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Theorem efgtlen 15321
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
efgtlen  |-  ( ( X  e.  W  /\  A  e.  ran  ( T `
 X ) )  ->  ( # `  A
)  =  ( (
# `  X )  +  2 ) )
Distinct variable groups:    y, z    v, n, w, y, z   
n, M, v, w   
n, W, v, w, y, z    y,  .~ , z    n, I, v, w, y, z
Allowed substitution hints:    A( y, z, w, v, n)    .~ ( w, v, n)    T( y, z, w, v, n)    M( y, z)    X( y, z, w, v, n)

Proof of Theorem efgtlen
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.w . . . . . . . 8  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
2 efgval.r . . . . . . . 8  |-  .~  =  ( ~FG  `  I )
3 efgval2.m . . . . . . . 8  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
4 efgval2.t . . . . . . . 8  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
51, 2, 3, 4efgtf 15317 . . . . . . 7  |-  ( 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 ) )
65simpld 446 . . . . . 6  |-  ( X  e.  W  ->  ( T `  X )  =  ( a  e.  ( 0 ... ( # `
 X ) ) ,  b  e.  ( I  X.  2o ) 
|->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) ) )
76rneqd 5064 . . . . 5  |-  ( X  e.  W  ->  ran  ( T `  X )  =  ran  ( a  e.  ( 0 ... ( # `  X
) ) ,  b  e.  ( I  X.  2o )  |->  ( X splice  <. a ,  a , 
<" b ( M `
 b ) "> >. ) ) )
87eleq2d 2479 . . . 4  |-  ( X  e.  W  ->  ( A  e.  ran  ( T `
 X )  <->  A  e.  ran  ( a  e.  ( 0 ... ( # `  X ) ) ,  b  e.  ( I  X.  2o )  |->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >. )
) ) )
9 eqid 2412 . . . . 5  |-  ( 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 ) "> >. )
)
10 ovex 6073 . . . . 5  |-  ( X splice  <. a ,  a , 
<" b ( M `
 b ) "> >. )  e.  _V
119, 10elrnmpt2 6150 . . . 4  |-  ( A  e.  ran  ( a  e.  ( 0 ... ( # `  X
) ) ,  b  e.  ( I  X.  2o )  |->  ( X splice  <. a ,  a , 
<" b ( M `
 b ) "> >. ) )  <->  E. a  e.  ( 0 ... ( # `
 X ) ) E. b  e.  ( I  X.  2o ) A  =  ( X splice  <. a ,  a , 
<" b ( M `
 b ) "> >. ) )
128, 11syl6bb 253 . . 3  |-  ( X  e.  W  ->  ( A  e.  ran  ( T `
 X )  <->  E. a  e.  ( 0 ... ( # `
 X ) ) E. b  e.  ( I  X.  2o ) A  =  ( X splice  <. a ,  a , 
<" b ( M `
 b ) "> >. ) ) )
13 fviss 5751 . . . . . . . . 9  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
141, 13eqsstri 3346 . . . . . . . 8  |-  W  C_ Word  ( I  X.  2o )
15 simpl 444 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  X  e.  W
)
1614, 15sseldi 3314 . . . . . . 7  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  X  e. Word  (
I  X.  2o ) )
17 elfzuz 11019 . . . . . . . . 9  |-  ( a  e.  ( 0 ... ( # `  X
) )  ->  a  e.  ( ZZ>= `  0 )
)
1817ad2antrl 709 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e.  (
ZZ>= `  0 ) )
19 eluzfz2b 11030 . . . . . . . 8  |-  ( a  e.  ( ZZ>= `  0
)  <->  a  e.  ( 0 ... a ) )
2018, 19sylib 189 . . . . . . 7  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e.  ( 0 ... a ) )
21 simprl 733 . . . . . . 7  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e.  ( 0 ... ( # `  X ) ) )
22 simprr 734 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  b  e.  ( I  X.  2o ) )
233efgmf 15308 . . . . . . . . . 10  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
2423ffvelrni 5836 . . . . . . . . 9  |-  ( b  e.  ( I  X.  2o )  ->  ( M `
 b )  e.  ( I  X.  2o ) )
2522, 24syl 16 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M `  b )  e.  ( I  X.  2o ) )
