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Theorem efgsval2 15357
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 15355 . . 3  |-  ( ( A concat  <" B "> )  e.  dom  S  ->  ( S `  ( A concat  <" B "> ) )  =  ( ( A concat  <" B "> ) `  (
( # `  ( A concat  <" B "> ) )  -  1 ) ) )
873ad2ant3 980 . 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 11727 . . . . . . 7  |-  ( A  e. Word  W  ->  ( # `
 A )  e. 
NN0 )
1093ad2ant1 978 . . . . . 6  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( # `  A
)  e.  NN0 )
1110nn0cnd 10268 . . . . 5  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( # `  A
)  e.  CC )
12 ax-1cn 9040 . . . . 5  |-  1  e.  CC
13 pncan 9303 . . . . 5  |-  ( ( ( # `  A
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( # `  A )  +  1 )  -  1 )  =  ( # `  A
) )
1411, 12, 13sylancl 644 . . . 4  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( (
( # `  A )  +  1 )  - 
1 )  =  (
# `  A )
)
15 simp1 957 . . . . . . 7  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  A  e. Word  W )
16 simp2 958 . . . . . . . 8  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  B  e.  W )
1716s1cld 11748 . . . . . . 7  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  <" B ">  e. Word  W )
18 ccatlen 11736 . . . . . . 7  |-  ( ( A  e. Word  W  /\  <" B ">  e. Word  W )  ->  ( # `
 ( A concat  <" B "> ) )  =  ( ( # `  A
)  +  ( # `  <" B "> ) ) )
1915, 17, 18syl2anc 643 . . . . . 6  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( # `  ( A concat  <" B "> ) )  =  ( ( # `  A
)  +  ( # `  <" B "> ) ) )
20 s1len 11750 . . . . . . 7  |-  ( # `  <" B "> )  =  1
2120oveq2i 6084 . . . . . 6  |-  ( (
# `  A )  +  ( # `  <" B "> )
)  =  ( (
# `  A )  +  1 )
2219, 21syl6eq 2483 . . . . 5  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( # `  ( A concat  <" B "> ) )  =  ( ( # `  A
)  +  1 ) )
2322oveq1d 6088 . . . 4  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( ( # `
 ( A concat  <" B "> ) )  - 
1 )  =  ( ( ( # `  A
)  +  1 )  -  1 ) )
2411addid2d 9259 . . . 4  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( 0  +  ( # `  A
) )  =  (
# `  A )
)
2514, 23, 243eqtr4d 2477 . . 3  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( ( # `
 ( A concat  <" B "> ) )  - 
1 )  =  ( 0  +  ( # `  A ) ) )
2625fveq2d 5724 . 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 10003 . . . . . . 7  |-  1  e.  NN
2820, 27eqeltri 2505 . . . . . 6  |-  ( # `  <" B "> )  e.  NN
2928a1i 11 . . . . 5  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( # `  <" B "> )  e.  NN )
30 lbfzo0 11162 . . . . 5  |-  ( 0  e.  ( 0..^ (
# `  <" B "> ) )  <->  ( # `  <" B "> )  e.  NN )
3129, 30sylibr 204 . . . 4  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  0  e.  ( 0..^ ( # `  <" B "> )
) )
32 ccatval3 11739 . . . 4  |-  ( ( A  e. Word  W  /\  <" B ">  e. Word  W  /\  0  e.  ( 0..^ ( # `  <" B "> ) ) )  -> 
( ( A concat  <" B "> ) `  (
0  +  ( # `  A ) ) )  =  ( <" B "> `  0 )
)
3315, 17, 31, 32syl3anc 1184 . . 3  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( ( A concat  <" B "> ) `  ( 0  +  ( # `  A
) ) )  =  ( <" B "> `  0 )
)
34 s1fv 11752 . . . 4  |-  ( B  e.  W  ->  ( <" B "> `  0 )  =  B )
35343ad2ant2 979 . . 3  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( <" B "> `  0
)  =  B )
3633, 35eqtrd 2467 . 2  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( ( A concat  <" B "> ) `  ( 0  +  ( # `  A
) ) )  =  B )
378, 26, 363eqtrd 2471 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701    \ cdif 3309   (/)c0 3620   {csn 3806   <.cop 3809   <.cotp 3810   U_ciun 4085    e. cmpt 4258    _I cid 4485    X. cxp 4868   dom cdm 4870   ran crn 4871   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1oc1o 6709   2oc2o 6710   CCcc 8980   0cc0 8982   1c1 8983    + caddc 8985    - cmin 9283   NNcn 9992   NN0cn0 10213   ...cfz 11035  ..^cfzo 11127   #chash 11610  Word cword 11709   concat cconcat 11710   <"cs1 11711   splice csplice 11713   <"cs2 11797   ~FG cefg 15330
This theorem is referenced by:  efgsfo  15363  efgredlemd  15368  efgrelexlemb  15374
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-hash 11611  df-word 11715  df-concat 11716  df-s1 11717
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