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Theorem efgtf 15309
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 5743 . . . . . . . . . 10  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3338 . . . . . . . . 9  |-  W  C_ Word  ( I  X.  2o )
4 simpl 444 . . . . . . . . 9  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  X  e.  W
)
53, 4sseldi 3306 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  X  e. Word  (
I  X.  2o ) )
6 simprr 734 . . . . . . . . 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 15300 . . . . . . . . . . 11  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
98ffvelrni 5828 . . . . . . . . . 10  |-  ( b  e.  ( I  X.  2o )  ->  ( M `
 b )  e.  ( I  X.  2o ) )
109ad2antll 710 . . . . . . . . 9  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M `  b )  e.  ( I  X.  2o ) )
116, 10s2cld 11788 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  <" b ( M `  b ) ">  e. Word  (
I  X.  2o ) )
12 splcl 11736 . . . . . . . 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 643 . . . . . . 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 15302 . . . . . . . . 9  |-  ( X  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
1514simprd 450 . . . . . . . 8  |-  ( X  e.  W  ->  W  = Word  ( I  X.  2o ) )
1615adantr 452 . . . . . . 7  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  W  = Word  (
I  X.  2o ) )
1713, 16eleqtrrd 2481 . . . . . 6  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
)  e.  W )
1817ralrimivva 2758 . . . . 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 2404 . . . . . 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 6377 . . . . 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 189 . . . 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 6065 . . . . 5  |-  ( 0 ... ( # `  X
) )  e.  _V
2314simpld 446 . . . . . 6  |-  ( X  e.  W  ->  I  e.  _V )
24 2on 6691 . . . . . 6  |-  2o  e.  On
25 xpexg 4948 . . . . . 6  |-  ( ( I  e.  _V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
2623, 24, 25sylancl 644 . . . . 5  |-  ( X  e.  W  ->  (
I  X.  2o )  e.  _V )
27 xpexg 4948 . . . . 5  |-  ( ( ( 0 ... ( # `
 X ) )  e.  _V  /\  (
I  X.  2o )  e.  _V )  -> 
( ( 0 ... ( # `  X
) )  X.  (
I  X.  2o ) )  e.  _V )
2822, 26, 27sylancr 645 . . . 4  |-  ( X  e.  W  ->  (
( 0 ... ( # `
 X ) )  X.  ( I  X.  2o ) )  e.  _V )
29 fvex 5701 . . . . . 6  |-  (  _I 
` Word  ( I  X.  2o ) )  e.  _V
301, 29eqeltri 2474 . . . . 5  |-  W  e. 
_V
3130a1i 11 . . . 4  |-  ( X  e.  W  ->  W  e.  _V )
32 fex2 5562 . . . 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 1184 . . 3  |-  ( X  e.  W  ->  (
a  e.  ( 0 ... ( # `  X
) ) ,  b  e.  ( I  X.  2o )  |->  ( X splice  <. a ,  a , 
<" b ( M `
 b ) "> >. ) )  e. 
_V )
34 fveq2 5687 . . . . . 6  |-  ( u  =  X  ->  ( # `
 u )  =  ( # `  X
) )
3534oveq2d 6056 . . . . 5  |-  ( u  =  X  ->  (
0 ... ( # `  u
) )  =  ( 0 ... ( # `  X ) ) )
36 eqidd 2405 . . . . 5  |-  ( u  =  X  ->  (
I  X.  2o )  =  ( I  X.  2o ) )
37 oveq1 6047 . . . . 5  |-  ( u  =  X  ->  (
u splice  <. a ,  a ,  <" b ( M `  b ) "> >. )  =  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) )
3835, 36, 37mpt2eq123dv 6095 . . . 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 3953 . . . . . . . . . 10  |-  ( n  =  a  ->  <. n ,  n ,  <" w
( M `  w
) "> >.  =  <. a ,  n ,  <" w ( M `  w ) "> >.