2622, 25s2cld 11796 . . . . . . 7  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  <" b ( M `  b ) ">  e. Word  (
I  X.  2o ) )
2716, 20, 21, 26spllen 11746 . . . . . 6  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >. )
)  =  ( (
# `  X )  +  ( ( # `  <" b ( M `  b ) "> )  -  ( a  -  a
) ) ) )
28 s2len 11814 . . . . . . . . . 10  |-  ( # `  <" b ( M `  b ) "> )  =  2
2928a1i 11 . . . . . . . . 9  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `  <" b ( M `  b ) "> )  =  2 )
30 eluzelz 10460 . . . . . . . . . . . 12  |-  ( a  e.  ( ZZ>= `  0
)  ->  a  e.  ZZ )
3130zcnd 10340 . . . . . . . . . . 11  |-  ( a  e.  ( ZZ>= `  0
)  ->  a  e.  CC )
3218, 31syl 16 . . . . . . . . . 10  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e.  CC )
3332subidd 9363 . . . . . . . . 9  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( a  -  a )  =  0 )
3429, 33oveq12d 6066 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( ( # `  <" b ( M `  b ) "> )  -  ( a  -  a
) )  =  ( 2  -  0 ) )
35 2cn 10034 . . . . . . . . 9  |-  2  e.  CC
3635subid1i 9336 . . . . . . . 8  |-  ( 2  -  0 )  =  2
3734, 36syl6eq 2460 . . . . . . 7  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( ( # `  <" b ( M `  b ) "> )  -  ( a  -  a
) )  =  2 )
3837oveq2d 6064 . . . . . 6  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( ( # `  X )  +  ( ( # `  <" b ( M `  b ) "> )  -  ( a  -  a ) ) )  =  ( (
# `  X )  +  2 ) )
3927, 38eqtrd 2444 . . . . 5  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >. )
)  =  ( (
# `  X )  +  2 ) )
40 fveq2 5695 . . . . . 6  |-  ( A  =  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
)  ->  ( # `  A
)  =  ( # `  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) ) )
4140eqeq1d 2420 . . . . 5  |-  ( A  =  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
)  ->  ( ( # `
 A )  =  ( ( # `  X
)  +  2 )  <-> 
( # `  ( X splice  <. a ,  a , 
<" b ( M `
 b ) "> >. ) )  =  ( ( # `  X
)  +  2 ) ) )
4239, 41syl5ibrcom 214 . . . 4  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( A  =  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
)  ->  ( # `  A
)  =  ( (
# `  X )  +  2 ) ) )
4342rexlimdvva 2805 . . 3  |-  ( X  e.  W  ->  ( E. a  e.  (
0 ... ( # `  X
) ) E. b  e.  ( I  X.  2o ) A  =  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >. )  ->  ( # `  A
)  =  ( (
# `  X )  +  2 ) ) )
4412, 43sylbid 207 . 2  |-  ( X  e.  W  ->  ( A  e.  ran  ( T `
 X )  -> 
( # `  A )  =  ( ( # `  X )  +  2 ) ) )
4544imp 419 1  |-  ( ( X  e.  W  /\  A  e.  ran  ( T `
 X ) )  ->  ( # `  A
)  =  ( (
# `  X )  +  2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2675    \ cdif 3285   <.cop 3785   <.cotp 3786    e. cmpt 4234    _I cid 4461    X. cxp 4843   ran crn 4846   -->wf 5417   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050   1oc1o 6684   2oc2o 6685   CCcc 8952   0cc0 8954    + caddc 8957    - cmin 9255   2c2 10013   ZZ>=cuz 10452   ...cfz 11007   #chash 11581  Word cword 11680   splice csplice 11684   <"cs2 11768   ~FG cefg 15301
This theorem is referenced by:  efgsfo  15334  efgredlemg  15337  efgredlemd  15339  efgredlem  15342  frgpnabllem1  15447
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-ot 3792  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-fzo 11099  df-hash 11582  df-word 11686  df-concat 11687  df-s1 11688  df-substr 11689  df-splice 11690  df-s2 11775
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