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

Theorem frgpval 16255
Description: Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
Hypotheses
Ref Expression
frgpval.m  |-  G  =  (freeGrp `  I )
frgpval.b  |-  M  =  (freeMnd `  ( I  X.  2o ) )
frgpval.r  |-  .~  =  ( ~FG  `  I )
Assertion
Ref Expression
frgpval  |-  ( I  e.  V  ->  G  =  ( M  /.s  .~  )
)

Proof of Theorem frgpval
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 frgpval.m . 2  |-  G  =  (freeGrp `  I )
2 elex 2981 . . 3  |-  ( I  e.  V  ->  I  e.  _V )
3 xpeq1 4854 . . . . . . 7  |-  ( i  =  I  ->  (
i  X.  2o )  =  ( I  X.  2o ) )
43fveq2d 5695 . . . . . 6  |-  ( i  =  I  ->  (freeMnd `  ( i  X.  2o ) )  =  (freeMnd `  ( I  X.  2o ) ) )
5 frgpval.b . . . . . 6  |-  M  =  (freeMnd `  ( I  X.  2o ) )
64, 5syl6eqr 2493 . . . . 5  |-  ( i  =  I  ->  (freeMnd `  ( i  X.  2o ) )  =  M )
7 fveq2 5691 . . . . . 6  |-  ( i  =  I  ->  ( ~FG  `  i
)  =  ( ~FG  `  I
) )
8 frgpval.r . . . . . 6  |-  .~  =  ( ~FG  `  I )
97, 8syl6eqr 2493 . . . . 5  |-  ( i  =  I  ->  ( ~FG  `  i
)  =  .~  )
106, 9oveq12d 6109 . . . 4  |-  ( i  =  I  ->  (
(freeMnd `  ( i  X.  2o ) )  /.s  ( ~FG  `  i
) )  =  ( M  /.s 
.~  ) )
11 df-frgp 16207 . . . 4  |- freeGrp  =  ( i  e.  _V  |->  ( (freeMnd `  ( i  X.  2o ) )  /.s  ( ~FG  `  i
) ) )
12 ovex 6116 . . . 4  |-  ( M 
/.s  .~  )  e.  _V
1310, 11, 12fvmpt 5774 . . 3  |-  ( I  e.  _V  ->  (freeGrp `  I )  =  ( M  /.s 
.~  ) )
142, 13syl 16 . 2  |-  ( I  e.  V  ->  (freeGrp `  I )  =  ( M  /.s 
.~  ) )
151, 14syl5eq 2487 1  |-  ( I  e.  V  ->  G  =  ( M  /.s  .~  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2972    X. cxp 4838   ` cfv 5418  (class class class)co 6091   2oc2o 6914    /.s cqus 14443  freeMndcfrmd 15525   ~FG cefg 16203  freeGrpcfrgp 16204
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-frgp 16207
This theorem is referenced by:  frgp0  16257  frgpeccl  16258  frgpadd  16260  frgpupf  16270  frgpup1  16272  frgpup3lem  16274  frgpnabllem2  16352
  Copyright terms: Public domain W3C validator