MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  efgtf Unicode version

Theorem efgtf 15047
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
efgtf  |-  ( X  e.  W  ->  (
( T `  X
)  =  ( a  e.  ( 0 ... ( # `  X
) ) ,  b  e.  ( I  X.  2o )  |->  ( X splice  <. a ,  a , 
<" b ( M `
 b ) "> >. ) )  /\  ( T `  X ) : ( ( 0 ... ( # `  X
) )  X.  (
I  X.  2o ) ) --> W ) )
Distinct variable groups:    a, b,
y, z    v, n, w, y, z, a    M, a    n, b, v, w, M    T, a, b    X, a, b    W, a, b, n, v, w, y, z    .~ , a, b, y, z    I, a, b, n, v, w, y, z
Allowed substitution hints:    .~ ( w, v, n)    T( y, z, w, v, n)    M( y,
z)    X( y, z, w, v, n)

Proof of Theorem efgtf
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 efgval.w . . . . . . . . . 10  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
2 fviss 5596 . . . . . . . . . 10  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3221 . . . . . . . . 9  |-  W  C_ Word  ( I  X.  2o )
4 simpl 443 . . . . . . . . 9  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  X  e.  W
)
53, 4sseldi 3191 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  X  e. Word  (
I  X.  2o ) )
6 simprr 733 . . . . . . . . 9  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  b  e.  ( I  X.  2o ) )
7 efgval2.m . . . . . . . . . . . 12  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
87efgmf 15038 . . . . . . . . . . 11  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
98ffvelrni 5680 . . . . . . . . . 10  |-  ( b  e.  ( I  X.  2o )  ->  ( M `
 b )  e.  ( I  X.  2o ) )
109ad2antll 709 . . . . . . . . 9  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M `  b )  e.  ( I  X.  2o ) )
116, 10s2cld 11535 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  <" b ( M `  b ) ">  e. Word  (
I  X.  2o ) )
12 splcl 11483 . . . . . . . 8  |-  ( ( X  e. Word  ( I  X.  2o )  /\  <" b ( M `
 b ) ">  e. Word  ( I  X.  2o ) )  -> 
( X splice  <. a ,  a ,  <" b
( M `  b
) "> >. )  e. Word  ( I  X.  2o ) )
135, 11, 12syl2anc 642 . . . . . . 7  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
)  e. Word  ( I  X.  2o ) )
141efgrcl 15040 . . . . . . . . 9  |-  ( X  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
1514simprd 449 . . . . . . . 8  |-  ( X  e.  W  ->  W  = Word  ( I  X.  2o ) )
1615adantr 451 . . . . . . 7  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  W  = Word  (
I  X.  2o ) )
1713, 16eleqtrrd 2373 . . . . . 6  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
)  e.  W )
1817ralrimivva 2648 . . . . 5  |-  ( X  e.  W  ->  A. a  e.  ( 0 ... ( # `
 X ) ) A. b  e.  ( I  X.  2o ) ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
)  e.  W )
19 eqid 2296 . . . . . 6  |-  ( a  e.  ( 0 ... ( # `  X
) ) ,  b  e.  ( I  X.  2o )  |->  ( X splice  <. a ,  a , 
<" b ( M `
 b ) "> >. ) )  =  ( a  e.  ( 0 ... ( # `  X ) ) ,  b  e.  ( I  X.  2o )  |->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >. )
)
2019fmpt2 6207 . . . . 5  |-  ( A. a  e.  ( 0 ... ( # `  X
) ) A. b  e.  ( I  X.  2o ) ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
)  e.  W  <->  ( a  e.  ( 0 ... ( # `
 X ) ) ,  b  e.  ( I  X.  2o ) 
|->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) ) : ( ( 0 ... ( # `
 X ) )  X.  ( I  X.  2o ) ) --> W )
2118, 20sylib 188 . . . 4  |-  ( X  e.  W  ->  (
a  e.  ( 0 ... ( # `  X
) ) ,  b  e.  ( I  X.  2o )  |->  ( X splice  <. a ,  a , 
<" b ( M `
 b ) "> >. ) ) : ( ( 0 ... ( # `  X
) )  X.  (
I  X.  2o ) ) --> W )
22 ovex 5899 . . . . 5  |-  ( 0 ... ( # `  X
) )  e.  _V
2314simpld 445 . . . . . 6  |-  ( X  e.  W  ->  I  e.  _V )
24 2on 6503 . . . . . 6  |-  2o  e.  On
25 xpexg 4816 . . . . . 6  |-  ( ( I  e.  _V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
2623, 24, 25sylancl 643 . . . . 5  |-  ( X  e.  W  ->  (
I  X.  2o )  e.  _V )
27 xpexg 4816 . . . . 5  |-  ( ( ( 0 ... ( # `
 X ) )  e.  _V  /\  (
I  X.  2o )  e.  _V )  -> 
( ( 0 ... ( # `  X
) )  X.  (
I  X.  2o ) )  e.  _V )
2822, 26, 27sylancr 644 . . . 4  |-  ( X  e.  W  ->  (
( 0 ... ( # `
 X ) )  X.  ( I  X.  2o ) )  e.  _V )
29 fvex 5555 . . . . . 6  |-  (  _I 
` Word  ( I  X.  2o ) )  e.  _V
301, 29eqeltri 2366 . . . . 5  |-  W  e. 
