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

Theorem frgpval 15345
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 2924 . . 3  |-  ( I  e.  V  ->  I  e.  _V )
3 xpeq1 4851 . . . . . . 7  |-  ( i  =  I  ->  (
i  X.  2o )  =  ( I  X.  2o ) )
43fveq2d 5691 . . . . . 6  |-  ( i  =  I  ->  (freeMnd `  ( i  X.  2o ) )  =  (freeMnd `  ( I  X.  2o ) ) )
5 frgpval.b . . . . . 6  |-  M  =  (freeMnd `  ( I  X.  2o ) )
64, 5syl6eqr 2454 . . . . 5  |-  ( i  =  I  ->  (freeMnd `  ( i  X.  2o ) )  =  M )
7 fveq2 5687 . . . . . 6  |-  ( i  =  I  ->  ( ~FG  `  i
)  =  ( ~FG  `  I
) )
8 frgpval.r . . . . . 6  |-  .~  =  ( ~FG  `  I )
97, 8syl6eqr 2454 . . . . 5  |-  ( i  =  I  ->  ( ~FG  `  i
)  =  .~  )
106, 9oveq12d 6058 . . . 4  |-  ( i  =  I  ->  (
(freeMnd `  ( i  X.  2o ) )  /.s  ( ~FG  `  i
) )  =  ( M  /.s 
.~  ) )
11 df-frgp 15297 . . . 4  |- freeGrp  =  ( i  e.  _V  |->  ( (freeMnd `  ( i  X.  2o ) )  /.s  ( ~FG  `  i
) ) )
12 ovex 6065 . . . 4  |-  ( M 
/.s  .~  )  e.  _V
1310, 11, 12fvmpt 5765 . . 3  |-  ( I  e.  _V  ->  (freeGrp `  I )  =  ( M  /.s 
.~  ) )
142, 13syl 16 . 2  |-  ( I  e.  V  ->  (freeGrp `  I )  =  ( M  /.s 
.~  ) )
151, 14syl5eq 2448 1  |-  ( I  e.  V  ->  G  =  ( M  /.s  .~  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   _Vcvv 2916    X. cxp 4835   ` cfv 5413  (class class class)co 6040   2oc2o 6677    /.s cqus 13686  freeMndcfrmd 14747   ~FG cefg 15293  freeGrpcfrgp 15294
This theorem is referenced by:  frgp0  15347  frgpeccl  15348  frgpadd  15350  frgpupf  15360  frgpup1  15362  frgpup3lem  15364  frgpnabllem2  15440
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-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  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-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-frgp 15297
  Copyright terms: Public domain W3C validator