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Theorem efgi2 15285
Description: Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-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
efgi2  |-  ( ( A  e.  W  /\  B  e.  ran  ( T `
 A ) )  ->  A  .~  B
)
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)    B( y, z, w, v, n)    .~ ( w, v, n)    T( y, z, w, v, n)    M( y, z)

Proof of Theorem efgi2
Dummy variables  a 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5669 . . . . . . . . . . 11  |-  ( a  =  A  ->  ( T `  a )  =  ( T `  A ) )
21rneqd 5038 . . . . . . . . . 10  |-  ( a  =  A  ->  ran  ( T `  a )  =  ran  ( T `
 A ) )
3 eceq1 6878 . . . . . . . . . 10  |-  ( a  =  A  ->  [ a ] r  =  [ A ] r )
42, 3sseq12d 3321 . . . . . . . . 9  |-  ( a  =  A  ->  ( ran  ( T `  a
)  C_  [ a ] r  <->  ran  ( T `
 A )  C_  [ A ] r ) )
54rspcv 2992 . . . . . . . 8  |-  ( A  e.  W  ->  ( A. a  e.  W  ran  ( T `  a
)  C_  [ a ] r  ->  ran  ( T `  A ) 
C_  [ A ]
r ) )
65adantr 452 . . . . . . 7  |-  ( ( A  e.  W  /\  B  e.  ran  ( T `
 A ) )  ->  ( A. a  e.  W  ran  ( T `
 a )  C_  [ a ] r  ->  ran  ( T `  A
)  C_  [ A ] r ) )
7 ssel 3286 . . . . . . . . 9  |-  ( ran  ( T `  A
)  C_  [ A ] r  ->  ( B  e.  ran  ( T `
 A )  ->  B  e.  [ A ] r ) )
87com12 29 . . . . . . . 8  |-  ( B  e.  ran  ( T `
 A )  -> 
( ran  ( T `  A )  C_  [ A ] r  ->  B  e.  [ A ] r ) )
9 simpl 444 . . . . . . . . . . 11  |-  ( ( B  e.  [ A ] r  /\  A  e.  W )  ->  B  e.  [ A ] r )
10 elecg 6880 . . . . . . . . . . 11  |-  ( ( B  e.  [ A ] r  /\  A  e.  W )  ->  ( B  e.  [ A ] r  <->  A r B ) )
119, 10mpbid 202 . . . . . . . . . 10  |-  ( ( B  e.  [ A ] r  /\  A  e.  W )  ->  A
r B )
12 df-br 4155 . . . . . . . . . 10  |-  ( A r B  <->  <. A ,  B >.  e.  r )
1311, 12sylib 189 . . . . . . . . 9  |-  ( ( B  e.  [ A ] r  /\  A  e.  W )  ->  <. A ,  B >.  e.  r )
1413expcom 425 . . . . . . . 8  |-  ( A  e.  W  ->  ( B  e.  [ A ] r  ->  <. A ,  B >.  e.  r ) )
158, 14sylan9r 640 . . . . . . 7  |-  ( ( A  e.  W  /\  B  e.  ran  ( T `
 A ) )  ->  ( ran  ( T `  A )  C_ 
[ A ] r  ->  <. A ,  B >.  e.  r ) )
166, 15syld 42 . . . . . 6  |-  ( ( A  e.  W  /\  B  e.  ran  ( T `
 A ) )  ->  ( A. a  e.  W  ran  ( T `
 a )  C_  [ a ] r  ->  <. A ,  B >.  e.  r ) )
1716adantld 454 . . . . 5  |-  ( ( A  e.  W  /\  B  e.  ran  ( T `
 A ) )  ->  ( ( r  Er  W  /\  A. a  e.  W  ran  ( T `  a ) 
C_  [ a ] r )  ->  <. A ,  B >.  e.  r ) )
1817alrimiv 1638 . . . 4  |-  ( ( A  e.  W  /\  B  e.  ran  ( T `
 A ) )  ->  A. r ( ( r  Er  W  /\  A. a  e.  W  ran  ( T `  a ) 
C_  [ a ] r )  ->  <. A ,  B >.  e.  r ) )
19 opex 4369 . . . . 5  |-  <. A ,  B >.  e.  _V
2019elintab 4004 . . . 4  |-  ( <. A ,  B >.  e. 
|^| { r  |  ( r  Er  W  /\  A. a  e.  W  ran  ( T `  a ) 
C_  [ a ] r ) }  <->  A. r
( ( r  Er  W  /\  A. a  e.  W  ran  ( T `
 a )  C_  [ a ] r )  ->  <. A ,  B >.  e.  r ) )
2118, 20sylibr 204 . . 3  |-  ( ( A  e.  W  /\  B  e.  ran  ( T `
 A ) )  ->  <. A ,  B >.  e.  |^| { r  |  ( r  Er  W  /\  A. a  e.  W  ran  ( T `  a
)  C_  [ a ] r ) } )
22 efgval.w . . . 4  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
23 efgval.r . . . 4  |-  .~  =  ( ~FG  `  I )
24 efgval2.m . . . 4  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
25 efgval2.t . . . 4  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
2622, 23, 24, 25efgval2 15284 . . 3  |-  .~  =  |^| { r  |  ( r  Er  W  /\  A. a  e.  W  ran  ( T `  a ) 
C_  [ a ] r ) }
2721, 26syl6eleqr 2479 . 2  |-  ( ( A  e.  W  /\  B  e.  ran  ( T `
 A ) )  ->  <. A ,  B >.  e.  .~  )
28 df-br 4155 . 2  |-  ( A  .~  B  <->  <. A ,  B >.  e.  .~  )
2927, 28sylibr 204 1  |-  ( ( A  e.  W  /\  B  e.  ran  ( T `
 A ) )  ->  A  .~  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1717   {cab 2374   A.wral 2650    \ cdif 3261    C_ wss 3264   <.cop 3761   <.cotp 3762   |^|cint 3993   class class class wbr 4154    e. cmpt 4208    _I cid 4435    X. cxp 4817   ran crn 4820   ` cfv 5395  (class class class)co 6021    e. cmpt2 6023   1oc1o 6654   2oc2o 6655    Er wer 6839   [cec 6840   0cc0 8924   ...cfz 10976   #chash 11546  Word cword 11645   splice csplice 11649   <"cs2 11733   ~FG cefg 15266
This theorem is referenced by:  efginvrel2  15287  efgsrel  15294  efgcpbllemb  15315
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-ot 3768  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-ec 6844  df-map 6957  df-pm 6958  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-card 7760  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-n0 10155  df-z 10216  df-uz 10422  df-fz 10977  df-fzo 11067  df-hash 11547  df-word 11651  df-concat 11652  df-s1 11653  df-substr 11654  df-splice 11655  df-s2 11740  df-efg 15269
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