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Theorem efgi2 15050
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 5541 . . . . . . . . . . 11  |-  ( a  =  A  ->  ( T `  a )  =  ( T `  A ) )
21rneqd 4922 . . . . . . . . . 10  |-  ( a  =  A  ->  ran  ( T `  a )  =  ran  ( T `
 A ) )
3 eceq1 6712 . . . . . . . . . 10  |-  ( a  =  A  ->  [ a ] r  =  [ A ] r )
42, 3sseq12d 3220 . . . . . . . . 9  |-  ( a  =  A  ->  ( ran  ( T `  a
)  C_  [ a ] r  <->  ran  ( T `
 A )  C_  [ A ] r ) )
54rspcv 2893 . . . . . . . 8  |-  ( A  e.  W  ->  ( A. a  e.  W  ran  ( T `  a
)  C_  [ a ] r  ->  ran  ( T `  A ) 
C_  [ A ]
r ) )
65adantr 451 . . . . . . 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 3187 . . . . . . . . 9  |-  ( ran  ( T `  A
)  C_  [ A ] r  ->  ( B  e.  ran  ( T `
 A )  ->  B  e.  [ A ] r ) )
87com12 27 . . . . . . . 8  |-  ( B  e.  ran  ( T `
 A )  -> 
( ran  ( T `  A )  C_  [ A ] r  ->  B  e.  [ A ] r ) )
9 simpl 443 . . . . . . . . . . 11  |-  ( ( B  e.  [ A ] r  /\  A  e.  W )  ->  B  e.  [ A ] r )
10 elecg 6714 . . . . . . . . . . 11  |-  ( ( B  e.  [ A ] r  /\  A  e.  W )  ->  ( B  e.  [ A ] r  <->  A r B ) )
119, 10mpbid 201 . . . . . . . . . 10  |-  ( ( B  e.  [ A ] r  /\  A  e.  W )  ->  A
r B )
12 df-br 4040 . . . . . . . . . 10  |-  ( A r B  <->  <. A ,  B >.  e.  r )
1311, 12sylib 188 . . . . . . . . 9  |-  ( ( B  e.  [ A ] r  /\  A  e.  W )  ->  <. A ,  B >.  e.  r )
1413expcom 424 . . . . . . . 8  |-  ( A  e.  W  ->  ( B  e.  [ A ] r  ->  <. A ,  B >.  e.  r ) )
158, 14sylan9r 639 . . . . . . 7  |-  ( ( A  e.  W  /\  B  e.  ran  ( T `
 A ) )  ->  ( ran  ( T `  A )  C_ 
[ A ] r  ->  <. A ,  B >.  e.  r ) )
166, 15syld 40 . . . . . 6  |-  ( ( A  e.  W  /\  B  e.  ran  ( T `
 A ) )  ->  ( A. a  e.  W  ran  ( T `
 a )  C_  [ a ] r  ->  <. A ,  B >.  e.  r ) )
1716adantld 453 . . . . 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 1621 . . . 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 4253 . . . . 5  |-  <. A ,  B >.  e.  _V
2019elintab 3889 . . . 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 203 . . 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 15049 . . 3  |-  .~  =  |^| { r  |  ( r  Er  W  /\  A. a  e.  W  ran  ( T `  a ) 
C_  [ a ] r ) }
2721, 26syl6eleqr 2387 . 2  |-  ( ( A  e.  W  /\  B  e.  ran  ( T `
 A ) )  ->  <. A ,  B >.  e.  .~  )
28 df-br 4040 . 2  |-  ( A  .~  B  <->  <. A ,  B >.  e.  .~  )
2927, 28sylibr 203 1  |-  ( ( A  e.  W  /\  B  e.  ran  ( T `
 A ) )  ->  A  .~  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556    \ cdif 3162    C_ wss 3165   <.cop 3656   <.cotp 3657   |^|cint 3878   class class class wbr 4039    e. cmpt 4093    _I cid 4320    X. cxp 4703   ran crn 4706   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1oc1o 6488   2oc2o 6489    Er wer 6673   [cec 6674   0cc0 8753   ...cfz 10798   #chash 11353  Word cword 11419   splice csplice 11423   <"cs2 11507   ~FG cefg 15031
This theorem is referenced by:  efginvrel2  15052  efgsrel  15059  efgcpbllemb  15080
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-ot 3663  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-ec 6678  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-hash 11354  df-word 11425  df-concat 11426  df-s1 11427  df-substr 11428  df-splice 11429  df-s2 11514  df-efg 15034
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