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Theorem efgmval 15264
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 3919 . 2  |-  ( a  =  A  ->  <. a ,  ( 1o  \ 
b ) >.  =  <. A ,  ( 1o  \ 
b ) >. )
2 difeq2 3395 . . 3  |-  ( b  =  B  ->  ( 1o  \  b )  =  ( 1o  \  B
) )
32opeq2d 3926 . 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 3919 . . . 4  |-  ( y  =  a  ->  <. y ,  ( 1o  \ 
z ) >.  =  <. a ,  ( 1o  \ 
z ) >. )
6 difeq2 3395 . . . . 5  |-  ( z  =  b  ->  ( 1o  \  z )  =  ( 1o  \  b
) )
76opeq2d 3926 . . . 4  |-  ( z  =  b  ->  <. a ,  ( 1o  \ 
z ) >.  =  <. a ,  ( 1o  \ 
b ) >. )
85, 7cbvmpt2v 6084 . . 3  |-  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. )  =  ( a  e.  I ,  b  e.  2o  |->  <. a ,  ( 1o  \  b )
>. )
94, 8eqtri 2400 . 2  |-  M  =  ( a  e.  I ,  b  e.  2o  |->  <. a ,  ( 1o 
\  b ) >.
)
10 opex 4361 . 2  |-  <. A , 
( 1o  \  B
) >.  e.  _V
111, 3, 9, 10ovmpt2 6141 1  |-  ( ( A  e.  I  /\  B  e.  2o )  ->  ( A M B )  =  <. A , 
( 1o  \  B
) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    \ cdif 3253   <.cop 3753  (class class class)co 6013    e. cmpt2 6015   1oc1o 6646   2oc2o 6647
This theorem is referenced by:  efgmnvl  15266  efgval2  15276  vrgpinv  15321  frgpuptinv  15323  frgpuplem  15324  frgpnabllem1  15404
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-iota 5351  df-fun 5389  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018
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