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Theorem efgi2 16213
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 5686 . . . . . . . . . . 11  |-  ( a  =  A  ->  ( T `  a )  =  ( T `  A ) )
21rneqd 5062 . . . . . . . . . 10  |-  ( a  =  A  ->  ran  ( T `  a )  =  ran  ( T `
 A ) )
3 eceq1 7129 . . . . . . . . . 10  |-  ( a  =  A  ->  [ a ] r  =  [ A ] r )
42, 3sseq12d 3380 . . . . . . . . 9  |-  ( a  =  A  ->  ( ran  ( T `  a
)  C_  [ a ] r  <->  ran  ( T `
 A )  C_  [ A ] r ) )
54rspcv 3064 . . . . . . . 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 3345 . . . . . . . . 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 7131 . . . . . . . . . . 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 4288 . . . . . . . . . 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 1685 . . . 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 4551 . . . . 5  |-  <. A ,  B >.  e.  _V
2019elintab 4134 . . . 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 16212 . . 3  |-  .~  =  |^| { r  |  ( r  Er  W  /\  A. a  e.  W  ran  ( T `  a ) 
C_  [ a ] r ) }
2721, 26syl6eleqr 2529 . 2  |-  ( ( A  e.  W  /\  B  e.  ran  ( T `
 A ) )  ->  <. A ,  B >.  e.  .~  )
28 df-br 4288 . 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 1367    = wceq 1369    e. wcel 1756   {cab 2424   A.wral 2710    \ cdif 3320    C_ wss 3323   <.cop 3878   <.cotp 3880   |^|cint 4123   class class class wbr 4287    e. cmpt 4345    _I cid 4626    X. cxp 4833   ran crn 4836   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   1oc1o 6905   2oc2o 6906    Er wer 7090   [cec 7091   0cc0 9274   ...cfz 11429   #chash 12095  Word cword 12213   splice csplice 12218   <"cs2 12460   ~FG cefg 16194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-ot 3881  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-ec 7095  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-fzo 11541  df-hash 12096  df-word 12221  df-concat 12223  df-s1 12224  df-substr 12225  df-splice 12226  df-s2 12467  df-efg 16197
This theorem is referenced by:  efginvrel2  16215  efgsrel  16222  efgcpbllemb  16243
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