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Theorem efgtf 15356
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 5786 . . . . . . . . . 10  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3380 . . . . . . . . 9  |-  W  C_ Word  ( I  X.  2o )
4 simpl 445 . . . . . . . . 9  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  X  e.  W
)
53, 4sseldi 3348 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  X  e. Word  (
I  X.  2o ) )
6 simprr 735 . . . . . . . . 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 15347 . . . . . . . . . . 11  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
98ffvelrni 5871 . . . . . . . . . 10  |-  ( b  e.  ( I  X.  2o )  ->  ( M `
 b )  e.  ( I  X.  2o ) )
109ad2antll 711 . . . . . . . . 9  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M `  b )  e.  ( I  X.  2o ) )
116, 10s2cld 11835 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  <" b ( M `  b ) ">  e. Word  (
I  X.  2o ) )
12 splcl 11783 . . . . . . . 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 644 . . . . . . 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 15349 . . . . . . . . 9  |-  ( X  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
1514simprd 451 . . . . . . . 8  |-  ( X  e.  W  ->  W  = Word  ( I  X.  2o ) )
1615adantr 453 . . . . . . 7  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  W  = Word  (
I  X.  2o ) )
1713, 16eleqtrrd 2515 . . . . . 6  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
)  e.  W )
1817ralrimivva 2800 . . . . 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 2438 . . . . . 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 6420 . . . . 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 190 . . . 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 6108 . . . . 5  |-  ( 0 ... ( # `  X
) )  e.  _V
2314simpld 447 . . . . . 6  |-  ( X  e.  W  ->  I  e.  _V )
24 2on 6734 . . . . . 6  |-  2o  e.  On
25 xpexg 4991 . . . . . 6  |-  ( ( I  e.  _V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
2623, 24, 25sylancl 645 . . . . 5  |-  ( X  e.  W  ->  (
I  X.  2o )  e.  _V )
27 xpexg 4991 . . . . 5  |-  ( ( ( 0 ... ( # `
 X ) )  e.  _V  /\  (
I  X.  2o )  e.  _V )  -> 
( ( 0 ... ( # `  X
) )  X.  (
I  X.  2o ) )  e.  _V )
2822, 26, 27sylancr 646 . . . 4  |-  ( X  e.  W  ->  (
( 0 ... ( # `
 X ) )  X.  ( I  X.  2o ) )  e.  _V )
29 fvex 5744 . . . . . 6  |-  (  _I 
` Word  ( I  X.  2o ) )  e.  _V
301, 29eqeltri 2508 . . . . 5  |-  W  e. 
_V
3130a1i 11 . . . 4  |-  ( X  e.  W  ->  W  e.  _V )
32 fex2 5605 . . . 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 1185 . . 3  |-  ( X  e.  W  ->  (
a  e.  ( 0 ... ( # `  X
) ) ,  b  e.  ( I  X.  2o )  |->  ( X splice  <. a ,  a , 
<" b ( M `
 b ) "> >. ) )  e. 
_V )
34 fveq2 5730 . . . . . 6  |-  ( u  =  X  ->  ( # `
 u )  =  ( # `  X
) )
3534oveq2d 6099 . . . . 5  |-  ( u  =  X  ->  (
0 ... ( # `  u
) )  =  ( 0 ... ( # `  X ) ) )
36 eqidd 2439 . . . . 5  |-  ( u  =  X  ->  (
I  X.  2o )  =  ( I  X.  2o ) )
37 oveq1 6090 . . . . 5  |-  ( u  =  X  ->  (
u splice  <. a ,  a ,  <" b ( M `  b ) "> >. )  =  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) )
3835, 36, 37mpt2eq123dv 6138 . . . 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 3995 . . . . . . . . . 10  |-  ( n  =  a  ->  <. n ,  n ,  <" w
( M `  w
) "> >.  =  <. a ,  n ,  <" w ( M `  w ) "> >.
