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Theorem efgredlemf 15050
Description: Lemma for efgredleme 15052. (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 15039 . . . . . . 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 970 . . . . . 6  |-  ( A  e.  dom  S  ->  A  e.  (Word  W  \  { (/) } ) )
101, 9syl 15 . . . . 5  |-  ( ph  ->  A  e.  (Word  W  \  { (/) } ) )
11 eldifi 3298 . . . . 5  |-  ( A  e.  (Word  W  \  { (/) } )  ->  A  e. Word  W )
1210, 11syl 15 . . . 4  |-  ( ph  ->  A  e. Word  W )
13 wrdf 11419 . . . 4  |-  ( A  e. Word  W  ->  A : ( 0..^ (
# `  A )
) --> W )
1412, 13syl 15 . . 3  |-  ( ph  ->  A : ( 0..^ ( # `  A
) ) --> W )
15 fzossfz 10892 . . . . 5  |-  ( 0..^ ( ( # `  A
)  -  1 ) )  C_  ( 0 ... ( ( # `  A )  -  1 ) )
16 lencl 11421 . . . . . . . 8  |-  ( A  e. Word  W  ->  ( # `
 A )  e. 
NN0 )
1712, 16syl 15 . . . . . . 7  |-  ( ph  ->  ( # `  A
)  e.  NN0 )
1817nn0zd 10115 . . . . . 6  |-  ( ph  ->  ( # `  A
)  e.  ZZ )
19 fzoval 10876 . . . . . 6  |-  ( (
# `  A )  e.  ZZ  ->  ( 0..^ ( # `  A
) )  =  ( 0 ... ( (
# `  A )  -  1 ) ) )
2018, 19syl 15 . . . . 5  |-  ( ph  ->  ( 0..^ ( # `  A ) )  =  ( 0 ... (
( # `  A )  -  1 ) ) )
2115, 20syl5sseqr 3227 . . . 4  |-  ( ph  ->  ( 0..^ ( (
# `  A )  -  1 ) ) 
C_  ( 0..^ (
# `  A )
) )
22 efgredlemb.k . . . . 5  |-  K  =  ( ( ( # `  A )  -  1 )  -  1 )
23 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 ) ) ) )
24 efgredlem.3 . . . . . . . 8  |-  ( ph  ->  B  e.  dom  S
)
25 efgredlem.4 . . . . . . . 8  |-  ( ph  ->  ( S `  A
)  =  ( S `
 B ) )
26 efgredlem.5 . . . . . . . 8  |-  ( ph  ->  -.  ( A ` 
0 )  =  ( B `  0 ) )
272, 3, 4, 5, 6, 7, 23, 1, 24, 25, 26efgredlema 15049 . . . . . . 7  |-  ( ph  ->  ( ( ( # `  A )  -  1 )  e.  NN  /\  ( ( # `  B
)  -  1 )  e.  NN ) )
2827simpld 445 . . . . . 6  |-  ( ph  ->  ( ( # `  A
)  -  1 )  e.  NN )
29 fzo0end 10915 . . . . . 6  |-  ( ( ( # `  A
)  -  1 )  e.  NN  ->  (
( ( # `  A
)  -  1 )  -  1 )  e.  ( 0..^ ( (
# `  A )  -  1 ) ) )
3028, 29syl 15 . . . . 5  |-  ( ph  ->  ( ( ( # `  A )  -  1 )  -  1 )  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) )
3122, 30syl5eqel 2367 . . . 4  |-  ( ph  ->  K  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) )
3221, 31sseldd 3181 . . 3  |-  ( ph  ->  K  e.  ( 0..^ ( # `  A
) ) )
33 ffvelrn 5663 . . 3  |-  ( ( A : ( 0..^ ( # `  A
) ) --> W  /\  K  e.  ( 0..^ ( # `  A
) ) )  -> 
( A `  K
)  e.  W )
3414, 32, 33syl2anc 642 . 2  |-  ( ph  ->  ( A `  K
)  e.  W )
352, 3, 4, 5, 6, 7efgsdm 15039 . . . . . . 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 ) ) ) ) )
3635simp1bi 970 . . . . . 6  |-  ( B  e.  dom  S  ->  B  e.  (Word  W  \  { (/) } ) )
3724, 36syl 15 . . . . 5  |-  ( ph  ->  B  e.  (Word  W  \  { (/) } ) )
38 eldifi 3298 . . . . 5  |-  ( B  e.  (Word  W  \  { (/) } )  ->  B  e. Word  W )
3937, 38syl 15 . . . 4  |-  ( ph  ->  B  e. Word  W )
40 wrdf 11419 . . . 4  |-  ( B  e. Word  W  ->  B : ( 0..^ (
# `  B )
) --> W )
4139, 40syl 15 . . 3  |-  ( ph  ->  B : ( 0..^ ( # `  B
) ) --> W )
42 fzossfz 10892 . . . . 5  |-  ( 0..^ ( ( # `  B
)  -  1 ) )  C_  ( 0 ... ( ( # `  B )  -  1 ) )
43 lencl 11421 . . . . . . . 8  |-  ( B  e. Word  W  ->  ( # `
 B )  e. 
