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Theorem frgpval 15083
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 2809 . . 3  |-  ( I  e.  V  ->  I  e.  _V )
3 xpeq1 4719 . . . . . . 7  |-  ( i  =  I  ->  (
i  X.  2o )  =  ( I  X.  2o ) )
43fveq2d 5545 . . . . . 6  |-  ( i  =  I  ->  (freeMnd `  ( i  X.  2o ) )  =  (freeMnd `  ( I  X.  2o ) ) )
5 frgpval.b . . . . . 6  |-  M  =  (freeMnd `  ( I  X.  2o ) )
64, 5syl6eqr 2346 . . . . 5  |-  ( i  =  I  ->  (freeMnd `  ( i  X.  2o ) )  =  M )
7 fveq2 5541 . . . . . 6  |-  ( i  =  I  ->  ( ~FG  `  i
)  =  ( ~FG  `  I
) )
8 frgpval.r . . . . . 6  |-  .~  =  ( ~FG  `  I )
97, 8syl6eqr 2346 . . . . 5  |-  ( i  =  I  ->  ( ~FG  `  i
)  =  .~  )
106, 9oveq12d 5892 . . . 4  |-  ( i  =  I  ->  (
(freeMnd `  ( i  X.  2o ) )  /.s  ( ~FG  `  i
) )  =  ( M  /.s 
.~  ) )
11 df-frgp 15035 . . . 4  |- freeGrp  =  ( i  e.  _V  |->  ( (freeMnd `  ( i  X.  2o ) )  /.s  ( ~FG  `  i
) ) )
12 ovex 5899 . . . 4  |-  ( M 
/.s  .~  )  e.  _V
1310, 11, 12fvmpt 5618 . . 3  |-  ( I  e.  _V  ->  (freeGrp `  I )  =  ( M  /.s 
.~  ) )
142, 13syl 15 . 2  |-  ( I  e.  V  ->  (freeGrp `  I )  =  ( M  /.s 
.~  ) )
151, 14syl5eq 2340 1  |-  ( I  e.  V  ->  G  =  ( M  /.s  .~  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801    X. cxp 4703   ` cfv 5271  (class class class)co 5874   2oc2o 6489    /.s cqus 13424  freeMndcfrmd 14485   ~FG cefg 15031  freeGrpcfrgp 15032
This theorem is referenced by:  frgp0  15085  frgpeccl  15086  frgpadd  15088  frgpupf  15098  frgpup1  15100  frgpup3lem  15102  frgpnabllem2  15178
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-frgp 15035
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