)
41 oteq2 3996 . . . . . . . . . 10  |-  ( n  =  a  ->  <. a ,  n ,  <" w
( M `  w
) "> >.  =  <. a ,  a ,  <" w ( M `  w ) "> >.
)
4240, 41eqtrd 2470 . . . . . . . . 9  |-  ( n  =  a  ->  <. n ,  n ,  <" w
( M `  w
) "> >.  =  <. a ,  a ,  <" w ( M `  w ) "> >.
)
4342oveq2d 6099 . . . . . . . 8  |-  ( n  =  a  ->  (
v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )  =  ( v splice  <. a ,  a ,  <" w ( M `  w ) "> >.
) )
44 id 21 . . . . . . . . . . 11  |-  ( w  =  b  ->  w  =  b )
45 fveq2 5730 . . . . . . . . . . 11  |-  ( w  =  b  ->  ( M `  w )  =  ( M `  b ) )
4644, 45s2eqd 11828 . . . . . . . . . 10  |-  ( w  =  b  ->  <" w
( M `  w
) ">  =  <" b ( M `
 b ) "> )
4746oteq3d 4000 . . . . . . . . 9  |-  ( w  =  b  ->  <. a ,  a ,  <" w ( M `  w ) "> >.  =  <. a ,  a ,  <" b ( M `  b ) "> >. )
4847oveq2d 6099 . . . . . . . 8  |-  ( w  =  b  ->  (
v splice  <. a ,  a ,  <" w ( M `  w ) "> >. )  =  ( v splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) )
4943, 48cbvmpt2v 6154 . . . . . . 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 5730 . . . . . . . . 9  |-  ( v  =  u  ->  ( # `
 v )  =  ( # `  u
) )
5150oveq2d 6099 . . . . . . . 8  |-  ( v  =  u  ->  (
0 ... ( # `  v
) )  =  ( 0 ... ( # `  u ) ) )
52 eqidd 2439 . . . . . . . 8  |-  ( v  =  u  ->  (
I  X.  2o )  =  ( I  X.  2o ) )
53 oveq1 6090 . . . . . . . 8  |-  ( v  =  u  ->  (
v splice  <. a ,  a ,  <" b ( M `  b ) "> >. )  =  ( u splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) )
5451, 52, 53mpt2eq123dv 6138 . . . . . . 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 2482 . . . . . 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 4302 . . . . 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 2458 . . . 4  |-  T  =  ( u  e.  W  |->  ( a  e.  ( 0 ... ( # `  u ) ) ,  b  e.  ( I  X.  2o )  |->  ( u splice  <. a ,  a ,  <" b ( M `  b ) "> >. )
) )
5838, 57fvmptg 5806 . . 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 651 . 2  |-  ( X  e.  W  ->  ( T `  X )  =  ( a  e.  ( 0 ... ( # `
 X ) ) ,  b  e.  ( I  X.  2o ) 
|->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) ) )
6059feq1d 5582 . . 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 225 . 2  |-  ( X  e.  W  ->  ( T `  X ) : ( ( 0 ... ( # `  X
) )  X.  (
I  X.  2o ) ) --> W )
6259, 61jca 520 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 360    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    \ cdif 3319   <.cop 3819   <.cotp 3820    e. cmpt 4268    _I cid 4495   Oncon0 4583    X. cxp 4878   -->wf 5452   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   1oc1o 6719   2oc2o 6720   0cc0 8992   ...cfz 11045   #chash 11620  Word cword 11719   splice csplice 11723   <"cs2 11807   ~FG cefg 15340
This theorem is referenced by:  efgtval  15357  efgval2  15358  efgtlen  15360  efginvrel2  15361  efgsp1  15371  efgredleme  15377  efgredlem  15381  efgrelexlemb  15384  efgcpbllemb  15389  frgpnabllem1  15486
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-ot 3826  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-fzo 11138  df-hash 11621  df-word 11725  df-concat 11726  df-s1 11727  df-substr 11728  df-splice 11729  df-s2 11814
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