NN0 )
4439, 43syl 15 . . . . . . 7  |-  ( ph  ->  ( # `  B
)  e.  NN0 )
4544nn0zd 10115 . . . . . 6  |-  ( ph  ->  ( # `  B
)  e.  ZZ )
46 fzoval 10876 . . . . . 6  |-  ( (
# `  B )  e.  ZZ  ->  ( 0..^ ( # `  B
) )  =  ( 0 ... ( (
# `  B )  -  1 ) ) )
4745, 46syl 15 . . . . 5  |-  ( ph  ->  ( 0..^ ( # `  B ) )  =  ( 0 ... (
( # `  B )  -  1 ) ) )
4842, 47syl5sseqr 3227 . . . 4  |-  ( ph  ->  ( 0..^ ( (
# `  B )  -  1 ) ) 
C_  ( 0..^ (
# `  B )
) )
49 efgredlemb.l . . . . 5  |-  L  =  ( ( ( # `  B )  -  1 )  -  1 )
5027simprd 449 . . . . . 6  |-  ( ph  ->  ( ( # `  B
)  -  1 )  e.  NN )
51 fzo0end 10915 . . . . . 6  |-  ( ( ( # `  B
)  -  1 )  e.  NN  ->  (
( ( # `  B
)  -  1 )  -  1 )  e.  ( 0..^ ( (
# `  B )  -  1 ) ) )
5250, 51syl 15 . . . . 5  |-  ( ph  ->  ( ( ( # `  B )  -  1 )  -  1 )  e.  ( 0..^ ( ( # `  B
)  -  1 ) ) )
5349, 52syl5eqel 2367 . . . 4  |-  ( ph  ->  L  e.  ( 0..^ ( ( # `  B
)  -  1 ) ) )
5448, 53sseldd 3181 . . 3  |-  ( ph  ->  L  e.  ( 0..^ ( # `  B
) ) )
55 ffvelrn 5663 . . 3  |-  ( ( B : ( 0..^ ( # `  B
) ) --> W  /\  L  e.  ( 0..^ ( # `  B
) ) )  -> 
( B `  L
)  e.  W )
5641, 54, 55syl2anc 642 . 2  |-  ( ph  ->  ( B `  L
)  e.  W )
5734, 56jca 518 1  |-  ( ph  ->  ( ( A `  K )  e.  W  /\  ( B `  L
)  e.  W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547    \ cdif 3149   (/)c0 3455   {csn 3640   <.cop 3643   <.cotp 3644   U_ciun 3905   class class class wbr 4023    e. cmpt 4077    _I cid 4304    X. cxp 4687   dom cdm 4689   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1oc1o 6472   2oc2o 6473   0cc0 8737   1c1 8738    < clt 8867    - cmin 9037   NNcn 9746   NN0cn0 9965   ZZcz 10024   ...cfz 10782  ..^cfzo 10870   #chash 11337  Word cword 11403   splice csplice 11407   <"cs2 11491   ~FG cefg 15015
This theorem is referenced by:  efgredlemg  15051  efgredleme  15052
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-hash 11338  df-word 11409
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