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Theorem efgredlema 16960
Description: The reduced word that forms the base of the sequence in efgsval 16951 is uniquely determined, given the ending representation. (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 ) ) )
efgredlem.1  |-  ( ph  ->  A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  ( # `  ( S `  A )
)  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) )
efgredlem.2  |-  ( ph  ->  A  e.  dom  S
)
efgredlem.3  |-  ( ph  ->  B  e.  dom  S
)
efgredlem.4  |-  ( ph  ->  ( S `  A
)  =  ( S `
 B ) )
efgredlem.5  |-  ( ph  ->  -.  ( A ` 
0 )  =  ( B `  0 ) )
Assertion
Ref Expression
efgredlema  |-  ( ph  ->  ( ( ( # `  A )  -  1 )  e.  NN  /\  ( ( # `  B
)  -  1 )  e.  NN ) )
Distinct variable groups:    a, b, A    y, a, z, b   
t, n, v, w, y, z    m, a, n, t, v, w, x, M, b    k,
a, T, b, m, t, x    W, a, b    k, n, v, w, y, z, W, m, t, x    .~ , a,
b, m, t, x, y, z    B, a, b    S, a, b    I,
a, b, m, n, t, v, w, x, y, z    D, a, b, m, t
Allowed substitution hints:    ph( x, y, z, w, v, t, k, m, n, a, b)    A( x, y, z, w, v, t, k, m, n)    B( 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 efgredlema
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 efgredlem.5 . . . . 5  |-  ( ph  ->  -.  ( A ` 
0 )  =  ( B `  0 ) )
2 efgredlem.3 . . . . . . . . 9  |-  ( ph  ->  B  e.  dom  S
)
3 efgval.w . . . . . . . . . 10  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
4 efgval.r . . . . . . . . . 10  |-  .~  =  ( ~FG  `  I )
5 efgval2.m . . . . . . . . . 10  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
6 efgval2.t . . . . . . . . . 10  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
7 efgred.d . . . . . . . . . 10  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
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 ) ) )
93, 4, 5, 6, 7, 8efgsval 16951 . . . . . . . . 9  |-  ( B  e.  dom  S  -> 
( S `  B
)  =  ( B `
 ( ( # `  B )  -  1 ) ) )
102, 9syl 16 . . . . . . . 8  |-  ( ph  ->  ( S `  B
)  =  ( B `
 ( ( # `  B )  -  1 ) ) )
11 efgredlem.4 . . . . . . . . 9  |-  ( ph  ->  ( S `  A
)  =  ( S `
 B ) )
12 efgredlem.2 . . . . . . . . . 10  |-  ( ph  ->  A  e.  dom  S
)
133, 4, 5, 6, 7, 8efgsval 16951 . . . . . . . . . 10  |-  ( A  e.  dom  S  -> 
( S `  A
)  =  ( A `
 ( ( # `  A )  -  1 ) ) )
1412, 13syl 16 . . . . . . . . 9  |-  ( ph  ->  ( S `  A
)  =  ( A `
 ( ( # `  A )  -  1 ) ) )
1511, 14eqtr3d 2497 . . . . . . . 8  |-  ( ph  ->  ( S `  B
)  =  ( A `
 ( ( # `  A )  -  1 ) ) )
1610, 15eqtr3d 2497 . . . . . . 7  |-  ( ph  ->  ( B `  (
( # `  B )  -  1 ) )  =  ( A `  ( ( # `  A
