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Theorem efgval2 15283
Description: Value of the free group construction. (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 ) "> >. )
) )
Assertion
Ref Expression
efgval2  |-  .~  =  |^| { r  |  ( r  Er  W  /\  A. x  e.  W  ran  ( T `  x ) 
C_  [ x ]
r ) }
Distinct variable groups:    y, r,
z    v, n, w, y, z, r, x    n, M    v, r, w, x, M    T, r, x    n, W, r, v, w    x, y, z, W    .~ , r, x, y, z    n, I, r, v, w, x, y, z
Allowed substitution hints:    .~ ( w, v, n)    T( y, z, w, v, n)    M( y,
z)

Proof of Theorem efgval2
Dummy variables  a 
b  u  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.w . . 3  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
2 efgval.r . . 3  |-  .~  =  ( ~FG  `  I )
31, 2efgval 15276 . 2  |-  .~  =  |^| { r  |  ( r  Er  W  /\  A. x  e.  W  A. m  e.  ( 0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. m ,  m ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) }
4 efgval2.m . . . . . . . . . . 11  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
5 efgval2.t . . . . . . . . . . 11  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
61, 2, 4, 5efgtf 15281 . . . . . . . . . 10  |-  ( x  e.  W  ->  (
( T `  x
)  =  ( m  e.  ( 0 ... ( # `  x
) ) ,  u  e.  ( I  X.  2o )  |->  ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )
)  /\  ( T `  x ) : ( ( 0 ... ( # `
 x ) )  X.  ( I  X.  2o ) ) --> W ) )
76simpld 446 . . . . . . . . 9  |-  ( x  e.  W  ->  ( T `  x )  =  ( m  e.  ( 0 ... ( # `
 x ) ) ,  u  e.  ( I  X.  2o ) 
|->  ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )
) )
87rneqd 5037 . . . . . . . 8  |-  ( x  e.  W  ->  ran  ( T `  x )  =  ran  ( m  e.  ( 0 ... ( # `  x
) ) ,  u  e.  ( I  X.  2o )  |->  ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )
) )
98sseq1d 3318 . . . . . . 7  |-  ( x  e.  W  ->  ( ran  ( T `  x
)  C_  [ x ] r  <->  ran  ( m  e.  ( 0 ... ( # `  x
) ) ,  u  e.  ( I  X.  2o )  |->  ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )
)  C_  [ x ] r ) )
10 dfss3 3281 . . . . . . . 8  |-  ( ran  ( m  e.  ( 0 ... ( # `  x ) ) ,  u  e.  ( I  X.  2o )  |->  ( x splice  <. m ,  m ,  <" u ( M `  u ) "> >. )
)  C_  [ x ] r  <->  A. a  e.  ran  ( m  e.  ( 0 ... ( # `
 x ) ) ,  u  e.  ( I  X.  2o ) 
|->  ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )
) a  e.  [
x ] r )
11 ovex 6045 . . . . . . . . . . 11  |-  ( x splice  <. m ,  m , 
<" u ( M `
 u ) "> >. )  e.  _V
1211rgen2w 2717 . . . . . . . . . 10  |-  A. m  e.  ( 0 ... ( # `
 x ) ) A. u  e.  ( I  X.  2o ) ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )  e.  _V
13 eqid 2387 . . . . . . . . . . 11  |-  ( m  e.  ( 0 ... ( # `  x
) ) ,  u  e.  ( I  X.  2o )  |->  ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )
)  =  ( m  e.  ( 0 ... ( # `  x
) ) ,  u  e.  ( I  X.  2o )  |->  ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )
)
14 vex 2902 . . . . . . . . . . . . 13  |-  a  e. 
_V
15 vex 2902 . . . . . . . . . . . . 13  |-  x  e. 
