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Theorem efgi2 16616
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 5872 . . . . . . . . . . 11  |-  ( a  =  A  ->  ( T `  a )  =  ( T `  A ) )
21rneqd 5236 . . . . . . . . . 10  |-  ( a  =  A  ->  ran  ( T `  a )  =  ran  ( T `
 A ) )
3 eceq1 7359 . . . . . . . . . 10  |-  ( a  =  A  ->  [ a ] r  =  [ A ] r )
42, 3sseq12d 3538 . . . . . . . . 9  |-  ( a  =  A  ->  ( ran  ( T `  a
)  C_  [ a ] r  <->  ran  ( T `
 A )  C_  [ A ] r ) )
54rspcv 3215 . . . . . . . 8  |-  ( A  e.  W  ->  ( A. a  e.  W  ran  ( T `  a
)  C_  [ a ] r  ->  ran  ( T `  A ) 
C_  [ A ]
r ) )
65adantr 465 . . . . . . 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 3503 . . . . . . . . 9  |-  ( ran  ( T `  A
)  C_  [ A ] r  ->  ( B  e.  ran  ( T `
 A )  ->  B  e.  [ A ] r ) )
87com12 31 . . . . . . . 8  |-  ( B  e.  ran  ( T `
 A )  -> 
( ran  ( T `  A )  C_  [ A ] r  ->  B  e.  [ A ] r ) )
9 simpl 457 . . . . . . . . . . 11  |-  ( ( B  e.  [ A ] r  /\  A  e.  W )  ->  B  e.  [ A ] r )
10 elecg 7362 . . . . . . . . . . 11  |-  ( ( B  e.  [ A ] r  /\  A  e.  W )  ->  ( B  e.  [ A ] r  <->  A r B ) )
119, 10mpbid 210 . . . . . . . . . 10  |-  ( ( B  e.  [ A ] r  /\  A  e.  W )  ->  A
r B )
12 df-br 4454 . . . . . . . . . 10  |-  ( A r B  <->  <. A ,  B >.  e.  r )
1311, 12sylib 196 . . . . . . . . 9  |-  ( ( B  e.  [ A ] r  /\  A  e.  W )  ->  <. A ,  B >.  e.  r )
1413expcom 435 . . . . . . . 8  |-  ( A  e.  W  ->  ( B  e.  [ A ] r  ->  <. A ,  B >.  e.  r ) )
158, 14sylan9r 658 . . . . . . 7  |-  ( ( A  e.  W  /\  B  e.  ran  ( T `
 A ) )  ->  ( ran  ( T `  A )  C_ 
[ A ] r  ->  <. A ,  B >.  e.  r ) )
166, 15syld 44 . . . . . 6  |-  ( ( A  e.  W  /\  B  e.  ran  ( T `
 A ) )  ->  ( A. a  e.  W  ran  ( T `
 a )  C_  [ a ] r  ->  <. A ,  B >.  e.  r ) )
1716adantld 467 . . . . 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 1695 . . . 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 4717 . . . . 5  |-  <. A ,  B >.  e.  _V
2019elintab 4299 . . . 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 212 . . 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 16615 . . 3  |-  .~  =  |^| { r  |  ( r  Er  W  /\  A. a  e.  W  ran  ( T `  a ) 
C_  [ a ] r ) }
2721, 26syl6eleqr 2566 . 2  |-  ( ( A  e.  W  /\  B  e.  ran  ( T `
 A ) )  ->  <. A ,  B >.  e.  .~  )
28 df-br 4454 . 2  |-  ( A  .~  B  <->  <. A ,  B >.  e.  .~  )
2927, 28sylibr 212 1  |-  ( ( A  e.  W  /\  B  e.  ran  ( T `
 A ) )  ->  A  .~  B
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1377    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2817    \ cdif 3478    C_ wss 3481   <.cop 4039   <.cotp 4041   |^|cint 4288   class class class wbr 4453    |-> cmpt 4511    _I cid 4796    X. cxp 5003   ran crn 5006   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   1oc1o 7135   2oc2o 7136    Er wer 7320   [cec 7321   0cc0 9504   ...cfz 11684   #chash 12385  Word cword 12515   splice csplice 12520   <"cs2 12786   ~FG cefg 16597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-ot 4042  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-ec 7325  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-hash 12386  df-word 12523  df-concat 12525  df-s1 12526  df-substr 12527  df-splice 12528  df-s2 12793  df-efg 16600
This theorem is referenced by:  efginvrel2  16618  efgsrel  16625  efgcpbllemb  16646
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