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Theorem efgtlen 15363
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 15359 . . . . . . 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 447 . . . . . 6  |-  ( X  e.  W  ->  ( T `  X )  =  ( a  e.  ( 0 ... ( # `
 X ) ) ,  b  e.  ( I  X.  2o ) 
|->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) ) )
76rneqd 5100 . . . . 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 2505 . . . 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 2438 . . . . 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 6109 . . . . 5  |-  ( X splice  <. a ,  a , 
<" b ( M `
 b ) "> >. )  e.  _V
119, 10elrnmpt2 6186 . . . 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 254 . . 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 5787 . . . . . . . . 9  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
141, 13eqsstri 3380 . . . . . . . 8  |-  W  C_ Word  ( I  X.  2o )
15 simpl 445 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  X  e.  W
)
1614, 15sseldi 3348 . . . . . . 7  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  X  e. Word  (
I  X.  2o ) )
17 elfzuz 11060 . . . . . . . . 9  |-  ( a  e.  ( 0 ... ( # `  X
) )  ->  a  e.  ( ZZ>= `  0 )
)
1817ad2antrl 710 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e.  (
ZZ>= `  0 ) )
19 eluzfz2b 11071 . . . . . . . 8  |-  ( a  e.  ( ZZ>= `  0
)  <->  a  e.  ( 0 ... a ) )
2018, 19sylib 190 . . . . . . 7  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e.  ( 0 ... a ) )
21 simprl 734 . . . . . . 7  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e.  ( 0 ... ( # `  X ) ) )
22 simprr 735 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  b  e.  ( I  X.  2o ) )
233efgmf 15350 . . . . . . . . . 10  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
2423ffvelrni 5872 . . . . . . . . 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 11838 . . . . . . 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 11788 . . . . . 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 11856 . . . . . . . . . 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 10501 . . . . . . . . . . . 12  |-  ( a  e.  ( ZZ>= `  0
)  ->  a  e.  ZZ )
3130zcnd 10381 . . . . . . . . . . 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 9404 . . . . . . . . 9  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( a  -  a )  =  0 )
3429, 33oveq12d 6102 . . . . . . . 8  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( ( # `  <" b ( M `  b ) "> )  -  ( a  -  a
) )  =  ( 2  -  0 ) )
35 2cn 10075 . . . . . . . . 9  |-  2  e.  CC
3635subid1i 9377 . . . . . . . 8  |-  ( 2  -  0 )  =  2
3734, 36syl6eq 2486 . . . . . . 7  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( ( # `  <" b ( M `  b ) "> )  -  ( a  -  a
) )  =  2 )
3837oveq2d 6100 . . . . . 6  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( ( # `  X )  +  ( ( # `  <" b ( M `  b ) "> )  -  ( a  -  a ) ) )  =  ( (
# `  X )  +  2 ) )
3927, 38eqtrd 2470 . . . . 5  |-  ( ( X  e.  W  /\  ( a  e.  ( 0 ... ( # `  X ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( # `  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >. )
)  =  ( (
# `  X )  +  2 ) )
40 fveq2 5731 . . . . . 6  |-  ( A  =  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
)  ->  ( # `  A
)  =  ( # `  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) ) )
4140eqeq1d 2446 . . . . 5  |-  ( A  =  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
)  ->  ( ( # `
 A )  =  ( ( # `  X
)  +  2 )  <-> 
( # `  ( X splice  <. a ,  a , 
<" b ( M `
 b ) "> >. ) )  =  ( ( # `  X
)  +  2 ) ) )
4239, 41syl5ibrcom 215 . . . 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 2839 . . 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 208 . 2  |-  ( X  e.  W  ->  ( A  e.  ran  ( T `
 X )  -> 
( # `  A )  =  ( ( # `  X )  +  2 ) ) )
4544imp 420 1  |-  ( ( X  e.  W  /\  A  e.  ran  ( T `
 X ) )  ->  ( # `  A
)  =  ( (
# `  X )  +  2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708    \ cdif 3319   <.cop 3819   <.cotp 3820    e. cmpt 4269    _I cid 4496    X. cxp 4879   ran crn 4882   -->wf 5453   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   1oc1o 6720   2oc2o 6721   CCcc 8993   0cc0 8995    + caddc 8998    - cmin 9296   2c2 10054   ZZ>=cuz 10493   ...cfz 11048   #chash 11623  Word cword 11722   splice csplice 11726   <"cs2 11810   ~FG cefg 15343
This theorem is referenced by:  efgsfo  15376  efgredlemg  15379  efgredlemd  15381  efgredlem  15384  frgpnabllem1  15489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
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 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-n0 10227  df-z 10288  df-uz 10494  df-fz 11049  df-fzo 11141  df-hash 11624  df-word 11728  df-concat 11729  df-s1 11730  df-substr 11731  df-splice 11732  df-s2 11817
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