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Theorem efgmval 15021
Description: Value of the formal inverse operation for the generating set of a free group. (Contributed by Mario Carneiro, 27-Sep-2015.)
Hypothesis
Ref Expression
efgmval.m  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
Assertion
Ref Expression
efgmval  |-  ( ( A  e.  I  /\  B  e.  2o )  ->  ( A M B )  =  <. A , 
( 1o  \  B
) >. )
Distinct variable group:    y, z, I
Allowed substitution hints:    A( y, z)    B( y, z)    M( y, z)

Proof of Theorem efgmval
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 3796 . 2  |-  ( a  =  A  ->  <. a ,  ( 1o  \ 
b ) >.  =  <. A ,  ( 1o  \ 
b ) >. )
2 difeq2 3288 . . 3  |-  ( b  =  B  ->  ( 1o  \  b )  =  ( 1o  \  B
) )
32opeq2d 3803 . 2  |-  ( b  =  B  ->  <. A , 
( 1o  \  b
) >.  =  <. A , 
( 1o  \  B
) >. )
4 efgmval.m . . 3  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
5 opeq1 3796 . . . 4  |-  ( y  =  a  ->  <. y ,  ( 1o  \ 
z ) >.  =  <. a ,  ( 1o  \ 
z ) >. )
6 difeq2 3288 . . . . 5  |-  ( z  =  b  ->  ( 1o  \  z )  =  ( 1o  \  b
) )
76opeq2d 3803 . . . 4  |-  ( z  =  b  ->  <. a ,  ( 1o  \ 
z ) >.  =  <. a ,  ( 1o  \ 
b ) >. )
85, 7cbvmpt2v 5926 . . 3  |-  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. )  =  ( a  e.  I ,  b  e.  2o  |->  <. a ,  ( 1o  \  b )
>. )
94, 8eqtri 2303 . 2  |-  M  =  ( a  e.  I ,  b  e.  2o  |->  <. a ,  ( 1o 
\  b ) >.
)
10 opex 4237 . 2  |-  <. A , 
( 1o  \  B
) >.  e.  _V
111, 3, 9, 10ovmpt2 5983 1  |-  ( ( A  e.  I  /\  B  e.  2o )  ->  ( A M B )  =  <. A , 
( 1o  \  B
) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    \ cdif 3149   <.cop 3643  (class class class)co 5858    e. cmpt2 5860   1oc1o 6472   2oc2o 6473
This theorem is referenced by:  efgmnvl  15023  efgval2  15033  vrgpinv  15078  frgpuptinv  15080  frgpuplem  15081  frgpnabllem1  15161
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863
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