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Theorem frgpval 16650
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 3104 . . 3  |-  ( I  e.  V  ->  I  e.  _V )
3 xpeq1 5003 . . . . . . 7  |-  ( i  =  I  ->  (
i  X.  2o )  =  ( I  X.  2o ) )
43fveq2d 5860 . . . . . 6  |-  ( i  =  I  ->  (freeMnd `  ( i  X.  2o ) )  =  (freeMnd `  ( I  X.  2o ) ) )
5 frgpval.b . . . . . 6  |-  M  =  (freeMnd `  ( I  X.  2o ) )
64, 5syl6eqr 2502 . . . . 5  |-  ( i  =  I  ->  (freeMnd `  ( i  X.  2o ) )  =  M )
7 fveq2 5856 . . . . . 6  |-  ( i  =  I  ->  ( ~FG  `  i
)  =  ( ~FG  `  I
) )
8 frgpval.r . . . . . 6  |-  .~  =  ( ~FG  `  I )
97, 8syl6eqr 2502 . . . . 5  |-  ( i  =  I  ->  ( ~FG  `  i
)  =  .~  )
106, 9oveq12d 6299 . . . 4  |-  ( i  =  I  ->  (
(freeMnd `  ( i  X.  2o ) )  /.s  ( ~FG  `  i
) )  =  ( M  /.s 
.~  ) )
11 df-frgp 16602 . . . 4  |- freeGrp  =  ( i  e.  _V  |->  ( (freeMnd `  ( i  X.  2o ) )  /.s  ( ~FG  `  i
) ) )
12 ovex 6309 . . . 4  |-  ( M 
/.s  .~  )  e.  _V
1310, 11, 12fvmpt 5941 . . 3  |-  ( I  e.  _V  ->  (freeGrp `  I )  =  ( M  /.s 
.~  ) )
142, 13syl 16 . 2  |-  ( I  e.  V  ->  (freeGrp `  I )  =  ( M  /.s 
.~  ) )
151, 14syl5eq 2496 1  |-  ( I  e.  V  ->  G  =  ( M  /.s  .~  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804   _Vcvv 3095    X. cxp 4987   ` cfv 5578  (class class class)co 6281   2oc2o 7126    /.s cqus 14779  freeMndcfrmd 15889   ~FG cefg 16598  freeGrpcfrgp 16599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-iota 5541  df-fun 5580  df-fv 5586  df-ov 6284  df-frgp 16602
This theorem is referenced by:  frgp0  16652  frgpeccl  16653  frgpadd  16655  frgpupf  16665  frgpup1  16667  frgpup3lem  16669  frgpnabllem2  16752
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