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Theorem efgredlemf 15300
Description: Lemma for efgredleme 15302. (Contributed by Mario Carneiro, 4-Jun-2016.)
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 )
Assertion
Ref Expression
efgredlemf  |-  ( ph  ->  ( ( A `  K )  e.  W  /\  ( B `  L
)  e.  W ) )
Distinct variable groups:    a, b, A    y, a, z, b    L, a, b    K, a, 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)    K( x, y, z, w, v, t, k, m, n)    L( x, y, z, w, v, t, k, m, n)    M( y, z, k)

Proof of Theorem efgredlemf
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 efgredlem.2 . . . . . 6  |-  ( ph  ->  A  e.  dom  S
)
2 efgval.w . . . . . . . 8  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
3 efgval.r . . . . . . . 8  |-  .~  =  ( ~FG  `  I )
4 efgval2.m . . . . . . . 8  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
5 efgval2.t . . . . . . . 8  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
6 efgred.d . . . . . . . 8  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
7 efgred.s . . . . . . . 8  |-  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 ) ) )
82, 3, 4, 5, 6, 7efgsdm 15289 . . . . . . 7  |-  ( A  e.  dom  S  <->  ( A  e.  (Word  W  \  { (/)
} )  /\  ( A `  0 )  e.  D  /\  A. i  e.  ( 1..^ ( # `  A ) ) ( A `  i )  e.  ran  ( T `
 ( A `  ( i  -  1 ) ) ) ) )
98simp1bi 972 . . . . . 6  |-  ( A  e.  dom  S  ->  A  e.  (Word  W  \  { (/) } ) )
101, 9syl 16 . . . . 5  |-  ( ph  ->  A  e.  (Word  W  \  { (/) } ) )
1110eldifad 3275 . . . 4  |-  ( ph  ->  A  e. Word  W )
12 wrdf 11660 . . . 4  |-  ( A  e. Word  W  ->  A : ( 0..^ (
# `  A )
) --> W )
1311, 12syl 16 . . 3  |-  ( ph  ->  A : ( 0..^ ( # `  A
) ) --> W )
14 fzossfz 11087 . . . . 5  |-  ( 0..^ ( ( # `  A
)  -  1 ) )  C_  ( 0 ... ( ( # `  A )  -  1 ) )
15 lencl 11662 . . . . . . . 8  |-  ( A  e. Word  W  ->  ( # `
 A )  e. 
NN0 )
1611, 15syl 16 . . . . . . 7  |-  ( ph  ->  ( # `  A
)  e.  NN0 )
1716nn0zd 10305 . . . . . 6  |-  ( ph  ->  ( # `  A
)  e.  ZZ )
18 fzoval 11071 . . . . . 6  |-  ( (
# `  A )  e.  ZZ  ->  ( 0..^ ( # `  A
) )  =  ( 0 ... ( (
# `  A )  -  1 ) ) )
1917, 18syl 16 . . . . 5  |-  ( ph  ->  ( 0..^ ( # `  A ) )  =  ( 0 ... (
( # `  A )  -  1 ) ) )
2014, 19syl5sseqr 3340 . . . 4  |-  ( ph  ->  ( 0..^ ( (
# `  A )  -  1 ) ) 
C_  ( 0..^ (
# `  A )
) )
21 efgredlemb.k . . . . 5  |-  K  =  ( ( ( # `  A )  -  1 )  -  1 )
22 efgredlem.1 . . . . . . . 8  |-  ( ph  ->  A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  ( # `  ( S `  A )
)  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) )
23 efgredlem.3 . . . . . . . 8  |-  ( ph  ->  B  e.  dom  S
)
24 efgredlem.4 . . . . . . . 8  |-  ( ph  ->  ( S `  A
)  =  ( S `
 B ) )
25 efgredlem.5 . . . . . . . 8  |-  ( ph  ->  -.  ( A ` 
0 )  =  ( B `  0 ) )
262, 3, 4, 5, 6, 7, 22, 1, 23, 24, 25efgredlema 15299 . . . . . . 7  |-  ( ph  ->  ( ( ( # `  A )  -  1 )  e.  NN  /\  ( ( # `  B
)  -  1 )  e.  NN ) )
2726simpld 446 . . . . . 6  |-  ( ph  ->  ( ( # `  A
)  -  1 )  e.  NN )
28 fzo0end 11115 . . . . . 6  |-  ( ( ( # `  A
)  -  1 )  e.  NN  ->  (
( ( # `  A
