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Theorem efgtlen 16216
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 16212 . . . . . . 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 456 . . . . . 6  |-  ( X  e.  W  ->  ( T `  X )  =  ( a  e.  ( 0 ... ( # `
 X ) ) ,  b  e.  ( I  X.  2o ) 
|->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) ) )
76rneqd 5063 . . . . 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 2508 . . . 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 2441 . . . . 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 6115 . . . . 5  |-  ( X splice  <. a ,  a , 
<" b ( M `
 b ) "> >. )  e.  _V
119, 10elrnmpt2 6202 . . . 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 261 . . 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 5746 . . . . . . . . 9  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
141, 13eqsstri 3383 . . . . . . . 8  |-  W  C_ Word  ( I  X.  2o )
15 simpl 454 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  X  e.  W
)
1614, 15sseldi 3351 . . . . . . 7  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  X  e. Word  (
I  X.  2o ) )
17 elfzuz 11445 . . . . . . . . 9  |-  ( a  e.  ( 0 ... ( # `  X
) )  ->  a  e.  ( ZZ>= `  0 )
)
1817ad2antrl 722 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e.  (
ZZ>= `  0 ) )
19 eluzfz2b 11456 . . . . . . . 8  |-  ( a  e.  ( ZZ>= `  0
)  <->  a  e.  ( 0 ... a ) )
2018, 19sylib 196 . . . . . . 7  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e.  ( 0 ... a ) )
21 simprl 750 . . . . . . 7  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e.  ( 0 ... ( # `  X ) ) )
22 simprr 751 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  b  e.  ( I  X.  2o ) )
233efgmf 16203 . . . . . . . . . 10  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
2423ffvelrni 5839 . . . . . . . . 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 12492 . . . . . . 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 12392 . . . . . 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 12510 . . . . . . . . . 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 10866 . . . . . . . . . . . 12  |-  ( a  e.  ( ZZ>= `  0
)  ->  a  e.  ZZ )
3130zcnd 10744 . . . . . . . . . . 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 9703 . . . . . . . . 9  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( a  -  a )  =  0 )
3429, 33oveq12d 6108 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( ( # `  <" b ( M `  b ) "> )  -  ( a  -  a
) )  =  ( 2  -  0 ) )
35 2cn 10388 . . . . . . . . 9  |-  2  e.  CC
3635subid1i 9676 . . . . . . . 8  |-  ( 2  -  0 )  =  2
3734, 36syl6eq 2489 . . . . . . 7  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( ( # `  <" b ( M `  b ) "> )  -  ( a  -  a
) )  =  2 )
3837oveq2d 6106 . . . . . 6  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( ( # `  X )  +  ( ( # `  <" b ( M `  b ) "> )  -  ( a  -  a ) ) )  =  ( (
# `  X )  +  2 ) )
3927, 38eqtrd 2473 . . . . 5  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >. )
)  =  ( (
# `  X )  +  2 ) )
40 fveq2 5688 . . . . . 6  |-  ( A  =  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
)  ->  ( # `  A
)  =  ( # `  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) ) )
4140eqeq1d 2449 . . . . 5  |-  ( A  =  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
)  ->  ( ( # `
 A )  =  ( ( # `  X
)  +  2 )  <-> 
( # `  ( X splice  <. a ,  a , 
<" b ( M `
 b ) "> >. ) )  =  ( ( # `  X
)  +  2 ) ) )
4239, 41syl5ibrcom 222 . . . 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 2846 . . 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 215 . 2  |-  ( X  e.  W  ->  ( A  e.  ran  ( T `
 X )  -> 
( # `  A )  =  ( ( # `  X )  +  2 ) ) )
4544imp 429 1  |-  ( ( X  e.  W  /\  A  e.  ran  ( T `
 X ) )  ->  ( # `  A
)  =  ( (
# `  X )  +  2 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   E.wrex 2714    \ cdif 3322   <.cop 3880   <.cotp 3882    e. cmpt 4347    _I cid 4627    X. cxp 4834   ran crn 4837   -->wf 5411   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   1oc1o 6909   2oc2o 6910   CCcc 9276   0cc0 9278    + caddc 9281    - cmin 9591   2c2 10367   ZZ>=cuz 10857   ...cfz 11433   #chash 12099  Word cword 12217   splice csplice 12222   <"cs2 12464   ~FG cefg 16196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-ot 3883  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-hash 12100  df-word 12225  df-concat 12227  df-s1 12228  df-substr 12229  df-splice 12230  df-s2 12471
This theorem is referenced by:  efgsfo  16229  efgredlemg  16232  efgredlemd  16234  efgredlem  16237  frgpnabllem1  16344
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