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Theorem efgsval2 15058
Description: Value of the auxiliary function  S defining a sequence of extensions (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 ) ) )
Assertion
Ref Expression
efgsval2  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( S `  ( A concat  <" B "> ) )  =  B )
Distinct variable groups:    y, z    t, n, v, w, y, z, m, x    m, M    x, n, M, t, v, w    k, m, t, x, T    k, n, v, w, y, z, W, m, t, x    .~ , m, t, x, y, z    m, I, n, t, v, w, x, y, z    D, m, t
Allowed substitution hints:    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 efgsval2
StepHypRef Expression
1 efgval.w . . . 4  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
2 efgval.r . . . 4  |-  .~  =  ( ~FG  `  I )
3 efgval2.m . . . 4  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
4 efgval2.t . . . 4  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
5 efgred.d . . . 4  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
6 efgred.s . . . 4  |-  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 ) ) )
71, 2, 3, 4, 5, 6efgsval 15056 . . 3  |-  ( ( A concat  <" B "> )  e.  dom  S  ->  ( S `  ( A concat  <" B "> ) )  =  ( ( A concat  <" B "> ) `  (
( # `  ( A concat  <" B "> ) )  -  1 ) ) )
873ad2ant3 978 . 2  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( S `  ( A concat  <" B "> ) )  =  ( ( A concat  <" B "> ) `  (
( # `  ( A concat  <" B "> ) )  -  1 ) ) )
9 lencl 11437 . . . . . . 7  |-  ( A  e. Word  W  ->  ( # `
 A )  e. 
NN0 )
1093ad2ant1 976 . . . . . 6  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( # `  A
)  e.  NN0 )
1110nn0cnd 10036 . . . . 5  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( # `  A
)  e.  CC )
12 ax-1cn 8811 . . . . 5  |-  1  e.  CC
13 pncan 9073 . . . . 5  |-  ( ( ( # `  A
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( # `  A )  +  1 )  -  1 )  =  ( # `  A
) )
1411, 12, 13sylancl 643 . . . 4  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( (
( # `  A )  +  1 )  - 
1 )  =  (
# `  A )
)
15 simp1 955 . . . . . . 7  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  A  e. Word  W )
16 simp2 956 . . . . . . . 8  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  B  e.  W )
1716s1cld 11458 . . . . . . 7  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  <" B ">  e. Word  W )
18 ccatlen 11446 . . . . . . 7  |-  ( ( A  e. Word  W  /\  <" B ">  e. Word  W )  ->  ( # `
 ( A concat  <" B "> ) )  =  ( ( # `  A
)  +  ( # `  <" B "> ) ) )
1915, 17, 18syl2anc 642 . . . . . 6  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( # `  ( A concat  <" B "> ) )  =  ( ( # `  A
)  +  ( # `  <" B "> ) ) )
20 s1len 11460 . . . . . . 7  |-  ( # `  <" B "> )  =  1
2120oveq2i 5885 . . . . . 6  |-  ( (
# `  A )  +  ( # `  <" B "> )
)  =  ( (
# `  A )  +  1 )
2219, 21syl6eq 2344 . . . . 5  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( # `  ( A concat  <" B "> ) )  =  ( ( # `  A
)  +  1 ) )
2322oveq1d 5889 . . . 4  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( ( # `
 ( A concat  <" B "> ) )  - 
1 )  =  ( ( ( # `  A
)  +  1 )  -  1 ) )
2411addid2d 9029 . . . 4  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( 0  +  ( # `  A
) )  =  (
# `  A )
)
2514, 23, 243eqtr4d 2338 . . 3  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( ( # `
 ( A concat  <" B "> ) )  - 
1 )  =  ( 0  +  ( # `  A ) ) )
2625fveq2d 5545 . 2  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( ( A concat  <" B "> ) `  ( (
# `  ( A concat  <" B "> ) )  -  1 ) )  =  ( ( A concat  <" B "> ) `  (
0  +  ( # `  A ) ) ) )
27 1nn 9773 . . . . . . 7  |-  1  e.  NN
2820, 27eqeltri 2366 . . . . . 6  |-  ( # `  <" B "> )  e.  NN
2928a1i 10 . . . . 5  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( # `  <" B "> )  e.  NN )
30 lbfzo0 10919 . . . . 5  |-  ( 0  e.  ( 0..^ (
# `  <" B "> ) )  <->  ( # `  <" B "> )  e.  NN )
3129, 30sylibr 203 . . . 4  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  0  e.  ( 0..^ ( # `  <" B "> )
) )
32 ccatval3 11449 . . . 4  |-  ( ( A  e. Word  W  /\  <" B ">  e. Word  W  /\  0  e.  ( 0..^ ( # `  <" B "> ) ) )  -> 
( ( A concat  <" B "> ) `  (
0  +  ( # `  A ) ) )  =  ( <" B "> `  0 )
)
3315, 17, 31, 32syl3anc 1182 . . 3  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( ( A concat  <" B "> ) `  ( 0  +  ( # `  A
) ) )  =  ( <" B "> `  0 )
)
34 s1fv 11462 . . . 4  |-  ( B  e.  W  ->  ( <" B "> `  0 )  =  B )
35343ad2ant2 977 . . 3  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( <" B "> `  0
)  =  B )
3633, 35eqtrd 2328 . 2  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( ( A concat  <" B "> ) `  ( 0  +  ( # `  A
) ) )  =  B )
378, 26, 363eqtrd 2332 1  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( S `  ( A concat  <" B "> ) )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560    \ cdif 3162   (/)c0 3468   {csn 3653   <.cop 3656   <.cotp 3657   U_ciun 3921    e. cmpt 4093    _I cid 4320    X. cxp 4703   dom cdm 4705   ran crn 4706   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1oc1o 6488   2oc2o 6489   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    - cmin 9053   NNcn 9762   NN0cn0 9981   ...cfz 10798  ..^cfzo 10886   #chash 11353  Word cword 11419   concat cconcat 11420   <"cs1 11421   splice csplice 11423   <"cs2 11507   ~FG cefg 15031
This theorem is referenced by:  efgsfo  15064  efgredlemd  15069  efgrelexlemb  15075
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-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-hash 11354  df-word 11425  df-concat 11426  df-s1 11427
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