_V
3130a1i 10 . . . 4  |-  ( X  e.  W  ->  W  e.  _V )
32 fex2 5417 . . . 4  |-  ( ( ( a  e.  ( 0 ... ( # `  X ) ) ,  b  e.  ( I  X.  2o )  |->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >. )
) : ( ( 0 ... ( # `  X ) )  X.  ( I  X.  2o ) ) --> W  /\  ( ( 0 ... ( # `  X
) )  X.  (
I  X.  2o ) )  e.  _V  /\  W  e.  _V )  ->  ( a  e.  ( 0 ... ( # `  X ) ) ,  b  e.  ( I  X.  2o )  |->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >. )
)  e.  _V )
3321, 28, 31, 32syl3anc 1182 . . 3  |-  ( X  e.  W  ->  (
a  e.  ( 0 ... ( # `  X
) ) ,  b  e.  ( I  X.  2o )  |->  ( X splice  <. a ,  a , 
<" b ( M `
 b ) "> >. ) )  e. 
_V )
34 fveq2 5541 . . . . . 6  |-  ( u  =  X  ->  ( # `
 u )  =  ( # `  X
) )
3534oveq2d 5890 . . . . 5  |-  ( u  =  X  ->  (
0 ... ( # `  u
) )  =  ( 0 ... ( # `  X ) ) )
36 eqidd 2297 . . . . 5  |-  ( u  =  X  ->  (
I  X.  2o )  =  ( I  X.  2o ) )
37 oveq1 5881 . . . . 5  |-  ( u  =  X  ->  (
u splice  <. a ,  a ,  <" b ( M `  b ) "> >. )  =  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) )
3835, 36, 37mpt2eq123dv 5926 . . . 4  |-  ( u  =  X  ->  (
a  e.  ( 0 ... ( # `  u
) ) ,  b  e.  ( I  X.  2o )  |->  ( u splice  <. a ,  a , 
<" b ( M `
 b ) "> >. ) )  =  ( a  e.  ( 0 ... ( # `  X ) ) ,  b  e.  ( I  X.  2o )  |->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >. )
) )
39 efgval2.t . . . . 5  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
40 oteq1 3821 . . . . . . . . . 10  |-  ( n  =  a  ->  <. n ,  n ,  <" w
( M `  w
) "> >.  =  <. a ,  n ,  <" w ( M `  w ) "> >.
)
41 oteq2 3822 . . . . . . . . . 10  |-  ( n  =  a  ->  <. a ,  n ,  <" w
( M `  w
) "> >.  =  <. a ,  a ,  <" w ( M `  w ) "> >.
)
4240, 41eqtrd 2328 . . . . . . . . 9  |-  ( n  =  a  ->  <. n ,  n ,  <" w
( M `  w
) "> >.  =  <. a ,  a ,  <" w ( M `  w ) "> >.
)
4342oveq2d 5890 . . . . . . . 8  |-  ( n  =  a  ->  (
v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )  =  ( v splice  <. a ,  a ,  <" w ( M `  w ) "> >.
) )
44 id 19 . . . . . . . . . . 11  |-  ( w  =  b  ->  w  =  b )
45 fveq2 5541 . . . . . . . . . . 11  |-  ( w  =  b  ->  ( M `  w )  =  ( M `  b ) )
4644, 45s2eqd 11528 . . . . . . . . . 10  |-  ( w  =  b  ->  <" w
( M `  w
) ">  =  <" b ( M `
 b ) "> )
47 oteq3 3823 . . . . . . . . . 10  |-  ( <" w ( M `
 w ) ">  =  <" b
( M `  b
) ">  ->  <.
a ,  a , 
<" w ( M `
 w ) "> >.  =  <. a ,  a ,  <" b ( M `  b ) "> >.
)
4846, 47syl 15 . . . . . . . . 9  |-  ( w  =  b  ->  <. a ,  a ,  <" w ( M `  w ) "> >.  =  <. a ,  a ,  <" b ( M `  b ) "> >. )
4948oveq2d 5890 . . . . . . . 8  |-  ( w  =  b  ->  (
v splice  <. a ,  a ,  <" w ( M `  w ) "> >. )  =  ( v splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) )
5043, 49cbvmpt2v 5942 . . . . . . 7  |-  ( n  e.  ( 0 ... ( # `  v
) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >.
