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Theorem efgredlema 15148
Description: The reduced word that forms the base of the sequence in efgsval 15139 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 15139 . . . . . . . . 9  |-  ( B  e.  dom  S  -> 
( S `  B
)  =  ( B `
 ( ( # `  B )  -  1 ) ) )
102, 9syl 15 . . . . . . . 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 15139 . . . . . . . . . 10  |-  ( A  e.  dom  S  -> 
( S `  A
)  =  ( A `
 ( ( # `  A )  -  1 ) ) )
1412, 13syl 15 . . . . . . . . 9  |-  ( ph  ->  ( S `  A
)  =  ( A `
 ( ( # `  A )  -  1 ) ) )
1511, 14eqtr3d 2392 . . . . . . . 8  |-  ( ph  ->  ( S `  B
)  =  ( A `
 ( ( # `  A )  -  1 ) ) )
1610, 15eqtr3d 2392 . . . . . . 7  |-  ( ph  ->  ( B `  (
( # `  B )  -  1 ) )  =  ( A `  ( ( # `  A
)  -  1 ) ) )
17 oveq1 5952 . . . . . . . . 9  |-  ( (
# `  A )  =  1  ->  (
( # `  A )  -  1 )  =  ( 1  -  1 ) )
18 1m1e0 9904 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
1917, 18syl6eq 2406 . . . . . . . 8  |-  ( (
# `  A )  =  1  ->  (
( # `  A )  -  1 )  =  0 )
2019fveq2d 5612 . . . . . . 7  |-  ( (
# `  A )  =  1  ->  ( A `  ( ( # `
 A )  - 
1 ) )  =  ( A `  0
) )
2116, 20sylan9eq 2410 . . . . . 6  |-  ( (
ph  /\  ( # `  A
)  =  1 )  ->  ( B `  ( ( # `  B
)  -  1 ) )  =  ( A `
 0 ) )
2211eleq1d 2424 . . . . . . . . 9  |-  ( ph  ->  ( ( S `  A )  e.  D  <->  ( S `  B )  e.  D ) )
233, 4, 5, 6, 7, 8efgs1b 15144 . . . . . . . . . 10  |-  ( A  e.  dom  S  -> 
( ( S `  A )  e.  D  <->  (
# `  A )  =  1 ) )
2412, 23syl 15 . . . . . . . . 9  |-  ( ph  ->  ( ( S `  A )  e.  D  <->  (
# `  A )  =  1 ) )
253, 4, 5, 6, 7, 8efgs1b 15144 . . . . . . . . . 10  |-  ( B  e.  dom  S  -> 
( ( S `  B )  e.  D  <->  (
# `  B )  =  1 ) )
262, 25syl 15 . . . . . . . . 9  |-  ( ph  ->  ( ( S `  B )  e.  D  <->  (
# `  B )  =  1 ) )
2722, 24, 263bitr3d 274 . . . . . . . 8  |-  ( ph  ->  ( ( # `  A
)  =  1  <->  ( # `
 B )  =  1 ) )
2827biimpa 470 . . . . . . 7  |-  ( (
ph  /\  ( # `  A
)  =  1 )  ->  ( # `  B
)  =  1 )
29 oveq1 5952 . . . . . . . . 9  |-  ( (
# `  B )  =  1  ->  (
( # `  B )  -  1 )  =  ( 1  -  1 ) )
3029, 18syl6eq 2406 . . . . . . . 8  |-  ( (
# `  B )  =  1  ->  (
( # `  B )  -  1 )  =  0 )
3130fveq2d 5612 . . . . . . 7  |-  ( (
# `  B )  =  1  ->  ( B `  ( ( # `
 B )  - 
1 ) )  =  ( B `  0
) )
3228, 31syl 15 . . . . . 6  |-  ( (
ph  /\  ( # `  A
)  =  1 )  ->  ( B `  ( ( # `  B
)  -  1 ) )  =  ( B `
 0 ) )
3321, 32eqtr3d 2392 . . . . 5  |-  ( (
ph  /\  ( # `  A
)  =  1 )  ->  ( A ` 
0 )  =  ( B `  0 ) )
341, 33mtand 640 . . . 4  |-  ( ph  ->  -.  ( # `  A
)  =  1 )
353, 4, 5, 6, 7, 8efgsdm 15138 . . . . . . . 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 970 . . . . . . 7  |-  ( A  e.  dom  S  ->  A  e.  (Word  W  \  { (/) } ) )
37 eldifsn 3825 . . . . . . . 8  |-  ( A  e.  (Word  W  \  { (/) } )  <->  ( A  e. Word  W  /\  A  =/=  (/) ) )
38 lennncl 11518 . . . . . . . 8  |-  ( ( A  e. Word  W  /\  A  =/=  (/) )  ->  ( # `
