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Theorem efgval2 15348
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 15341 . 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 15346 . . . . . . . . . 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 5089 . . . . . . . 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 3367 . . . . . . 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 3330 . . . . . . . 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 6098 . . . . . . . . . . 11  |-  ( x splice  <. m ,  m , 
<" u ( M `
 u ) "> >. )  e.  _V
1211rgen2w 2766 . . . . . . . . . 10  |-  A. m  e.  ( 0 ... ( # `
 x ) ) A. u  e.  ( I  X.  2o ) ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )  e.  _V
13 eqid 2435 . . . . . . . . . . 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 2951 . . . . . . . . . . . . 13  |-  a  e. 
_V
15 vex 2951 . . . . . . . . . . . . 13  |-  x  e. 
_V
1614, 15elec 6936 . . . . . . . . . . . 12  |-  ( a  e.  [ x ]
r  <->  x r a )
17 breq2 4208 . . . . . . . . . . . 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 6176 . . . . . . . . . 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 5720 . . . . . . . . . . . . . . . . 17  |-  ( u  =  <. a ,  b
>.  ->  ( M `  u )  =  ( M `  <. a ,  b >. )
)
23 df-ov 6076 . . . . . . . . . . . . . . . . 17  |-  ( a M b )  =  ( M `  <. a ,  b >. )
2422, 23syl6eqr 2485 . . . . . . . . . . . . . . . 16  |-  ( u  =  <. a ,  b
>.  ->  ( M `  u )  =  ( a M b ) )
2521, 24s2eqd 11818 . . . . . . . . . . . . . . 15  |-  ( u  =  <. a ,  b
>.  ->  <" u ( M `  u ) ">  =  <"
<. a ,  b >.
( a M b ) "> )
2625oteq3d 3990 . . . . . . . . . . . . . 14  |-  ( u  =  <. a ,  b
>.  ->  <. m ,  m ,  <" u ( M `  u ) "> >.  =  <. m ,  m ,  <"
<. a ,  b >.
( a M b ) "> >. )
2726oveq2d 6089 . . . . . . . . . . . . 13  |-  ( u  =  <. a ,  b
>.  ->  ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )  =  ( x splice  <. m ,  m ,  <" <. a ,  b >. (
a M b ) "> >. )
)
2827breq2d 4216 . . . . . . . . . . . 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 5008 . . . . . . . . . . 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 2436 . . . . . . . . . . . . . . . . 17  |-  ( ( a  e.  I  /\  b  e.  2o )  -> 
<. a ,  b >.  =  <. a ,  b
>. )
314efgmval 15336 . . . . . . . . . . . . . . . . 17  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( a M b )  =  <. a ,  ( 1o  \ 
b ) >. )
3230, 31s2eqd 11818 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  <" <. a ,  b >. (
a M b ) ">  =  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> )
3332oteq3d 3990 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  I  /\  b  e.  2o )  -> 
<. m ,  m , 
<" <. a ,  b
>. ( a M b ) "> >.  =  <. m ,  m ,  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> >. )
3433oveq2d 6089 . . . . . . . . . . . . . 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 4216 . . . . . . . . . . . . 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 2713 . . . . . . . . . . . 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 2729 . . . . . . . . . . 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 2721 . . . . . . . . 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 2729 . . . . 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 2547 . . 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 4046 . 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 2458 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 1652    e. wcel 1725   {cab 2421   A.wral 2697   _Vcvv 2948    \ cdif 3309    C_ wss 3312   <.cop 3809   <.cotp 3810   |^|cint 4042   class class class wbr 4204    e. cmpt 4258    _I cid 4485    X. cxp 4868   ran crn 4871   -->wf 5442   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1oc1o 6709   2oc2o 6710    Er wer 6894   [cec 6895   0cc0 8982   ...cfz 11035   #chash 11610  Word cword 11709   splice csplice 11713   <"cs2 11797   ~FG cefg 15330
This theorem is referenced by:  efgi2  15349  efgrelexlemb  15374  efgcpbllemb  15379
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-ot 3816  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-2o 6717  df-oadd 6720  df-er 6897  df-ec 6899  df-map 7012  df-pm 7013  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  df-substr 11718  df-splice 11719  df-s2 11804  df-efg 15333
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