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Theorem efgval2 15033
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 15026 . 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 15031 . . . . . . . . . 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 445 . . . . . . . . 9  |-  ( x  e.  W  ->  ( T `  x )  =  ( m  e.  ( 0 ... ( # `
 x ) ) ,  u  e.  ( I  X.  2o ) 
|->  ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )
) )
87rneqd 4906 . . . . . . . 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 3205 . . . . . . 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 3170 . . . . . . . 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 5883 . . . . . . . . . . 11  |-  ( x splice  <. m ,  m , 
<" u ( M `
 u ) "> >. )  e.  _V
1211rgen2w 2611 . . . . . . . . . 10  |-  A. m  e.  ( 0 ... ( # `
 x ) ) A. u  e.  ( I  X.  2o ) ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )  e.  _V
13 eqid 2283 . . . . . . . . . . 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 2791 . . . . . . . . . . . . 13  |-  a  e. 
_V
15 vex 2791 . . . . . . . . . . . . 13  |-  x  e. 
_V
1614, 15elec 6699 . . . . . . . . . . . 12  |-  ( a  e.  [ x ]
r  <->  x r a )
17 breq2 4027 . . . . . . . . . . . 12  |-  ( a  =  ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )  ->  ( x r a  <-> 
x r ( x splice  <. m ,  m , 
<" u ( M `
 u ) "> >. ) ) )
1816, 17syl5bb 248 . . . . . . . . . . 11  |-  ( a  =  ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )  ->  ( a  e.  [
x ] r  <->  x r
( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )
) )
1913, 18ralrnmpt2 5958 . . . . . . . . . 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 19 . . . . . . . . . . . . . . . 16  |-  ( u  =  <. a ,  b
>.  ->  u  =  <. a ,  b >. )
22 fveq2 5525 . . . . . . . . . . . . . . . . 17  |-  ( u  =  <. a ,  b
>.  ->  ( M `  u )  =  ( M `  <. a ,  b >. )
)
23 df-ov 5861 . . . . . . . . . . . . . . . . 17  |-  ( a M b )  =  ( M `  <. a ,  b >. )
2422, 23syl6eqr 2333 . . . . . . . . . . . . . . . 16  |-  ( u  =  <. a ,  b
>.  ->  ( M `  u )  =  ( a M b ) )
2521, 24s2eqd 11512 . . . . . . . . . . . . . . 15  |-  ( u  =  <. a ,  b
>.  ->  <" u ( M `  u ) ">  =  <"
<. a ,  b >.
( a M b ) "> )
26 oteq3 3807 . . . . . . . . . . . . . . 15  |-  ( <" u ( M `
 u ) ">  =  <" <. a ,  b >. (
a M b ) ">  ->  <. m ,  m ,  <" u
( M `  u
) "> >.  =  <. m ,  m ,  <"
<. a ,  b >.
( a M b ) "> >. )
2725, 26syl 15 . . . . . . . . . . . . . 14  |-  ( u  =  <. a ,  b
>.  ->  <. m ,  m ,  <" u ( M `  u ) "> >.  =  <. m ,  m ,  <"
<. a ,  b >.
( a M b ) "> >. )
2827oveq2d 5874 . . . . . . . . . . . . 13  |-  ( u  =  <. a ,  b
>.  ->  ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )  =  ( x splice  <. m ,  m ,  <" <. a ,  b >. (
a M b ) "> >. )
)
2928breq2d 4035 . . . . . . . . . . . 12  |-  ( u  =  <. a ,  b
>.  ->  ( x r ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )  <->  x r ( x splice  <. m ,  m ,  <" <. a ,  b >. (
a M b ) "> >. )
) )
3029ralxp 4827 . . . . . . . . . . 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 ) "> >. )
)
31 eqidd 2284 . . . . . . . . . . . . . . . . 17  |-  ( ( a  e.  I  /\  b  e.  2o )  -> 
<. a ,  b >.  =  <. a ,  b
>. )
324efgmval 15021 . . . . . . . . . . . . . . . . 17  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( a M b )  =  <. a ,  ( 1o  \ 
b ) >. )
3331, 32s2eqd 11512 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  <" <. a ,  b >. (
a M b ) ">  =  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> )
34 oteq3 3807 . . . . . . . . . . . . . . . 16  |-  ( <" <. a ,  b
>. ( a M b ) ">  =  <" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. ">  ->  <. m ,  m ,  <" <. a ,  b >. (
a M b ) "> >.  =  <. m ,  m ,  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> >. )
3533, 34syl 15 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  I  /\  b  e.  2o )  -> 
<. m ,  m , 
<" <. a ,  b
>. ( a M b ) "> >.  =  <. m ,  m ,  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> >. )
3635oveq2d 5874 . . . . . . . . . . . . . 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 ) >. "> >.
) )
3736breq2d 4035 . . . . . . . . . . . . 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 ) >. "> >.
) ) )
3837ralbidva 2559 . . . . . . . . . . . 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 ) >. "> >.
) ) )
3938ralbiia 2575 . . . . . . . . . . 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 ) >. "> >.
) )
4030, 39bitri 240 . . . . . . . . . 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 ) >. "> >.
) )
4140ralbii 2567 . . . . . . . . 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 ) >. "> >.
) )
4220, 41bitri 240 . . . . . . . 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 )
>. "> >. )
)
4310, 42bitri 240 . . . . . . 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 )
>. "> >. )
)
449, 43syl6bb 252 . . . . . 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 )
>. "> >. )
) )
4544ralbiia 2575 . . . . 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 ) >. "> >.
) )
4645anbi2i 675 . . . 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 )
>. "> >. )
) )
4746abbii 2395 . . 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 )
>. "> >. )
) }
4847inteqi 3866 . 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 )
>. "> >. )
) }
493, 48eqtr4i 2306 1  |-  .~  =  |^| { r  |  ( r  Er  W  /\  A. x  e.  W  ran  ( T `  x ) 
C_  [ x ]
r ) }
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   _Vcvv 2788    \ cdif 3149    C_ wss 3152   <.cop 3643   <.cotp 3644   |^|cint 3862   class class class wbr 4023    e. cmpt 4077    _I cid 4304    X. cxp 4687   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1oc1o 6472   2oc2o 6473    Er wer 6657   [cec 6658   0cc0 8737   ...cfz 10782   #chash 11337  Word cword 11403   splice csplice 11407   <"cs2 11491   ~FG cefg 15015
This theorem is referenced by:  efgi2  15034  efgrelexlemb  15059  efgcpbllemb  15064
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-ot 3650  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-ec 6662  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-substr 11412  df-splice 11413  df-s2 11498  df-efg 15018
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