 A )  e.  NN )
3937, 38sylbi 187 . . . . . . 7  |-  ( A  e.  (Word  W  \  { (/) } )  -> 
( # `  A )  e.  NN )
4012, 36, 393syl 18 . . . . . 6  |-  ( ph  ->  ( # `  A
)  e.  NN )
41 elnn1uz2 10386 . . . . . 6  |-  ( (
# `  A )  e.  NN  <->  ( ( # `  A )  =  1  \/  ( # `  A
)  e.  ( ZZ>= ` 
2 ) ) )
4240, 41sylib 188 . . . . 5  |-  ( ph  ->  ( ( # `  A
)  =  1  \/  ( # `  A
)  e.  ( ZZ>= ` 
2 ) ) )
4342ord 366 . . . 4  |-  ( ph  ->  ( -.  ( # `  A )  =  1  ->  ( # `  A
)  e.  ( ZZ>= ` 
2 ) ) )
4434, 43mpd 14 . . 3  |-  ( ph  ->  ( # `  A
)  e.  ( ZZ>= ` 
2 ) )
45 uz2m1nn 10384 . . 3  |-  ( (
# `  A )  e.  ( ZZ>= `  2 )  ->  ( ( # `  A
)  -  1 )  e.  NN )
4644, 45syl 15 . 2  |-  ( ph  ->  ( ( # `  A
)  -  1 )  e.  NN )
4734, 27mtbid 291 . . . 4  |-  ( ph  ->  -.  ( # `  B
)  =  1 )
483, 4, 5, 6, 7, 8efgsdm 15138 . . . . . . . 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 970 . . . . . . 7  |-  ( B  e.  dom  S  ->  B  e.  (Word  W  \  { (/) } ) )
50 eldifsn 3825 . . . . . . . 8  |-  ( B  e.  (Word  W  \  { (/) } )  <->  ( B  e. Word  W  /\  B  =/=  (/) ) )
51 lennncl 11518 . . . . . . . 8  |-  ( ( B  e. Word  W  /\  B  =/=  (/) )  ->  ( # `
 B )  e.  NN )
5250, 51sylbi 187 . . . . . . 7  |-  ( B  e.  (Word  W  \  { (/) } )  -> 
( # `  B )  e.  NN )
532, 49, 523syl 18 . . . . . 6  |-  ( ph  ->  ( # `  B
)  e.  NN )
54 elnn1uz2 10386 . . . . . 6  |-  ( (
# `  B )  e.  NN  <->  ( ( # `  B )  =  1  \/  ( # `  B
)  e.  ( ZZ>= ` 
2 ) ) )
5553, 54sylib 188 . . . . 5  |-  ( ph  ->  ( ( # `  B
)  =  1  \/  ( # `  B
)  e.  ( ZZ>= ` 
2 ) ) )
5655ord 366 . . . 4  |-  ( ph  ->  ( -.  ( # `  B )  =  1  ->  ( # `  B
)  e.  ( ZZ>= ` 
2 ) ) )
5747, 56mpd 14 . . 3  |-  ( ph  ->  ( # `  B
)  e.  ( ZZ>= ` 
2 ) )
58 uz2m1nn 10384 . . 3  |-  ( (
# `  B )  e.  ( ZZ>= `  2 )  ->  ( ( # `  B
)  -  1 )  e.  NN )
5957, 58syl 15 . 2  |-  ( ph  ->  ( ( # `  B
)  -  1 )  e.  NN )
6046, 59jca 518 1  |-  ( ph  ->  ( ( ( # `  A )  -  1 )  e.  NN  /\  ( ( # `  B
)  -  1 )  e.  NN ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   A.wral 2619   {crab 2623    \ cdif 3225   (/)c0 3531   {csn 3716   <.cop 3719   <.cotp 3720   U_ciun 3986   class class class wbr 4104    e. cmpt 4158    _I cid 4386    X. cxp 4769   dom cdm 4771   ran crn 4772   ` cfv 5337  (class class class)co 5945    e. cmpt2 5947   1oc1o 6559   2oc2o 6560   0cc0 8827   1c1 8828    < clt 8957    - cmin 9127   NNcn 9836   2c2 9885   ZZ>=cuz 10322   ...cfz 10874  ..^cfzo 10962   #chash 11430  Word cword 11499   splice csplice 11503   <"cs2 11587   ~FG cefg 15114
This theorem is referenced by:  efgredlemf  15149  efgredlemg  15150  efgredlemd  15152  efgredlemc  15153  efgredlem  15155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-card 7662  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-n0 10058  df-z 10117  df-uz 10323  df-fz 10875  df-fzo 10963  df-hash 11431  df-word 11505
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