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Theorem efgredlemb 15378
Description: The reduced word that forms the base of the sequence in efgsval 15363 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 30-Sep-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 ) )
efgredlemb.k  |-  K  =  ( ( ( # `  A )  -  1 )  -  1 )
efgredlemb.l  |-  L  =  ( ( ( # `  B )  -  1 )  -  1 )
efgredlemb.p  |-  ( ph  ->  P  e.  ( 0 ... ( # `  ( A `  K )
) ) )
efgredlemb.q  |-  ( ph  ->  Q  e.  ( 0 ... ( # `  ( B `  L )
) ) )
efgredlemb.u  |-  ( ph  ->  U  e.  ( I  X.  2o ) )
efgredlemb.v  |-  ( ph  ->  V  e.  ( I  X.  2o ) )
efgredlemb.6  |-  ( ph  ->  ( S `  A
)  =  ( P ( T `  ( A `  K )
) U ) )
efgredlemb.7  |-  ( ph  ->  ( S `  B
)  =  ( Q ( T `  ( B `  L )
) V ) )
efgredlemb.8  |-  ( ph  ->  -.  ( A `  K )  =  ( B `  L ) )
Assertion
Ref Expression
efgredlemb  |-  -.  ph
Distinct variable groups:    a, b, A    y, a, z, b    L, a, b    K, a, b    t, n, v, w, y, z, P   
m, a, n, t, v, w, x, M, b    U, n, v, w, y, z    k, a, T, b, m, t, x    n, V, v, w, y, z    Q, n, t, v, w, y, z    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)    P( x, k, m, a, b)    Q( x, k, m, a, b)    .~ ( w, v, k, n)    S( x, y, z, w, v, t, k, m, n)    T( y, z, w, v, n)    U( x, t, k, m, a, b)    I( k)    K( x, y, z, w, v, t, k, m, n)    L( x, y, z, w, v, t, k, m, n)    M( y, z, k)    V( x, t, k, m, a, b)

Proof of Theorem efgredlemb
StepHypRef Expression
1 efgval.w . . . . 5  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
2 efgval.r . . . . 5  |-  .~  =  ( ~FG  `  I )
3 efgval2.m . . . . 5  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
4 efgval2.t . . . . 5  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
5 efgred.d . . . . 5  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
6 efgred.s . . . . 5  |-  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 ) ) )
7 efgredlem.1 . . . . . 6  |-  ( ph  ->  A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  ( # `  ( S `  A )
)  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) )
8 efgredlem.4 . . . . . . 7  |-  ( ph  ->  ( S `  A
)  =  ( S `
 B ) )
9 fveq2 5728 . . . . . . . . . 10  |-  ( ( S `  A )  =  ( S `  B )  ->  ( # `
 ( S `  A ) )  =  ( # `  ( S `  B )
) )
109breq2d 4224 . . . . . . . . 9  |-  ( ( S `  A )  =  ( S `  B )  ->  (
( # `  ( S `
 a ) )  <  ( # `  ( S `  A )
)  <->  ( # `  ( S `  a )
)  <  ( # `  ( S `  B )
) ) )
1110imbi1d 309 . . . . . . . 8  |-  ( ( S `  A )  =  ( S `  B )  ->  (
( ( # `  ( S `  a )
)  <  ( # `  ( S `  A )
)  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) )  <-> 
( ( # `  ( S `  a )
)  <  ( # `  ( S `  B )
)  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) ) )
12112ralbidv 2747 . . . . . . 7  |-  ( ( S `  A )  =  ( S `  B )  ->  ( A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  ( # `  ( S `  A )
)  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) )  <->  A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  ( # `  ( S `  B )
)  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) ) )
138, 12syl 16 . . . . . 6  |-  ( ph  ->  ( A. a  e. 
dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  < 
( # `  ( S `
 A ) )  ->  ( ( S `
 a )  =  ( S `  b
)  ->  ( a `  0 )  =  ( b `  0
) ) )  <->  A. a  e.  dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  < 
( # `  ( S `
 B ) )  ->  ( ( S `
 a )  =  ( S `  b
)  ->  ( a `  0 )  =  ( b `  0
) ) ) ) )
147, 13mpbid 202 . . . . 5  |-  ( ph  ->  A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  ( # `  ( S `  B )
)  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) )
15 efgredlem.3 . . . . 5  |-  ( ph  ->  B  e.  dom  S
)
16 efgredlem.2 . . . . 5  |-  ( ph  ->  A  e.  dom  S
)
178eqcomd 2441 . . . . 5  |-  ( ph  ->  ( S `  B
)  =  ( S `
 A ) )
18 efgredlem.5 . . . . . 6  |-  ( ph  ->  -.  ( A ` 
0 )  =  ( B `  0 ) )
19 eqcom 2438 . . . . . 6  |-  ( ( A `  0 )  =  ( B ` 
0 )  <->  ( B `  0 )  =  ( A `  0
) )
2018, 19sylnib 296 . . . . 5  |-  ( ph  ->  -.  ( B ` 
0 )  =  ( A `  0 ) )
21 efgredlemb.l . . . . 5  |-  L  =  ( ( ( # `  B )  -  1 )  -  1 )
22 efgredlemb.k . . . . 5  |-  K  =  ( ( ( # `  A )  -  1 )  -  1 )
23 efgredlemb.q . . . . 5  |-  ( ph  ->  Q  e.  ( 0 ... ( # `  ( B `  L )
) ) )
24 efgredlemb.p . . . . 5  |-  ( ph  ->  P  e.  ( 0 ... ( # `  ( A `  K )
) ) )
25 efgredlemb.v . . . . 5  |-  ( ph  ->  V  e.  ( I  X.  2o ) )
26 efgredlemb.u . . . . 5  |-  ( ph  ->  U  e.  ( I  X.  2o ) )
27 efgredlemb.7 . . . . 5  |-  ( ph  ->  ( S `  B
)  =  ( Q ( T `  ( B `  L )
) V ) )
28 efgredlemb.6 . . . . 5  |-  ( ph  ->  ( S `  A
)  =  ( P ( T `  ( A `  K )
) U ) )
29 efgredlemb.8 . . . . . 6  |-  ( ph  ->  -.  ( A `  K )  =  ( B `  L ) )
30 eqcom 2438 . . . . . 6  |-  ( ( A `  K )  =  ( B `  L )  <->  ( B `  L )  =  ( A `  K ) )
3129, 30sylnib 296 . . . . 5  |-  ( ph  ->  -.  ( B `  L )  =  ( A `  K ) )
321, 2, 3, 4, 5, 6, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, 28, 31efgredlemc 15377 . . . 4  |-  ( ph  ->  ( Q  e.  (
ZZ>= `  P )  -> 
( B `  0
)  =  ( A `
 0 ) ) )
3332, 19syl6ibr 219 . . 3  |-  ( ph  ->  ( Q  e.  (
ZZ>= `  P )  -> 
( A `  0
)  =  ( B `
 0 ) ) )
341, 2, 3, 4, 5, 6, 7, 16, 15, 8, 18, 22, 21, 24, 23, 26, 25, 28, 27, 29efgredlemc 15377 . . 3  |-  ( ph  ->  ( P  e.  (
ZZ>= `  Q )  -> 
( A `  0
)  =  ( B `
 0 ) ) )
35 elfzelz 11059 . . . . 5  |-  ( P  e.  ( 0 ... ( # `  ( A `  K )
) )  ->  P  e.  ZZ )
3624, 35syl 16 . . . 4  |-  ( ph  ->  P  e.  ZZ )
37 elfzelz 11059 . . . . 5  |-  ( Q  e.  ( 0 ... ( # `  ( B `  L )
) )  ->  Q  e.  ZZ )
3823, 37syl 16 . . . 4  |-  ( ph  ->  Q  e.  ZZ )
39 uztric 10507 . . . 4  |-  ( ( P  e.  ZZ  /\  Q  e.  ZZ )  ->  ( Q  e.  (
ZZ>= `  P )  \/  P  e.  ( ZZ>= `  Q ) ) )
4036, 38, 39syl2anc 643 . . 3  |-  ( ph  ->  ( Q  e.  (
ZZ>= `  P )  \/  P  e.  ( ZZ>= `  Q ) ) )
4133, 34, 40mpjaod 371 . 2  |-  ( ph  ->  ( A `  0
)  =  ( B `
 0 ) )
4241, 18pm2.65i 167 1  |-  -.  ph
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709    \ cdif 3317   (/)c0 3628   {csn 3814   <.cop 3817   <.cotp 3818   U_ciun 4093   class class class wbr 4212    e. cmpt 4266    _I cid 4493    X. cxp 4876   dom cdm 4878   ran crn 4879   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   1oc1o 6717   2oc2o 6718   0cc0 8990   1c1 8991    < clt 9120    - cmin 9291   ZZcz 10282   ZZ>=cuz 10488   ...cfz 11043  ..^cfzo 11135   #chash 11618  Word cword 11717   splice csplice 11721   <"cs2 11805   ~FG cefg 15338
This theorem is referenced by:  efgredlem  15379
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-ot 3824  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-hash 11619  df-word 11723  df-concat 11724  df-s1 11725  df-substr 11726  df-splice 11727  df-s2 11812
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