)
41 oteq2 3954 . . . . . . . . . 10  |-  ( n  =  a  ->  <. a ,  n ,  <" w
( M `  w
) "> >.  =  <. a ,  a ,  <" w ( M `  w ) "> >.
)
4240, 41eqtrd 2436 . . . . . . . . 9  |-  ( n  =  a  ->  <. n ,  n ,  <" w
( M `  w
) "> >.  =  <. a ,  a ,  <" w ( M `  w ) "> >.
)
4342oveq2d 6056 . . . . . . . 8  |-  ( n  =  a  ->  (
v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )  =  ( v splice  <. a ,  a ,  <" w ( M `  w ) "> >.
) )
44 id 20 . . . . . . . . . . 11  |-  ( w  =  b  ->  w  =  b )
45 fveq2 5687 . . . . . . . . . . 11  |-  ( w  =  b  ->  ( M `  w )  =  ( M `  b ) )
4644, 45s2eqd 11781 . . . . . . . . . 10  |-  ( w  =  b  ->  <" w
( M `  w
) ">  =  <" b ( M `
 b ) "> )
4746oteq3d 3958 . . . . . . . . 9  |-  ( w  =  b  ->  <. a ,  a ,  <" w ( M `  w ) "> >.  =  <. a ,  a ,  <" b ( M `  b ) "> >. )
4847oveq2d 6056 . . . . . . . 8  |-  ( w  =  b  ->  (
v splice  <. a ,  a ,  <" w ( M `  w ) "> >. )  =  ( v splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) )
4943, 48cbvmpt2v 6111 . . . . . . 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 ) "> >. ) )
50 fveq2 5687 . . . . . . . . 9  |-  ( v  =  u  ->  ( # `
 v )  =  ( # `  u
) )
5150oveq2d 6056 . . . . . . . 8  |-  ( v  =  u  ->  (
0 ... ( # `  v
) )  =  ( 0 ... ( # `  u ) ) )
52 eqidd 2405 . . . . . . . 8  |-  ( v  =  u  ->  (
I  X.  2o )  =  ( I  X.  2o ) )
53 oveq1 6047 . . . . . . . 8  |-  ( v  =  u  ->  (
v splice  <. a ,  a ,  <" b ( M `  b ) "> >. )  =  ( u splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) )
5451, 52, 53mpt2eq123dv 6095 . . . . . . 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 ) "> >. )
) )
5549, 54syl5eq 2448 . . . . . 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 ) "> >. ) ) )
5655cbvmptv 4260 . . . . 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 ) "> >. )
) )
5739, 56eqtri 2424 . . . 4  |-  T  =  ( u  e.  W  |->  ( a  e.  ( 0 ... ( # `  u ) ) ,  b  e.  ( I  X.  2o )  |->  ( u splice  <. a ,  a ,  <" b ( M `  b ) "> >. )
) )
5838, 57fvmptg 5763 . . 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 ) "> >. ) ) )
5933, 58mpdan 650 . 2  |-  ( X  e.  W  ->  ( T `  X )  =  ( a  e.  ( 0 ... ( # `
 X ) ) ,  b  e.  ( I  X.  2o ) 
|->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) ) )
6059feq1d 5539 . . 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 ) )
6121, 60mpbird 224 . 2  |-  ( X  e.  W  ->  ( T `  X ) : ( ( 0 ... ( # `  X
) )  X.  (
I  X.  2o ) ) --> W )
6259, 61jca 519 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 359    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    \ cdif 3277   <.cop 3777   <.cotp 3778    e. cmpt 4226    _I cid 4453   Oncon0 4541    X. cxp 4835   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1oc1o 6676   2oc2o 6677   0cc0 8946   ...cfz 10999   #chash 11573  Word cword 11672   splice csplice 11676   <"cs2 11760   ~FG cefg 15293
This theorem is referenced by:  efgtval  15310  efgval2  15311  efgtlen  15313  efginvrel2  15314  efgsp1  15324  efgredleme  15330  efgredlem  15334  efgrelexlemb  15337  efgcpbllemb  15342  frgpnabllem1  15439
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-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
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