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Theorem efgval2 15311
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 15304 . 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 15309 . . . . . . . . . 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 5056 . . . . . . . 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 3335 . . . . . . 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 3298 . . . . . . . 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 6065 . . . . . . . . . . 11  |-  ( x splice  <. m ,  m , 
<" u ( M `
 u ) "> >. )  e.  _V
1211rgen2w 2734 . . . . . . . . . 10  |-  A. m  e.  ( 0 ... ( # `
 x ) ) A. u  e.  ( I  X.  2o ) ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )  e.  _V
13 eqid 2404 . . . . . . . . . . 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 2919 . . . . . . . . . . . . 13  |-  a  e. 
_V
15 vex 2919 . . . . . . . . . . . . 13  |-  x  e. 
_V
1614, 15elec 6903 . . . . . . . . . . . 12  |-  ( a  e.  [ x ]
r  <->  x r a )
17 breq2 4176 . . . . . . . . . . . 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 6143 . . . . . . . . . 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 5687 . . . . . . . . . . . . . . . . 17  |-  ( u  =  <. a ,  b
>.  ->  ( M `  u )  =  ( M `  <. a ,  b >. )
)
23 df-ov 6043 . . . . . . . . . . . . . . . . 17  |-  ( a M b )  =  ( M `  <. a ,  b >. )
2422, 23syl6eqr 2454 . . . . . . . . . . . . . . . 16  |-  ( u  =  <. a ,  b
>.  ->  ( M `  u )  =  ( a M b ) )
2521, 24s2eqd 11781 . . . . . . . . . . . . . . 15  |-  ( u  =  <. a ,  b
>.  ->  <" u ( M `  u ) ">  =  <"
<. a ,  b >.
( a M b ) "> )
2625oteq3d 3958 . . . . . . . . . . . . . 14  |-  ( u  =  <. a ,  b
>.  ->  <. m ,  m ,  <" u ( M `  u ) "> >.  =  <. m ,  m ,  <"
<. a ,  b >.
( a M b ) "> >. )
2726oveq2d 6056 . . . . . . . . . . . . 13  |-  ( u  =  <. a ,  b
>.  ->  ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )  =  ( x splice  <. m ,  m ,  <" <. a ,  b >. (
a M b ) "> >. )
)
2827breq2d 4184 . . . . . . . . . . . 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 4975 . . . . . . . . . . 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 2405 . . . . . . . . . . . . . . . . 17  |-  ( ( a  e.  I  /\  b  e.  2o )  -> 
<. a ,  b >.  =  <. a ,  b
>. )
314efgmval 15299 . . . . . . . . . . . . . . . . 17  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( a M b )  =  <. a ,  ( 1o  \ 
b ) >. )
3230, 31s2eqd 11781 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  <" <. a ,  b >. (
a M b ) ">  =  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> )
3332oteq3d 3958 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  I  /\  b  e.  2o )  -> 
<. m ,  m , 
<" <. a ,  b
>. ( a M b ) "> >.  =  <. m ,  m ,  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> >. )
3433oveq2d 6056 . . . . . . . . . . . . . 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 4184 . . . . . . . . . . . . 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 2682 . . . . . . . . . . . 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 2698 . . . . . . . . . . 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 2690 . . . . . . . . 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 2698 . . . . 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 2516 . . 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 4014 . 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 2427 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 1721   {cab 2390   A.wral 2666   _Vcvv 2916    \ cdif 3277    C_ wss 3280   <.cop 3777   <.cotp 3778   |^|cint 4010   class class class wbr 4172    e. cmpt 4226    _I cid 4453    X. cxp 4835   ran crn 4838   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1oc1o 6676   2oc2o 6677    Er wer 6861   [cec 6862   0cc0 8946   ...cfz 10999   #chash 11573  Word cword 11672   splice csplice 11676   <"cs2 11760   ~FG cefg 15293
This theorem is referenced by:  efgi2  15312  efgrelexlemb  15337  efgcpbllemb  15342
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-ec 6866  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-hash 11574  df-word 11678  df-concat 11679  df-s1 11680  df-substr 11681  df-splice 11682  df-s2 11767  df-efg 15296
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