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Theorem efgval2 15049
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 15042 . 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 15047 . . . . . . . . . 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 4922 . . . . . . . 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 3218 . . . . . . 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 3183 . . . . . . . 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 5899 . . . . . . . . . . 11  |-  ( x splice  <. m ,  m , 
<" u ( M `
 u ) "> >. )  e.  _V
1211rgen2w 2624 . . . . . . . . . 10  |-  A. m  e.  ( 0 ... ( # `
 x ) ) A. u  e.  ( I  X.  2o ) ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )  e.  _V
13 eqid 2296 . . . . . . . . . . 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 2804 . . . . . . . . . . . . 13  |-  a  e. 
_V
15 vex 2804 . . . . . . . . . . . . 13  |-  x  e. 
_V
1614, 15elec 6715 . . . . . . . . . . . 12  |-  ( a  e.  [ x ]
r  <->  x r a )
17 breq2 4043 . . . . . . . . . . . 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 5974 . . . . . . . . . 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 5541 . . . . . . . . . . . . . . . . 17  |-  ( u  =  <. a ,  b
>.  ->  ( M `  u )  =  ( M `  <. a ,  b >. )
)
23 df-ov 5877 . . . . . . . . . . . . . . . . 17  |-  ( a M b )  =  ( M `  <. a ,  b >. )
2422, 23syl6eqr 2346 . . . . . . . . . . . . . . . 16  |-  ( u  =  <. a ,  b
>.  ->  ( M `  u )  =  ( a M b ) )
2521, 24s2eqd 11528 . . . . . . . . . . . . . . 15  |-  ( u  =  <. a ,  b
>.  ->  <" u ( M `  u ) ">  =  <"
<. a ,  b >.
( a M b ) "> )
26 oteq3 3823 . . . . . . . . . . . . . . 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 5890 . . . . . . . . . . . . 13  |-  ( u  =  <. a ,  b
>.  ->  ( x splice  <. m ,  m ,  <" u
( M `  u
) "> >. )  =  ( x splice  <. m ,  m ,  <" <. a ,  b >. (
a M b ) "> >. )
)
2928breq2d 4051 . . . . . . . . . . . 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 4843 . . . . . . . . . . 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 2297 . . . . . . . . . . . . . . . . 17  |-  ( ( a  e.  I  /\  b  e.  2o )  -> 
<. a ,  b >.  =  <. a ,  b
>. )
324efgmval 15037 . . . . . . . . . . . . . . . . 17  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( a M b )  =  <. a ,  ( 1o  \ 
b ) >. )
3331, 32s2eqd 11528 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  <" <. a ,  b >. (
a M b ) ">  =  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> )
34 oteq3 3823 . . . . . . . . . . . . . . . 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 5890 . . . . . . . . . . . . . 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 4051 . . . . . . . . . . . . 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 2572 . . . . . . . . . . . 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 2588 . . . . . . . . . . 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 2580 . . . . . . . . 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 2588 . . . . 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 2408 . . 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 3882 . 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 2319 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 1632    e. wcel 1696   {cab 2282   A.wral 2556   _Vcvv 2801    \ cdif 3162    C_ wss 3165   <.cop 3656   <.cotp 3657   |^|cint 3878   class class class wbr 4039    e. cmpt 4093    _I cid 4320    X. cxp 4703   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1oc1o 6488   2oc2o 6489    Er wer 6673   [cec 6674   0cc0 8753   ...cfz 10798   #chash 11353  Word cword 11419   splice csplice 11423   <"cs2 11507   ~FG cefg 15031
This theorem is referenced by:  efgi2  15050  efgrelexlemb  15075  efgcpbllemb  15080
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-ot 3663  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-ec 6678  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-hash 11354  df-word 11425  df-concat 11426  df-s1 11427  df-substr 11428  df-splice 11429  df-s2 11514  df-efg 15034
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