)  -  1 ) ) )
17 oveq1 6277 . . . . . . . . 9  |-  ( (
# `  A )  =  1  ->  (
( # `  A )  -  1 )  =  ( 1  -  1 ) )
18 1m1e0 10600 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
1917, 18syl6eq 2511 . . . . . . . 8  |-  ( (
# `  A )  =  1  ->  (
( # `  A )  -  1 )  =  0 )
2019fveq2d 5852 . . . . . . 7  |-  ( (
# `  A )  =  1  ->  ( A `  ( ( # `
 A )  - 
1 ) )  =  ( A `  0
) )
2116, 20sylan9eq 2515 . . . . . 6  |-  ( (
ph  /\  ( # `  A
)  =  1 )  ->  ( B `  ( ( # `  B
)  -  1 ) )  =  ( A `
 0 ) )
2211eleq1d 2523 . . . . . . . . 9  |-  ( ph  ->  ( ( S `  A )  e.  D  <->  ( S `  B )  e.  D ) )
233, 4, 5, 6, 7, 8efgs1b 16956 . . . . . . . . . 10  |-  ( A  e.  dom  S  -> 
( ( S `  A )  e.  D  <->  (
# `  A )  =  1 ) )
2412, 23syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( S `  A )  e.  D  <->  (
# `  A )  =  1 ) )
253, 4, 5, 6, 7, 8efgs1b 16956 . . . . . . . . . 10  |-  ( B  e.  dom  S  -> 
( ( S `  B )  e.  D  <->  (
# `  B )  =  1 ) )
262, 25syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( S `  B )  e.  D  <->  (
# `  B )  =  1 ) )
2722, 24, 263bitr3d 283 . . . . . . . 8  |-  ( ph  ->  ( ( # `  A
)  =  1  <->  ( # `
 B )  =  1 ) )
2827biimpa 482 . . . . . . 7  |-  ( (
ph  /\  ( # `  A
)  =  1 )  ->  ( # `  B
)  =  1 )
29 oveq1 6277 . . . . . . . . 9  |-  ( (
# `  B )  =  1  ->  (
( # `  B )  -  1 )  =  ( 1  -  1 ) )
3029, 18syl6eq 2511 . . . . . . . 8  |-  ( (
# `  B )  =  1  ->  (
( # `  B )  -  1 )  =  0 )
3130fveq2d 5852 . . . . . . 7  |-  ( (
# `  B )  =  1  ->  ( B `  ( ( # `
 B )  - 
1 ) )  =  ( B `  0
) )
3228, 31syl 16 . . . . . 6  |-  ( (
ph  /\  ( # `  A
)  =  1 )  ->  ( B `  ( ( # `  B
)  -  1 ) )  =  ( B `
 0 ) )
3321, 32eqtr3d 2497 . . . . 5  |-  ( (
ph  /\  ( # `  A
)  =  1 )  ->  ( A ` 
0 )  =  ( B `  0 ) )
341, 33mtand 657 . . . 4  |-  ( ph  ->  -.  ( # `  A
)  =  1 )
353, 4, 5, 6, 7, 8efgsdm 16950 . . . . . . . 8  |-  ( A  e.  dom  S  <->  ( A  e.  (Word  W  \  { (/)
} )  /\  ( A `  0 )  e.  D  /\  A. u  e.  ( 1..^ ( # `  A ) ) ( A `  u )  e.  ran  ( T `
 ( A `  ( u  -  1
) ) ) ) )
3635simp1bi 1009 . . . . . . 7  |-  ( A  e.  dom  S  ->  A  e.  (Word  W  \  { (/) } ) )
37 eldifsn 4141 . . . . . . . 8  |-  ( A  e.  (Word  W  \  { (/) } )  <->  ( A  e. Word  W  /\  A  =/=  (/) ) )
38 lennncl 12553 . . . . . . . 8  |-  ( ( A  e. Word  W  /\  A  =/=  (/) )  ->  ( # `
 A )  e.  NN )
3937, 38sylbi 195 . . . . . . 7  |-  ( A  e.  (Word  W  \  { (/) } )  -> 
( # `  A )  e.  NN )
4012, 36, 393syl 20 . . . . . 6  |-  ( ph  ->  ( # `  A
)  e.  NN )
41 elnn1uz2 11159 . . . . . 6  |-  ( (
# `  A )  e.  NN  <->  ( ( # `  A )  =  1  \/  ( # `  A
)  e.  ( ZZ>= ` 
2 ) ) )