)  -  1 )  -  1 )  e.  ( 0..^ ( (
# `  A )  -  1 ) ) )
2927, 28syl 16 . . . . 5  |-  ( ph  ->  ( ( ( # `  A )  -  1 )  -  1 )  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) )
3021, 29syl5eqel 2471 . . . 4  |-  ( ph  ->  K  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) )
3120, 30sseldd 3292 . . 3  |-  ( ph  ->  K  e.  ( 0..^ ( # `  A
) ) )
3213, 31ffvelrnd 5810 . 2  |-  ( ph  ->  ( A `  K
)  e.  W )
332, 3, 4, 5, 6, 7efgsdm 15289 . . . . . . 7  |-  ( B  e.  dom  S  <->  ( B  e.  (Word  W  \  { (/)
} )  /\  ( B `  0 )  e.  D  /\  A. i  e.  ( 1..^ ( # `  B ) ) ( B `  i )  e.  ran  ( T `
 ( B `  ( i  -  1 ) ) ) ) )
3433simp1bi 972 . . . . . 6  |-  ( B  e.  dom  S  ->  B  e.  (Word  W  \  { (/) } ) )
3523, 34syl 16 . . . . 5  |-  ( ph  ->  B  e.  (Word  W  \  { (/) } ) )
3635eldifad 3275 . . . 4  |-  ( ph  ->  B  e. Word  W )
37 wrdf 11660 . . . 4  |-  ( B  e. Word  W  ->  B : ( 0..^ (
# `  B )
) --> W )
3836, 37syl 16 . . 3  |-  ( ph  ->  B : ( 0..^ ( # `  B
) ) --> W )
39 fzossfz 11087 . . . . 5  |-  ( 0..^ ( ( # `  B
)  -  1 ) )  C_  ( 0 ... ( ( # `  B )  -  1 ) )
40 lencl 11662 . . . . . . . 8  |-  ( B  e. Word  W  ->  ( # `
 B )  e. 
NN0 )
4136, 40syl 16 . . . . . . 7  |-  ( ph  ->  ( # `  B
)  e.  NN0 )
4241nn0zd 10305 . . . . . 6  |-  ( ph  ->  ( # `  B
)  e.  ZZ )
43 fzoval 11071 . . . . . 6  |-  ( (
# `  B )  e.  ZZ  ->  ( 0..^ ( # `  B
) )  =  ( 0 ... ( (
# `  B )  -  1 ) ) )
4442, 43syl 16 . . . . 5  |-  ( ph  ->  ( 0..^ ( # `  B ) )  =  ( 0 ... (
( # `  B )  -  1 ) ) )
4539, 44syl5sseqr 3340 . . . 4  |-  ( ph  ->  ( 0..^ ( (
# `  B )  -  1 ) ) 
C_  ( 0..^ (
# `  B )
) )
46 efgredlemb.l . . . . 5  |-  L  =  ( ( ( # `  B )  -  1 )  -  1 )
4726simprd 450 . . . . . 6  |-  ( ph  ->  ( ( # `  B
)  -  1 )  e.  NN )
48 fzo0end 11115 . . . . . 6  |-  ( ( ( # `  B
)  -  1 )  e.  NN  ->  (
( ( # `  B
)  -  1 )  -  1 )  e.  ( 0..^ ( (
# `  B )  -  1 ) ) )
4947, 48syl 16 . . . . 5  |-  ( ph  ->  ( ( ( # `  B )  -  1 )  -  1 )  e.  ( 0..^ ( ( # `  B
)  -  1 ) ) )
5046, 49syl5eqel 2471 . . . 4  |-  ( ph  ->  L  e.  ( 0..^ ( ( # `  B
)  -  1 ) ) )
5145, 50sseldd 3292 . . 3  |-  ( ph  ->  L  e.  ( 0..^ ( # `  B
) ) )
5238, 51ffvelrnd 5810 . 2  |-  ( ph  ->  ( B `  L
)  e.  W )
5332, 52jca 519 1  |-  ( ph  ->  ( ( A `  K )  e.  W  /\  ( B `  L
)  e.  W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649   {crab 2653    \ cdif 3260   (/)c0 3571   {csn 3757   <.cop 3760   <.cotp 3761   U_ciun 4035   class class class wbr 4153    e. cmpt 4207    _I cid 4434    X. cxp 4816   dom cdm 4818   ran crn 4819   -->wf 5390   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022   1oc1o 6653   2oc2o 6654   0cc0 8923   1c1 8924    < clt 9053    - cmin 9223   NNcn 9932   NN0cn0 10153   ZZcz 10214   ...cfz 10975  ..^cfzo 11065   #chash 11545  Word cword 11644   splice csplice 11648   <"cs2 11732   ~FG cefg 15265
This theorem is referenced by:  efgredlemg  15301  efgredleme  15302
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-n0 10154  df-z 10215  df-uz 10421  df-fz 10976  df-fzo 11066  df-hash 11546  df-word 11650
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