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Theorem frgpval 15390
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 2964 . . 3  |-  ( I  e.  V  ->  I  e.  _V )
3 xpeq1 4892 . . . . . . 7  |-  ( i  =  I  ->  (
i  X.  2o )  =  ( I  X.  2o ) )
43fveq2d 5732 . . . . . 6  |-  ( i  =  I  ->  (freeMnd `  ( i  X.  2o ) )  =  (freeMnd `  ( I  X.  2o ) ) )
5 frgpval.b . . . . . 6  |-  M  =  (freeMnd `  ( I  X.  2o ) )
64, 5syl6eqr 2486 . . . . 5  |-  ( i  =  I  ->  (freeMnd `  ( i  X.  2o ) )  =  M )
7 fveq2 5728 . . . . . 6  |-  ( i  =  I  ->  ( ~FG  `  i
)  =  ( ~FG  `  I
) )
8 frgpval.r . . . . . 6  |-  .~  =  ( ~FG  `  I )
97, 8syl6eqr 2486 . . . . 5  |-  ( i  =  I  ->  ( ~FG  `  i
)  =  .~  )
106, 9oveq12d 6099 . . . 4  |-  ( i  =  I  ->  (
(freeMnd `  ( i  X.  2o ) )  /.s  ( ~FG  `  i
) )  =  ( M  /.s 
.~  ) )
11 df-frgp 15342 . . . 4  |- freeGrp  =  ( i  e.  _V  |->  ( (freeMnd `  ( i  X.  2o ) )  /.s  ( ~FG  `  i
) ) )
12 ovex 6106 . . . 4  |-  ( M 
/.s  .~  )  e.  _V
1310, 11, 12fvmpt 5806 . . 3  |-  ( I  e.  _V  ->  (freeGrp `  I )  =  ( M  /.s 
.~  ) )
142, 13syl 16 . 2  |-  ( I  e.  V  ->  (freeGrp `  I )  =  ( M  /.s 
.~  ) )
151, 14syl5eq 2480 1  |-  ( I  e.  V  ->  G  =  ( M  /.s  .~  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2956    X. cxp 4876   ` cfv 5454  (class class class)co 6081   2oc2o 6718    /.s cqus 13731  freeMndcfrmd 14792   ~FG cefg 15338  freeGrpcfrgp 15339
This theorem is referenced by:  frgp0  15392  frgpeccl  15393  frgpadd  15395  frgpupf  15405  frgpup1  15407  frgpup3lem  15409  frgpnabllem2  15485
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-frgp 15342
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