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Theorem efgtf 15031
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 5580 . . . . . . . . . 10  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3208 . . . . . . . . 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 3178 . . . . . . . 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 15022 . . . . . . . . . . 11  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
98ffvelrni 5664 . . . . . . . . . 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 11519 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  <" b ( M `  b ) ">  e. Word  (
I  X.  2o ) )
12 splcl 11467 . . . . . . . 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 15024 . . . . . . . . 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 2360 . . . . . 6  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
)  e.  W )
1817ralrimivva 2635 . . . . 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 2283 . . . . . 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 6191 . . . . 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 5883 . . . . 5  |-  ( 0 ... ( # `  X
) )  e.  _V
2314simpld 445 . . . . . 6  |-  ( X  e.  W  ->  I  e.  _V )
24 2on 6487 . . . . . 6  |-  2o  e.  On
25 xpexg 4800 . . . . . 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 4800 . . . . 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 5539 . . . . . 6  |-  (  _I 
` Word  ( I  X.  2o ) )  e.  _V
301, 29eqeltri 2353 . . . . 5  |-  W  e. 
_V
3130a1i 10 . . . 4  |-  ( X  e.  W  ->  W  e.  _V )
32 fex2 5401 . . . 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 5525 . . . . . 6  |-  ( u  =  X  ->  ( # `
 u )  =  ( # `  X
) )
3534oveq2d 5874 . . . . 5  |-  ( u  =  X  ->  (
0 ... ( # `  u
) )  =  ( 0 ... ( # `  X ) ) )
36 eqidd 2284 . . . . 5  |-  ( u  =  X  ->  (
I  X.  2o )  =  ( I  X.  2o ) )
37 oveq1 5865 . . . . 5  |-  ( u  =  X  ->  (
u splice  <. a ,  a ,  <" b ( M `  b ) "> >. )  =  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) )
3835, 36, 37mpt2eq123dv 5910 . . . 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 3805 . . . . . . . . . 10  |-  ( n  =  a  ->  <. n ,  n ,  <" w
( M `  w
) "> >.  =  <. a ,  n ,  <" w ( M `  w ) "> >.
)
41 oteq2 3806 . . . . . . . . . 10  |-  ( n  =  a  ->  <. a ,  n ,  <" w
( M `  w
) "> >.  =  <. a ,  a ,  <" w ( M `  w ) "> >.
)
4240, 41eqtrd 2315 . . . . . . . . 9  |-  ( n  =  a  ->  <. n ,  n ,  <" w
( M `  w
) "> >.  =  <. a ,  a ,  <" w ( M `  w ) "> >.
)
4342oveq2d 5874 . . . . . . . 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 5525 . . . . . . . . . . 11  |-  ( w  =  b  ->  ( M `  w )  =  ( M `  b ) )
4644, 45s2eqd 11512 . . . . . . . . . 10  |-  ( w  =  b  ->  <" w
( M `  w
) ">  =  <" b ( M `
 b ) "> )
47 oteq3 3807 . . . . . . . . . 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 5874 . . . . . . . 8  |-  ( w  =  b  ->  (
v splice  <. a ,  a ,  <" w ( M `  w ) "> >. )  =  ( v splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) )
5043, 49cbvmpt2v 5926 . . . . . . 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 5525 . . . . . . . . 9  |-  ( v  =  u  ->  ( # `
 v )  =  ( # `  u
) )
5251oveq2d 5874 . . . . . . . 8  |-  ( v  =  u  ->  (
0 ... ( # `  v
) )  =  ( 0 ... ( # `  u ) ) )
53 eqidd 2284 . . . . . . . 8  |-  ( v  =  u  ->  (
I  X.  2o )  =  ( I  X.  2o ) )
54 oveq1 5865 . . . . . . . 8  |-  ( v  =  u  ->  (
v splice  <. a ,  a ,  <" b ( M `  b ) "> >. )  =  ( u splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) )
5552, 53, 54mpt2eq123dv 5910 . . . . . . 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 2327 . . . . . 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 4111 . . . . 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 2303 . . . 4  |-  T  =  ( u  e.  W  |->  ( a  e.  ( 0 ... ( # `  u ) ) ,  b  e.  ( I  X.  2o )  |->  ( u splice  <. a ,  a ,  <" b ( M `  b ) "> >. )
) )
5938, 58fvmptg 5600 . . 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 5379 . . 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 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    \ cdif 3149   <.cop 3643   <.cotp 3644    e. cmpt 4077    _I cid 4304   Oncon0 4392    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1oc1o 6472   2oc2o 6473   0cc0 8737   ...cfz 10782   #chash 11337  Word cword 11403   splice csplice 11407   <"cs2 11491   ~FG cefg 15015
This theorem is referenced by:  efgtval  15032  efgval2  15033  efgtlen  15035  efginvrel2  15036  efgsp1  15046  efgredleme  15052  efgredlem  15056  efgrelexlemb  15059  efgcpbllemb  15064  frgpnabllem1  15161
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-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
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