4240, 41sylib 196 . . . . 5  |-  ( ph  ->  ( ( # `  A
)  =  1  \/  ( # `  A
)  e.  ( ZZ>= ` 
2 ) ) )
4342ord 375 . . . 4  |-  ( ph  ->  ( -.  ( # `  A )  =  1  ->  ( # `  A
)  e.  ( ZZ>= ` 
2 ) ) )
4434, 43mpd 15 . . 3  |-  ( ph  ->  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )
45 uz2m1nn 11157 . . 3  |-  ( (
# `  A )  e.  ( ZZ>= `  2 )  ->  ( ( # `  A
)  -  1 )  e.  NN )
4644, 45syl 16 . 2  |-  ( ph  ->  ( ( # `  A
)  -  1 )  e.  NN )
4734, 27mtbid 298 . . . 4  |-  ( ph  ->  -.  ( # `  B
)  =  1 )
483, 4, 5, 6, 7, 8efgsdm 16950 . . . . . . . 8  |-  ( B  e.  dom  S  <->  ( B  e.  (Word  W  \  { (/)
} )  /\  ( B `  0 )  e.  D  /\  A. u  e.  ( 1..^ ( # `  B ) ) ( B `  u )  e.  ran  ( T `
 ( B `  ( u  -  1
) ) ) ) )
4948simp1bi 1009 . . . . . . 7  |-  ( B  e.  dom  S  ->  B  e.  (Word  W  \  { (/) } ) )
50 eldifsn 4141 . . . . . . . 8  |-  ( B  e.  (Word  W  \  { (/) } )  <->  ( B  e. Word  W  /\  B  =/=  (/) ) )
51 lennncl 12553 . . . . . . . 8  |-  ( ( B  e. Word  W  /\  B  =/=  (/) )  ->  ( # `
 B )  e.  NN )
5250, 51sylbi 195 . . . . . . 7  |-  ( B  e.  (Word  W  \  { (/) } )  -> 
( # `  B )  e.  NN )
532, 49, 523syl 20 . . . . . 6  |-  ( ph  ->  ( # `  B
)  e.  NN )
54 elnn1uz2 11159 . . . . . 6  |-  ( (
# `  B )  e.  NN  <->  ( ( # `  B )  =  1  \/  ( # `  B
)  e.  ( ZZ>= ` 
2 ) ) )
5553, 54sylib 196 . . . . 5  |-  ( ph  ->  ( ( # `  B
)  =  1  \/  ( # `  B
)  e.  ( ZZ>= ` 
2 ) ) )
5655ord 375 . . . 4  |-  ( ph  ->  ( -.  ( # `  B )  =  1  ->  ( # `  B
)  e.  ( ZZ>= ` 
2 ) ) )
5747, 56mpd 15 . . 3  |-  ( ph  ->  ( # `  B
)  e.  ( ZZ>= ` 
2 ) )
58 uz2m1nn 11157 . . 3  |-  ( (
# `  B )  e.  ( ZZ>= `  2 )  ->  ( ( # `  B
)  -  1 )  e.  NN )
5957, 58syl 16 . 2  |-  ( ph  ->  ( ( # `  B
)  -  1 )  e.  NN )
6046, 59jca 530 1  |-  ( ph  ->  ( ( ( # `  A )  -  1 )  e.  NN  /\  ( ( # `  B
)  -  1 )  e.  NN ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   {crab 2808    \ cdif 3458   (/)c0 3783   {csn 4016   <.cop 4022   <.cotp 4024   U_ciun 4315   class class class wbr 4439    |-> cmpt 4497    _I cid 4779    X. cxp 4986   dom cdm 4988   ran crn 4989   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   1oc1o 7115   2oc2o 7116   0cc0 9481   1c1 9482    < clt 9617    - cmin 9796   NNcn 10531   2c2 10581   ZZ>=cuz 11082   ...cfz 11675  ..^cfzo 11799   #chash 12390  Word cword 12521   splice csplice 12526   <"cs2 12800   ~FG cefg 16926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12391  df-word 12529
This theorem is referenced by:  efgredlemf  16961  efgredlemg  16962  efgredlemd  16964  efgredlemc  16965  efgredlem  16967
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