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Theorem efgredlemf 15066
Description: Lemma for efgredleme 15068. (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 15055 . . . . . . 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 3311 . . . . 5  |-  ( A  e.  (Word  W  \  { (/) } )  ->  A  e. Word  W )
1210, 11syl 15 . . . 4  |-  ( ph  ->  A  e. Word  W )
13 wrdf 11435 . . . 4  |-  ( A  e. Word  W  ->  A : ( 0..^ (
# `  A )
) --> W )
1412, 13syl 15 . . 3  |-  ( ph  ->  A : ( 0..^ ( # `  A
) ) --> W )
15 fzossfz 10908 . . . . 5  |-  ( 0..^ ( ( # `  A
)  -  1 ) )  C_  ( 0 ... ( ( # `  A )  -  1 ) )
16 lencl 11437 . . . . . . . 8  |-  ( A  e. Word  W  ->  ( # `
 A )  e. 
NN0 )
1712, 16syl 15 . . . . . . 7  |-  ( ph  ->  ( # `  A
)  e.  NN0 )
1817nn0zd 10131 . . . . . 6  |-  ( ph  ->  ( # `  A
)  e.  ZZ )
19 fzoval 10892 . . . . . 6  |-  ( (
# `  A )  e.  ZZ  ->  ( 0..^ ( # `  A
) )  =  ( 0 ... ( (
# `  A )  -  1 ) ) )
2018, 19syl 15 . . . . 5  |-  ( ph  ->  ( 0..^ ( # `  A ) )  =  ( 0 ... (
( # `  A )  -  1 ) ) )
2115, 20syl5sseqr 3240 . . . 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 15065 . . . . . . 7  |-  ( ph  ->  ( ( ( # `  A )  -  1 )  e.  NN  /\  ( ( # `  B
)  -  1 )  e.  NN ) )
2827simpld 445 . . . . . 6  |-  ( ph  ->  ( ( # `  A
)  -  1 )  e.  NN )
29 fzo0end 10931 . . . . . 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 2380 . . . 4  |-  ( ph  ->  K  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) )
3221, 31sseldd 3194 . . 3  |-  ( ph  ->  K  e.  ( 0..^ ( # `  A
) ) )
33 ffvelrn 5679 . . 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 15055 . . . . . . 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 3311 . . . . 5  |-  ( B  e.  (Word  W  \  { (/) } )  ->  B  e. Word  W )
3937, 38syl 15 . . . 4  |-  ( ph  ->  B  e. Word  W )
40 wrdf 11435 . . . 4  |-  ( B  e. Word  W  ->  B : ( 0..^ (
# `  B )
) --> W )
4139, 40syl 15 . . 3  |-  ( ph  ->  B : ( 0..^ ( # `  B
) ) --> W )
42 fzossfz 10908 . . . . 5  |-  ( 0..^ ( ( # `  B
)  -  1 ) )  C_  ( 0 ... ( ( # `  B )  -  1 ) )
43 lencl 11437 . . . . . . . 8  |-  ( B  e. Word  W  ->  ( # `
 B )  e. 
NN0 )
4439, 43syl 15 . . . . . . 7  |-  ( ph  ->  ( # `  B
)  e.  NN0 )
4544nn0zd 10131 . . . . . 6  |-  ( ph  ->  ( # `  B
)  e.  ZZ )
46 fzoval 10892 . . . . . 6  |-  ( (
# `  B )  e.  ZZ  ->  ( 0..^ ( # `  B
) )  =  ( 0 ... ( (
# `  B )  -  1 ) ) )
4745, 46syl 15 . . . . 5  |-  ( ph  ->  ( 0..^ ( # `  B ) )  =  ( 0 ... (
( # `  B )  -  1 ) ) )
4842, 47syl5sseqr 3240 . . . 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 10931 . . . . . 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 2380 . . . 4  |-  ( ph  ->  L  e.  ( 0..^ ( ( # `  B
)  -  1 ) ) )
5448, 53sseldd 3194 . . 3  |-  ( ph  ->  L  e.  ( 0..^ ( # `  B
) ) )
55 ffvelrn 5679 . . 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 1632    e. wcel 1696   A.wral 2556   {crab 2560    \ cdif 3162   (/)c0 3468   {csn 3653   <.cop 3656   <.cotp 3657   U_ciun 3921   class class class wbr 4039    e. cmpt 4093    _I cid 4320    X. cxp 4703   dom cdm 4705   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1oc1o 6488   2oc2o 6489   0cc0 8753   1c1 8754    < clt 8883    - cmin 9053   NNcn 9762   NN0cn0 9981   ZZcz 10040   ...cfz 10798  ..^cfzo 10886   #chash 11353  Word cword 11419   splice csplice 11423   <"cs2 11507   ~FG cefg 15031
This theorem is referenced by:  efgredlemg  15067  efgredleme  15068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-hash 11354  df-word 11425
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