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Theorem efgtlen 15128
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 15124 . . . . . . 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 445 . . . . . 6  |-  ( X  e.  W  ->  ( T `  X )  =  ( a  e.  ( 0 ... ( # `
 X ) ) ,  b  e.  ( I  X.  2o ) 
|->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) ) )
76rneqd 4985 . . . . 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 2425 . . . 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 2358 . . . . 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 5967 . . . . 5  |-  ( X splice  <. a ,  a , 
<" b ( M `
 b ) "> >. )  e.  _V
119, 10elrnmpt2 6041 . . . 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 252 . . 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 5660 . . . . . . . . 9  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
141, 13eqsstri 3284 . . . . . . . 8  |-  W  C_ Word  ( I  X.  2o )
15 simpl 443 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  X  e.  W
)
1614, 15sseldi 3254 . . . . . . 7  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  X  e. Word  (
I  X.  2o ) )
17 elfzuz 10883 . . . . . . . . 9  |-  ( a  e.  ( 0 ... ( # `  X
) )  ->  a  e.  ( ZZ>= `  0 )
)
1817ad2antrl 708 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e.  (
ZZ>= `  0 ) )
19 eluzfz2b 10894 . . . . . . . 8  |-  ( a  e.  ( ZZ>= `  0
)  <->  a  e.  ( 0 ... a ) )
2018, 19sylib 188 . . . . . . 7  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e.  ( 0 ... a ) )
21 simprl 732 . . . . . . 7  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e.  ( 0 ... ( # `  X ) ) )
22 simprr 733 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  b  e.  ( I  X.  2o ) )
233efgmf 15115 . . . . . . . . . 10  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
2423ffvelrni 5744 . . . . . . . . 9  |-  ( b  e.  ( I  X.  2o )  ->  ( M `
 b )  e.  ( I  X.  2o ) )
2522, 24syl 15 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M `  b )  e.  ( I  X.  2o ) )
2622, 25s2cld 11609 . . . . . . 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 11559 . . . . . 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 11627 . . . . . . . . . 10  |-  ( # `  <" b ( M `  b ) "> )  =  2
2928a1i 10 . . . . . . . . 9  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `  <" b ( M `  b ) "> )  =  2 )
30 eluzelz 10327 . . . . . . . . . . . 12  |-  ( a  e.  ( ZZ>= `  0
)  ->  a  e.  ZZ )
3130zcnd 10207 . . . . . . . . . . 11  |-  ( a  e.  ( ZZ>= `  0
)  ->  a  e.  CC )
3218, 31syl 15 . . . . . . . . . 10  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e.  CC )
3332subidd 9232 . . . . . . . . 9  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( a  -  a )  =  0 )
3429, 33oveq12d 5960 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( ( # `  <" b ( M `  b ) "> )  -  ( a  -  a
) )  =  ( 2  -  0 ) )
35 2cn 9903 . . . . . . . . 9  |-  2  e.  CC
3635subid1i 9205 . . . . . . . 8  |-  ( 2  -  0 )  =  2
3734, 36syl6eq 2406 . . . . . . 7  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( ( # `  <" b ( M `  b ) "> )  -  ( a  -  a
) )  =  2 )
3837oveq2d 5958 . . . . . 6  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( ( # `  X )  +  ( ( # `  <" b ( M `  b ) "> )  -  ( a  -  a ) ) )  =  ( (
# `  X )  +  2 ) )
3927, 38eqtrd 2390 . . . . 5  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >. )
)  =  ( (
# `  X )  +  2 ) )
40 fveq2 5605 . . . . . 6  |-  ( A  =  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
)  ->  ( # `  A
)  =  ( # `  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) ) )
4140eqeq1d 2366 . . . . 5  |-  ( A  =  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
)  ->  ( ( # `
 A )  =  ( ( # `  X
)  +  2 )  <-> 
( # `  ( X splice  <. a ,  a , 
<" b ( M `
 b ) "> >. ) )  =  ( ( # `  X
)  +  2 ) ) )
4239, 41syl5ibrcom 213 . . . 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 2750 . . 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 206 . 2  |-  ( X  e.  W  ->  ( A  e.  ran  ( T `
 X )  -> 
( # `  A )  =  ( ( # `  X )  +  2 ) ) )
4544imp 418 1  |-  ( ( X  e.  W  /\  A  e.  ran  ( T `
 X ) )  ->  ( # `  A
)  =  ( (
# `  X )  +  2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   E.wrex 2620    \ cdif 3225   <.cop 3719   <.cotp 3720    e. cmpt 4156    _I cid 4383    X. cxp 4766   ran crn 4769   -->wf 5330   ` cfv 5334  (class class class)co 5942    e. cmpt2 5944   1oc1o 6556   2oc2o 6557   CCcc 8822   0cc0 8824    + caddc 8827    - cmin 9124   2c2 9882   ZZ>=cuz 10319   ...cfz 10871   #chash 11427  Word cword 11493   splice csplice 11497   <"cs2 11581   ~FG cefg 15108
This theorem is referenced by:  efgsfo  15141  efgredlemg  15144  efgredlemd  15146  efgredlem  15149  frgpnabllem1  15254
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-ot 3726  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-2o 6564  df-oadd 6567  df-er 6744  df-map 6859  df-pm 6860  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-card 7659  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-nn 9834  df-2 9891  df-n0 10055  df-z 10114  df-uz 10320  df-fz 10872  df-fzo 10960  df-hash 11428  df-word 11499  df-concat 11500  df-s1 11501  df-substr 11502  df-splice 11503  df-s2 11588
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