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Theorem efgmval 16322
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 4160 . 2  |-  ( a  =  A  ->  <. a ,  ( 1o  \ 
b ) >.  =  <. A ,  ( 1o  \ 
b ) >. )
2 difeq2 3569 . . 3  |-  ( b  =  B  ->  ( 1o  \  b )  =  ( 1o  \  B
) )
32opeq2d 4167 . 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 4160 . . . 4  |-  ( y  =  a  ->  <. y ,  ( 1o  \ 
z ) >.  =  <. a ,  ( 1o  \ 
z ) >. )
6 difeq2 3569 . . . . 5  |-  ( z  =  b  ->  ( 1o  \  z )  =  ( 1o  \  b
) )
76opeq2d 4167 . . . 4  |-  ( z  =  b  ->  <. a ,  ( 1o  \ 
z ) >.  =  <. a ,  ( 1o  \ 
b ) >. )
85, 7cbvmpt2v 6268 . . 3  |-  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. )  =  ( a  e.  I ,  b  e.  2o  |->  <. a ,  ( 1o  \  b )
>. )
94, 8eqtri 2480 . 2  |-  M  =  ( a  e.  I ,  b  e.  2o  |->  <. a ,  ( 1o 
\  b ) >.
)
10 opex 4657 . 2  |-  <. A , 
( 1o  \  B
) >.  e.  _V
111, 3, 9, 10ovmpt2 6329 1  |-  ( ( A  e.  I  /\  B  e.  2o )  ->  ( A M B )  =  <. A , 
( 1o  \  B
) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    \ cdif 3426   <.cop 3984  (class class class)co 6193    |-> cmpt2 6195   1oc1o 7016   2oc2o 7017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-iota 5482  df-fun 5521  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198
This theorem is referenced by:  efgmnvl  16324  efgval2  16334  vrgpinv  16379  frgpuptinv  16381  frgpuplem  16382  frgpnabllem1  16464
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