_V
1614, 15elec 6880 . . . . . . . . . . . 12  |-  ( a  e.  [ x ]
r  <->  x r a )
17 breq2 4157 . . . . . . . . . . . 12  |-  ( a  =  ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )  ->  ( x r a  <-> 
x r ( x splice  <. m ,  m , 
<" u ( M `
 u ) "> >. ) ) )
1816, 17syl5bb 249 . . . . . . . . . . 11  |-  ( a  =  ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )  ->  ( a  e.  [
x ] r  <->  x r
( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )
) )
1913, 18ralrnmpt2 6123 . . . . . . . . . 10  |-  ( A. m  e.  ( 0 ... ( # `  x
) ) A. u  e.  ( I  X.  2o ) ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )  e.  _V  ->  ( A. a  e.  ran  ( m  e.  ( 0 ... ( # `  x
) ) ,  u  e.  ( I  X.  2o )  |->  ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )
) a  e.  [
x ] r  <->  A. m  e.  ( 0 ... ( # `
 x ) ) A. u  e.  ( I  X.  2o ) x r ( x splice  <. m ,  m , 
<" u ( M `
 u ) "> >. ) ) )
2012, 19ax-mp 8 . . . . . . . . 9  |-  ( A. a  e.  ran  ( m  e.  ( 0 ... ( # `  x
) ) ,  u  e.  ( I  X.  2o )  |->  ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )
) a  e.  [
x ] r  <->  A. m  e.  ( 0 ... ( # `
 x ) ) A. u  e.  ( I  X.  2o ) x r ( x splice  <. m ,  m , 
<" u ( M `
 u ) "> >. ) )
21 id 20 . . . . . . . . . . . . . . . 16  |-  ( u  =  <. a ,  b
>.  ->  u  =  <. a ,  b >. )
22 fveq2 5668 . . . . . . . . . . . . . . . . 17  |-  ( u  =  <. a ,  b
>.  ->  ( M `  u )  =  ( M `  <. a ,  b >. )
)
23 df-ov 6023 . . . . . . . . . . . . . . . . 17  |-  ( a M b )  =  ( M `  <. a ,  b >. )
2422, 23syl6eqr 2437 . . . . . . . . . . . . . . . 16  |-  ( u  =  <. a ,  b
>.  ->  ( M `  u )  =  ( a M b ) )
2521, 24s2eqd 11753 . . . . . . . . . . . . . . 15  |-  ( u  =  <. a ,  b
>.  ->  <" u ( M `  u ) ">  =  <"
<. a ,  b >.
( a M b ) "> )
2625oteq3d 3940 . . . . . . . . . . . . . 14  |-  ( u  =  <. a ,  b
>.  ->  <. m ,  m ,  <" u ( M `  u ) "> >.  =  <. m ,  m ,  <"
<. a ,  b >.
( a M b ) "> >. )
2726oveq2d 6036 . . . . . . . . . . . . 13  |-  ( u  =  <. a ,  b
>.  ->  ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )  =  ( x splice  <. m ,  m ,  <" <. a ,  b >. (
a M b ) "> >. )
)
2827breq2d 4165 . . . . . . . . . . . 12  |-  ( u  =  <. a ,  b
>.  ->  ( x r ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )  <->  x r ( x splice  <. m ,  m ,  <" <. a ,  b >. (
a M b ) "> >. )
) )
2928ralxp 4956 . . . . . . . . . . 11  |-  ( A. u  e.  ( I  X.  2o ) x r ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )  <->  A. a  e.  I  A. b  e.  2o  x
r ( x splice  <. m ,  m ,  <" <. a ,  b >. (
a M b ) "> >. )
)
30 eqidd 2388 . . . . . . . . . . . . . . . . 17  |-  ( ( a  e.  I  /\  b  e.  2o )  -> 
<. a ,  b >.  =  <. a ,  b
>. )
314efgmval 15271 . . . . . . . . . . . . . . . . 17  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( a M b )  =  <. a ,  ( 1o  \ 
b ) >. )
3230, 31s2eqd 11753 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  <" <. a ,  b >. (
a M b ) ">  =  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> )
3332oteq3d 3940 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  I  /\  b  e.  2o )  -> 
<. m ,  m , 
<" <. a ,  b
>. ( a M b ) "> >.  =  <. m ,  m ,  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> >. )
3433oveq2d 6036 . . . . . . . . . . . . . 14  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( x splice  <. m ,  m ,  <" <. a ,  b >. (
a M b ) "> >. )  =  ( x splice  <. m ,  m ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) )
3534breq2d 4165 . . . . . . . . . . . . 13  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( x r ( x splice  <. m ,  m ,  <" <. a ,  b >. (
a M b ) "> >. )  <->  x r ( x splice  <. m ,  m ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) )
3635ralbidva 2665 . . . . . . . . . . . 12  |-  ( a  e.  I  ->  ( A. b  e.  2o  x r ( x splice  <. m ,  m , 
<" <. a ,  b
>. ( a M b ) "> >. )  <->  A. b  e.  2o  x
r ( x splice  <. m ,  m ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) )
