ( T `
 a )  <->  E. a  e.  W  ( S `  A )  e.  ran  ( T `  a ) )
5149, 50sylibr 216 . . . . . . 7  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( S `  A
)  e.  U_ a  e.  W  ran  ( T `
 a ) )
52 fveq2 5863 . . . . . . . . 9  |-  ( a  =  x  ->  ( T `  a )  =  ( T `  x ) )
5352rneqd 5061 . . . . . . . 8  |-  ( a  =  x  ->  ran  ( T `  a )  =  ran  ( T `
 x ) )
5453cbviunv 4316 . . . . . . 7  |-  U_ a  e.  W  ran  ( T `
 a )  = 
U_ x  e.  W  ran  ( T `  x
)
5551, 54syl6eleq 2538 . . . . . 6  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( S `  A
)  e.  U_ x  e.  W  ran  ( T `
 x ) )
5655ex 436 . . . . 5  |-  ( A  e.  dom  S  -> 
( ( # `  A
)  e.  ( ZZ>= ` 
2 )  ->  ( S `  A )  e.  U_ x  e.  W  ran  ( T `  x
) ) )
5717, 56syld 45 . . . 4  |-  ( A  e.  dom  S  -> 
( -.  ( # `  A )  =  1  ->  ( S `  A )  e.  U_ x  e.  W  ran  ( T `  x ) ) )
5857con1d 128 . . 3  |-  ( A  e.  dom  S  -> 
( -.  ( S `
 A )  e. 
U_ x  e.  W  ran  ( T `  x
)  ->  ( # `  A
)  =  1 ) )
593, 58syl5 33 . 2  |-  ( A  e.  dom  S  -> 
( ( S `  A )  e.  D  ->  ( # `  A
)  =  1 ) )
609simp2bi 1023 . . . 4  |-  ( A  e.  dom  S  -> 
( A `  0
)  e.  D )
61 oveq1 6295 . . . . . . 7  |-  ( (
# `  A )  =  1  ->  (
( # `  A )  -  1 )  =  ( 1  -  1 ) )
62 1m1e0 10675 . . . . . . 7  |-  ( 1  -  1 )  =  0
6361, 62syl6eq 2500 . . . . . 6  |-  ( (
# `  A )  =  1  ->  (
( # `  A )  -  1 )  =  0 )
6463fveq2d 5867 . . . . 5  |-  ( (
# `  A )  =  1  ->  ( A `  ( ( # `
 A )  - 
1 ) )  =  ( A `  0
) )
6564eleq1d 2512 . . . 4  |-  ( (
# `  A )  =  1  ->  (
( A `  (
( # `  A )  -  1 ) )  e.  D  <->  ( A `  0 )  e.  D ) )
6660, 65syl5ibrcom 226 . . 3  |-  ( A  e.  dom  S  -> 
( ( # `  A
)  =  1  -> 
( A `  (
( # `  A )  -  1 ) )  e.  D ) )
674, 5, 6, 7, 2, 8efgsval 17374 . . . 4  |-  ( A  e.  dom  S  -> 
( S `  A
)  =  ( A `
 ( ( # `  A )  -  1 ) ) )
6867eleq1d 2512 . . 3  |-  ( A  e.  dom  S  -> 
( ( S `  A )  e.  D  <->  ( A `  ( (
# `  A )  -  1 ) )  e.  D ) )
6966, 68sylibrd 238 . 2  |-  ( A  e.  dom  S  -> 
( ( # `  A
)  =  1  -> 
( S `  A
)  e.  D ) )
7059, 69impbid 194 1  |-  ( A  e.  dom  S  -> 
( ( S `  A )  e.  D  <->  (
# `  A )  =  1 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    = wceq 1443    e. wcel 1886    =/= wne 2621   A.wral 2736   E.wrex 2737   {crab 2740    \ cdif 3400   (/)c0 3730   {csn 3967   <.cop 3973   <.cotp 3975   U_ciun 4277    |-> cmpt 4460    _I cid 4743    X. cxp 4831   dom cdm 4833   ran crn 4834   -->wf 5577   ` cfv 5581  (class class class)co 6288    |-> cmpt2 6290   1oc1o 7172   2oc2o 7173   0cc0 9536   1c1 9537    + caddc 9539    - cmin 9857   NNcn 10606   2c2 10656   ZZcz 10934   ZZ>=cuz 11156   ...cfz 11781  ..^cfzo 11912   #chash 12512  Word cword 12653   splice csplice 12658   <"cs2 12932   ~FG cefg 17349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-n0 10867  df-z 10935  df-uz 11157  df-fz 11782  df-fzo 11913  df-hash 12513  df-word 12661
This theorem is referenced by:  efgredlema  17383  efgredeu  17395
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