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Theorem efgsval 15365
Description: Value of the auxiliary function  S defining a sequence of extensions (Contributed by Mario Carneiro, 27-Sep-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
efgsval  |-  ( F  e.  dom  S  -> 
( S `  F
)  =  ( F `
 ( ( # `  F )  -  1 ) ) )
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:    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)    F( x, y, z, w, v, t, k, m, n)    I( k)    M( y, z, k)

Proof of Theorem efgsval
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 id 21 . . . 4  |-  ( f  =  F  ->  f  =  F )
2 fveq2 5730 . . . . 5  |-  ( f  =  F  ->  ( # `
 f )  =  ( # `  F
) )
32oveq1d 6098 . . . 4  |-  ( f  =  F  ->  (
( # `  f )  -  1 )  =  ( ( # `  F
)  -  1 ) )
41, 3fveq12d 5736 . . 3  |-  ( f  =  F  ->  (
f `  ( ( # `
 f )  - 
1 ) )  =  ( F `  (
( # `  F )  -  1 ) ) )
5 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 ) ) )
6 id 21 . . . . . 6  |-  ( m  =  f  ->  m  =  f )
7 fveq2 5730 . . . . . . 7  |-  ( m  =  f  ->  ( # `
 m )  =  ( # `  f
) )
87oveq1d 6098 . . . . . 6  |-  ( m  =  f  ->  (
( # `  m )  -  1 )  =  ( ( # `  f
)  -  1 ) )
96, 8fveq12d 5736 . . . . 5  |-  ( m  =  f  ->  (
m `  ( ( # `
 m )  - 
1 ) )  =  ( f `  (
( # `  f )  -  1 ) ) )
109cbvmptv 4302 . . . 4  |-  ( 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 ) ) )  =  ( f  e.  { t  e.  (Word  W  \  { (/)
} )  |  ( ( t `  0
)  e.  D  /\  A. k  e.  ( 1..^ ( # `  t
) ) ( t `
 k )  e. 
ran  ( T `  ( t `  (
k  -  1 ) ) ) ) } 
|->  ( f `  (
( # `  f )  -  1 ) ) )
115, 10eqtri 2458 . . 3  |-  S  =  ( f  e.  {
t  e.  (Word  W  \  { (/) } )  |  ( ( t ` 
0 )  e.  D  /\  A. k  e.  ( 1..^ ( # `  t
) ) ( t `
 k )  e. 
ran  ( T `  ( t `  (
k  -  1 ) ) ) ) } 
|->  ( f `  (
( # `  f )  -  1 ) ) )
12 fvex 5744 . . 3  |-  ( F `
 ( ( # `  F )  -  1 ) )  e.  _V
134, 11, 12fvmpt 5808 . 2  |-  ( F  e.  { t  e.  (Word  W  \  { (/)
} )  |  ( ( t `  0
)  e.  D  /\  A. k  e.  ( 1..^ ( # `  t
) ) ( t `
 k )  e. 
ran  ( T `  ( t `  (
k  -  1 ) ) ) ) }  ->  ( S `  F )  =  ( F `  ( (
# `  F )  -  1 ) ) )
14 efgval.w . . . 4  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
15 efgval.r . . . 4  |-  .~  =  ( ~FG  `  I )
16 efgval2.m . . . 4  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
17 efgval2.t . . . 4  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
18 efgred.d . . . 4  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
1914, 15, 16, 17, 18, 5efgsf 15363 . . 3  |-  S : { t  e.  (Word 
W  \  { (/) } )  |  ( ( t `
 0 )  e.  D  /\  A. k  e.  ( 1..^ ( # `  t ) ) ( t `  k )  e.  ran  ( T `
 ( t `  ( k  -  1 ) ) ) ) } --> W
2019fdmi 5598 . 2  |-  dom  S  =  { t  e.  (Word 
W  \  { (/) } )  |  ( ( t `
 0 )  e.  D  /\  A. k  e.  ( 1..^ ( # `  t ) ) ( t `  k )  e.  ran  ( T `
 ( t `  ( k  -  1 ) ) ) ) }
2113, 20eleq2s 2530 1  |-  ( F  e.  dom  S  -> 
( S `  F
)  =  ( F `
 ( ( # `  F )  -  1 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711    \ cdif 3319   (/)c0 3630   {csn 3816   <.cop 3819   <.cotp 3820   U_ciun 4095    e. cmpt 4268    _I cid 4495    X. cxp 4878   dom cdm 4880   ran crn 4881   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   1oc1o 6719   2oc2o 6720   0cc0 8992   1c1 8993    - cmin 9293   ...cfz 11045  ..^cfzo 11137   #chash 11620  Word cword 11719   splice csplice 11723   <"cs2 11807   ~FG cefg 15340
This theorem is referenced by:  efgsdmi  15366  efgsval2  15367  efgsrel  15368  efgs1b  15370  efgsp1  15371  efgsfo  15373  efgredlema  15374  efgredlemd  15378  efgredlem  15381  efgredeu  15386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-fzo 11138  df-hash 11621  df-word 11725
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