MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  efgi2 Unicode version

Theorem efgi2 15312
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 5687 . . . . . . . . . . 11  |-  ( a  =  A  ->  ( T `  a )  =  ( T `  A ) )
21rneqd 5056 . . . . . . . . . 10  |-  ( a  =  A  ->  ran  ( T `  a )  =  ran  ( T `
 A ) )
3 eceq1 6900 . . . . . . . . . 10  |-  ( a  =  A  ->  [ a ] r  =  [ A ] r )
42, 3sseq12d 3337 . . . . . . . . 9  |-  ( a  =  A  ->  ( ran  ( T `  a
)  C_  [ a ] r  <->  ran  ( T `
 A )  C_  [ A ] r ) )
54rspcv 3008 . . . . . . . 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 3302 . . . . . . . . 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 6902 . . . . . . . . . . 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 4173 . . . . . . . . . 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 4387 . . . . 5  |-  <. A ,  B >.  e.  _V
2019elintab 4021 . . . 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 15311 . . 3  |-  .~  =  |^| { r  |  ( r  Er  W  /\  A. a  e.  W  ran  ( T `  a ) 
C_  [ a ] r ) }
2721, 26syl6eleqr 2495 . 2  |-  ( ( A  e.  W  /\  B  e.  ran  ( T `
 A ) )  ->  <. A ,  B >.  e.  .~  )
28 df-br 4173 . 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 1721   {cab 2390   A.wral 2666    \ cdif 3277    C_ wss 3280   <.cop 3777   <.cotp 3778   |^|cint 4010   class class class wbr 4172    e. cmpt 4226    _I cid 4453    X. cxp 4835   ran crn 4838   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1oc1o 6676   2oc2o 6677    Er wer 6861   [cec 6862   0cc0 8946   ...cfz 10999   #chash 11573  Word cword 11672   splice csplice 11676   <"cs2 11760   ~FG cefg 15293
This theorem is referenced by:  efginvrel2  15314  efgsrel  15321  efgcpbllemb  15342
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-ec 6866  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-hash 11574  df-word 11678  df-concat 11679  df-s1 11680  df-substr 11681  df-splice 11682  df-s2 11767  df-efg 15296
  Copyright terms: Public domain W3C validator