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Theorem efgs1b 15370
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 3472 . . . 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 2530 . . 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 15364 . . . . . . . . 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 973 . . . . . . . 8  |-  ( A  e.  dom  S  ->  A  e.  (Word  W  \  { (/) } ) )
11 eldifsn 3929 . . . . . . . . 9  |-  ( A  e.  (Word  W  \  { (/) } )  <->  ( A  e. Word  W  /\  A  =/=  (/) ) )
12 lennncl 11738 . . . . . . . . 9  |-  ( ( A  e. Word  W  /\  A  =/=  (/) )  ->  ( # `
 A )  e.  NN )
1311, 12sylbi 189 . . . . . . . 8  |-  ( A  e.  (Word  W  \  { (/) } )  -> 
( # `  A )  e.  NN )
1410, 13syl 16 . . . . . . 7  |-  ( A  e.  dom  S  -> 
( # `  A )  e.  NN )
15 elnn1uz2 10554 . . . . . . 7  |-  ( (
# `  A )  e.  NN  <->  ( ( # `  A )  =  1  \/  ( # `  A
)  e.  ( ZZ>= ` 
2 ) ) )
1614, 15sylib 190 . . . . . 6  |-  ( A  e.  dom  S  -> 
( ( # `  A
)  =  1  \/  ( # `  A
)  e.  ( ZZ>= ` 
2 ) ) )
1716ord 368 . . . . 5  |-  ( A  e.  dom  S  -> 
( -.  ( # `  A )  =  1  ->  ( # `  A
)  e.  ( ZZ>= ` 
2 ) ) )
1810eldifad 3334 . . . . . . . . . . . 12  |-  ( A  e.  dom  S  ->  A  e. Word  W )
1918adantr 453 . . . . . . . . . . 11  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  ->  A  e. Word  W )
20 wrdf 11735 . . . . . . . . . . 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 10313 . . . . . . . . . . . . . . 15  |-  1  e.  ZZ
23 simpr 449 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( # `  A )  e.  ( ZZ>= `  2
) )
24 df-2 10060 . . . . . . . . . . . . . . . . 17  |-  2  =  ( 1  +  1 )
2524fveq2i 5733 . . . . . . . . . . . . . . . 16  |-  ( ZZ>= ` 
2 )  =  (
ZZ>= `  ( 1  +  1 ) )
2623, 25syl6eleq 2528 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( # `  A )  e.  ( ZZ>= `  (
1  +  1 ) ) )
27 eluzp1m1 10511 . . . . . . . . . . . . . . 15  |-  ( ( 1  e.  ZZ  /\  ( # `  A )  e.  ( ZZ>= `  (
1  +  1 ) ) )  ->  (
( # `  A )  -  1 )  e.  ( ZZ>= `  1 )
)
2822, 26, 27sylancr 646 . . . . . . . . . . . . . 14  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( ( # `  A
)  -  1 )  e.  ( ZZ>= `  1
) )
29 nnuz 10523 . . . . . . . . . . . . . 14  |-  NN  =  ( ZZ>= `  1 )
3028, 29syl6eleqr 2529 . . . . . . . . . . . . 13  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( ( # `  A
)  -  1 )  e.  NN )
31 lbfzo0 11172 . . . . . . . . . . . . 13  |-  ( 0  e.  ( 0..^ ( ( # `  A
)  -  1 ) )  <->  ( ( # `  A )  -  1 )  e.  NN )
3230, 31sylibr 205 . . . . . . . . . . . 12  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
0  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) )
33 fzoend 11189 . . . . . . . . . . . 12  |-  ( 0  e.  ( 0..^ ( ( # `  A
)  -  1 ) )  ->  ( (
( # `  A )  -  1 )  - 
1 )  e.  ( 0..^ ( ( # `  A )  -  1 ) ) )
34 elfzofz 11156 . . . . . . . . . . . 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 10498 . . . . . . . . . . . . 13  |-  ( (
# `  A )  e.  ( ZZ>= `  2 )  ->  ( # `  A
)  e.  ZZ )
3736adantl 454 . . . . . . . . . . . 12  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( # `  A )  e.  ZZ )
38 fzoval 11143 . . . . . . . . . . . 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 2515 . . . . . . . . . 10  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( ( ( # `  A )  -  1 )  -  1 )  e.  ( 0..^ (
# `  A )
) )
4121, 40ffvelrnd 5873 . . . . . . . . 9  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( A `  (
( ( # `  A
)  -  1 )  -  1 ) )  e.  W )
42 uz2m1nn 10552 . . . . . . . . . 10  |-  ( (
# `  A )  e.  ( ZZ>= `  2 )  ->  ( ( # `  A
)  -  1 )  e.  NN )
434, 5, 6, 7, 2, 8efgsdmi 15366 . . . . . . . . . 10  |-  ( ( A  e.  dom  S  /\  ( ( # `  A
)  -  1 )  e.  NN )  -> 
( S `  A
)  e.  ran  ( T `  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) ) ) )
4442, 43sylan2 462 . . . . . . . . 9  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( S `  A