) )  =  ( a  e.  ( 0 ... ( # `  v
) ) ,  b  e.  ( I  X.  2o )  |->  ( v splice  <. a ,  a , 
<" b ( M `
 b ) "> >. ) )
51 fveq2 5541 . . . . . . . . 9  |-  ( v  =  u  ->  ( # `
 v )  =  ( # `  u
) )
5251oveq2d 5890 . . . . . . . 8  |-  ( v  =  u  ->  (
0 ... ( # `  v
) )  =  ( 0 ... ( # `  u ) ) )
53 eqidd 2297 . . . . . . . 8  |-  ( v  =  u  ->  (
I  X.  2o )  =  ( I  X.  2o ) )
54 oveq1 5881 . . . . . . . 8  |-  ( v  =  u  ->  (
v splice  <. a ,  a ,  <" b ( M `  b ) "> >. )  =  ( u splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) )
5552, 53, 54mpt2eq123dv 5926 . . . . . . 7  |-  ( v  =  u  ->  (
a  e.  ( 0 ... ( # `  v
) ) ,  b  e.  ( I  X.  2o )  |->  ( v splice  <. a ,  a , 
<" b ( M `
 b ) "> >. ) )  =  ( a  e.  ( 0 ... ( # `  u ) ) ,  b  e.  ( I  X.  2o )  |->  ( u splice  <. a ,  a ,  <" b ( M `  b ) "> >. )
) )
5650, 55syl5eq 2340 . . . . . 6  |-  ( v  =  u  ->  (
n  e.  ( 0 ... ( # `  v
) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >.
) )  =  ( a  e.  ( 0 ... ( # `  u
) ) ,  b  e.  ( I  X.  2o )  |->  ( u splice  <. a ,  a , 
<" b ( M `
 b ) "> >. ) ) )
5756cbvmptv 4127 . . . . 5  |-  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v
) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >.
) ) )  =  ( u  e.  W  |->  ( a  e.  ( 0 ... ( # `  u ) ) ,  b  e.  ( I  X.  2o )  |->  ( u splice  <. a ,  a ,  <" b ( M `  b ) "> >. )
) )
5839, 57eqtri 2316 . . . 4  |-  T  =  ( u  e.  W  |->  ( a  e.  ( 0 ... ( # `  u ) ) ,  b  e.  ( I  X.  2o )  |->  ( u splice  <. a ,  a ,  <" b ( M `  b ) "> >. )
) )
5938, 58fvmptg 5616 . . 3  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) ) ,  b  e.  ( I  X.  2o )  |->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >. )
)  e.  _V )  ->  ( T `  X
)  =  ( a  e.  ( 0 ... ( # `  X
) ) ,  b  e.  ( I  X.  2o )  |->  ( X splice  <. a ,  a , 
<" b ( M `
 b ) "> >. ) ) )
6033, 59mpdan 649 . 2  |-  ( X  e.  W  ->  ( T `  X )  =  ( a  e.  ( 0 ... ( # `
 X ) ) ,  b  e.  ( I  X.  2o ) 
|->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) ) )
6160feq1d 5395 . . 3  |-  ( X  e.  W  ->  (
( T `  X
) : ( ( 0 ... ( # `  X ) )  X.  ( I  X.  2o ) ) --> W  <->  ( a  e.  ( 0 ... ( # `
 X ) ) ,  b  e.  ( I  X.  2o ) 
|->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) ) : ( ( 0 ... ( # `
 X ) )  X.  ( I  X.  2o ) ) --> W ) )
6221, 61mpbird 223 . 2  |-  ( X  e.  W  ->  ( T `  X ) : ( ( 0 ... ( # `  X
) )  X.  (
I  X.  2o ) ) --> W )
6360, 62jca 518 1  |-  ( X  e.  W  ->  (
( T `  X
)  =  ( a  e.  ( 0 ... ( # `  X
) ) ,  b  e.  ( I  X.  2o )  |->  ( X splice  <. a ,  a , 
<" b ( M `
 b ) "> >. ) )  /\  ( T `  X ) : ( ( 0 ... ( # `  X
) )  X.  (
I  X.  2o ) ) --> W ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    \ cdif 3162   <.cop 3656   <.cotp 3657    e. cmpt 4093    _I cid 4320   Oncon0 4408    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1oc1o 6488   2oc2o 6489   0cc0 8753   ...cfz 10798   #chash 11353  Word cword 11419   splice csplice 11423   <"cs2 11507   ~FG cefg 15031
This theorem is referenced by:  efgtval  15048  efgval2  15049  efgtlen  15051  efginvrel2  15052  efgsp1  15062  efgredleme  15068  efgredlem  15072  efgrelexlemb  15075  efgcpbllemb  15080  frgpnabllem1  15177
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-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
  Copyright terms: Public domain W3C validator