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Theorem efgmval 17301
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 4190 . 2  |-  ( a  =  A  ->  <. a ,  ( 1o  \ 
b ) >.  =  <. A ,  ( 1o  \ 
b ) >. )
2 difeq2 3583 . . 3  |-  ( b  =  B  ->  ( 1o  \  b )  =  ( 1o  \  B
) )
32opeq2d 4197 . 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 4190 . . . 4  |-  ( y  =  a  ->  <. y ,  ( 1o  \ 
z ) >.  =  <. a ,  ( 1o  \ 
z ) >. )
6 difeq2 3583 . . . . 5  |-  ( z  =  b  ->  ( 1o  \  z )  =  ( 1o  \  b
) )
76opeq2d 4197 . . . 4  |-  ( z  =  b  ->  <. a ,  ( 1o  \ 
z ) >.  =  <. a ,  ( 1o  \ 
b ) >. )
85, 7cbvmpt2v 6385 . . 3  |-  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. )  =  ( a  e.  I ,  b  e.  2o  |->  <. a ,  ( 1o  \  b )
>. )
94, 8eqtri 2458 . 2  |-  M  =  ( a  e.  I ,  b  e.  2o  |->  <. a ,  ( 1o 
\  b ) >.
)
10 opex 4686 . 2  |-  <. A , 
( 1o  \  B
) >.  e.  _V
111, 3, 9, 10ovmpt2 6446 1  |-  ( ( A  e.  I  /\  B  e.  2o )  ->  ( A M B )  =  <. A , 
( 1o  \  B
) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870    \ cdif 3439   <.cop 4008  (class class class)co 6305    |-> cmpt2 6307   1oc1o 7183   2oc2o 7184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310
This theorem is referenced by:  efgmnvl  17303  efgval2  17313  vrgpinv  17358  frgpuptinv  17360  frgpuplem  17361  frgpnabllem1  17448
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