)  e.  ran  ( T `  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) ) ) )
45 fveq2 5730 . . . . . . . . . . . 12  |-  ( a  =  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) )  ->  ( T `  a )  =  ( T `  ( A `
 ( ( (
# `  A )  -  1 )  - 
1 ) ) ) )
4645rneqd 5099 . . . . . . . . . . 11  |-  ( a  =  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) )  ->  ran  ( T `
 a )  =  ran  ( T `  ( A `  ( ( ( # `  A
)  -  1 )  -  1 ) ) ) )
4746eleq2d 2505 . . . . . . . . . 10  |-  ( a  =  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) )  ->  ( ( S `  A )  e.  ran  ( T `  a )  <->  ( S `  A )  e.  ran  ( T `  ( A `
 ( ( (
# `  A )  -  1 )  - 
1 ) ) ) ) )
4847rspcev 3054 . . . . . . . . 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 644 . . . . . . . 8  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  ->  E. a  e.  W  ( S `  A )  e.  ran  ( T `
 a ) )
50 eliun 4099 . . . . . . . 8  |-  ( ( S `  A )  e.  U_ a  e.  W  ran  ( T `
 a )  <->  E. a  e.  W  ( S `  A )  e.  ran  ( T `  a ) )
5149, 50sylibr 205 . . . . . . 7  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( S `  A
)  e.  U_ a  e.  W  ran  ( T `
 a ) )
52 fveq2 5730 . . . . . . . . 9  |-  ( a  =  x  ->  ( T `  a )  =  ( T `  x ) )
5352rneqd 5099 . . . . . . . 8  |-  ( a  =  x  ->  ran  ( T `  a )  =  ran  ( T `
 x ) )
5453cbviunv 4132 . . . . . . 7  |-  U_ a  e.  W  ran  ( T `
 a )  = 
U_ x  e.  W  ran  ( T `  x
)
5551, 54syl6eleq 2528 . . . . . 6  |-  ( ( A  e.  dom  S  /\  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )  -> 
( S `  A
)  e.  U_ x  e.  W  ran  ( T `
 x ) )
5655ex 425 . . . . 5  |-  ( A  e.  dom  S  -> 
( ( # `  A
)  e.  ( ZZ>= ` 
2 )  ->  ( S `  A )  e.  U_ x  e.  W  ran  ( T `  x
) ) )
5717, 56syld 43 . . . 4  |-  ( A  e.  dom  S  -> 
( -.  ( # `  A )  =  1  ->  ( S `  A )  e.  U_ x  e.  W  ran  ( T `  x ) ) )
5857con1d 119 . . 3  |-  ( A  e.  dom  S  -> 
( -.  ( S `
 A )  e. 
U_ x  e.  W  ran  ( T `  x
)  ->  ( # `  A
)  =  1 ) )
593, 58syl5 31 . 2  |-  ( A  e.  dom  S  -> 
( ( S `  A )  e.  D  ->  ( # `  A
)  =  1 ) )
609simp2bi 974 . . . 4  |-  ( A  e.  dom  S  -> 
( A `  0
)  e.  D )
61 oveq1 6090 . . . . . . 7  |-  ( (
# `  A )  =  1  ->  (
( # `  A )  -  1 )  =  ( 1  -  1 ) )
62 1m1e0 10070 . . . . . . 7  |-  ( 1  -  1 )  =  0
6361, 62syl6eq 2486 . . . . . 6  |-  ( (
# `  A )  =  1  ->  (
( # `  A )  -  1 )  =  0 )
6463fveq2d 5734 . . . . 5  |-  ( (
# `  A )  =  1  ->  ( A `  ( ( # `
 A )  - 
1 ) )  =  ( A `  0
) )
6564eleq1d 2504 . . . 4  |-  ( (
# `  A )  =  1  ->  (
( A `  (
( # `  A )  -  1 ) )  e.  D  <->  ( A `  0 )  e.  D ) )
6660, 65syl5ibrcom 215 . . 3  |-  ( A  e.  dom  S  -> 
( ( # `  A
)  =  1  -> 
( A `  (
( # `  A )  -  1 ) )  e.  D ) )
674, 5, 6, 7, 2, 8efgsval 15365 . . . 4  |-  ( A  e.  dom  S  -> 
( S `  A
)  =  ( A `
 ( ( # `  A )  -  1 ) ) )
6867eleq1d 2504 . . 3  |-  ( A  e.  dom  S  -> 
( ( S `  A )  e.  D  <->  ( A `  ( (
# `  A )  -  1 ) )  e.  D ) )
6966, 68sylibrd 227 . 2  |-  ( A  e.  dom  S  -> 
( ( # `  A
)  =  1  -> 
( S `  A
)  e.  D ) )
7059, 69impbid 185 1  |-  ( A  e.  dom  S  -> 
( ( S `  A )  e.  D  <->  (
# `  A )  =  1 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   {crab 2711    \ cdif 3319   (/)c0 3630   {csn 3816   <.cop 3819   <.cotp 3820   U_ciun 4095    e. cmpt 4268    _I cid 4495    X. cxp 4878   dom cdm 4880   ran crn 4881   -->wf 5452   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   1oc1o 6719   2oc2o 6720   0cc0 8992   1c1 8993    + caddc 8995    - cmin 9293   NNcn 10002   2c2 10051   ZZcz 10284   ZZ>=cuz 10490   ...cfz 11045  ..^cfzo 11137   #chash 11620  Word cword 11719   splice csplice 11723   <"cs2 11807   ~FG cefg 15340
This theorem is referenced by:  efgredlema  15374  efgredeu  15386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-fzo 11138  df-hash 11621  df-word 11725
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