3736ralbiia 2681 . . . . . . . . . . 11  |-  ( A. a  e.  I  A. b  e.  2o  x
r ( x splice  <. m ,  m ,  <" <. a ,  b >. (
a M b ) "> >. )  <->  A. a  e.  I  A. b  e.  2o  x
r ( x splice  <. m ,  m ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) )
3829, 37bitri 241 . . . . . . . . . 10  |-  ( A. u  e.  ( I  X.  2o ) x r ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )  <->  A. a  e.  I  A. b  e.  2o  x
r ( x splice  <. m ,  m ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) )
3938ralbii 2673 . . . . . . . . 9  |-  ( A. m  e.  ( 0 ... ( # `  x
) ) A. u  e.  ( I  X.  2o ) x r ( x splice  <. m ,  m ,  <" u ( M `  u ) "> >. )  <->  A. m  e.  ( 0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. m ,  m ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) )
4020, 39bitri 241 . . . . . . . 8  |-  ( A. a  e.  ran  ( m  e.  ( 0 ... ( # `  x
) ) ,  u  e.  ( I  X.  2o )  |->  ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )
) a  e.  [
x ] r  <->  A. m  e.  ( 0 ... ( # `
 x ) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. m ,  m , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )
)
4110, 40bitri 241 . . . . . . 7  |-  ( ran  ( m  e.  ( 0 ... ( # `  x ) ) ,  u  e.  ( I  X.  2o )  |->  ( x splice  <. m ,  m ,  <" u ( M `  u ) "> >. )
)  C_  [ x ] r  <->  A. m  e.  ( 0 ... ( # `
 x ) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. m ,  m , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )
)
429, 41syl6bb 253 . . . . . 6  |-  ( x  e.  W  ->  ( ran  ( T `  x
)  C_  [ x ] r  <->  A. m  e.  ( 0 ... ( # `
 x ) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. m ,  m , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )
) )
4342ralbiia 2681 . . . . 5  |-  ( A. x  e.  W  ran  ( T `  x ) 
C_  [ x ]
r  <->  A. x  e.  W  A. m  e.  (
0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. m ,  m ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) )
4443anbi2i 676 . . . 4  |-  ( ( r  Er  W  /\  A. x  e.  W  ran  ( T `  x ) 
C_  [ x ]
r )  <->  ( r  Er  W  /\  A. x  e.  W  A. m  e.  ( 0 ... ( # `
 x ) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. m ,  m , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )
) )
4544abbii 2499 . . 3  |-  { r  |  ( r  Er  W  /\  A. x  e.  W  ran  ( T `
 x )  C_  [ x ] r ) }  =  { r  |  ( r  Er  W  /\  A. x  e.  W  A. m  e.  ( 0 ... ( # `
 x ) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. m ,  m , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )
) }
4645inteqi 3996 . 2  |-  |^| { r  |  ( r  Er  W  /\  A. x  e.  W  ran  ( T `
 x )  C_  [ x ] r ) }  =  |^| { r  |  ( r  Er  W  /\  A. x  e.  W  A. m  e.  ( 0 ... ( # `
 x ) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. m ,  m , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )
) }
473, 46eqtr4i 2410 1  |-  .~  =  |^| { r  |  ( r  Er  W  /\  A. x  e.  W  ran  ( T `  x ) 
C_  [ x ]
r ) }
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2373   A.wral 2649   _Vcvv 2899    \ cdif 3260    C_ wss 3263   <.cop 3760   <.cotp 3761   |^|cint 3992   class class class wbr 4153    e. cmpt 4207    _I cid 4434    X. cxp 4816   ran crn 4819   -->wf 5390   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022   1oc1o 6653   2oc2o 6654    Er wer 6838   [cec 6839   0cc0 8923   ...cfz 10975   #chash 11545  Word cword 11644   splice csplice 11648   <"cs2 11732   ~FG cefg 15265
This theorem is referenced by:  efgi2  15284  efgrelexlemb  15309  efgcpbllemb  15314
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-ot 3767  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-ec 6843  df-map 6956  df-pm 6957  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-n0 10154  df-z 10215  df-uz 10421  df-fz 10976  df-fzo 11066  df-hash 11546  df-word 11650  df-concat 11651  df-s1 11652  df-substr 11653  df-splice 11654  df-s2 11739  df-efg 15268
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