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Theorem efgmval 15336
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 3976 . 2  |-  ( a  =  A  ->  <. a ,  ( 1o  \ 
b ) >.  =  <. A ,  ( 1o  \ 
b ) >. )
2 difeq2 3451 . . 3  |-  ( b  =  B  ->  ( 1o  \  b )  =  ( 1o  \  B
) )
32opeq2d 3983 . 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 3976 . . . 4  |-  ( y  =  a  ->  <. y ,  ( 1o  \ 
z ) >.  =  <. a ,  ( 1o  \ 
z ) >. )
6 difeq2 3451 . . . . 5  |-  ( z  =  b  ->  ( 1o  \  z )  =  ( 1o  \  b
) )
76opeq2d 3983 . . . 4  |-  ( z  =  b  ->  <. a ,  ( 1o  \ 
z ) >.  =  <. a ,  ( 1o  \ 
b ) >. )
85, 7cbvmpt2v 6144 . . 3  |-  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. )  =  ( a  e.  I ,  b  e.  2o  |->  <. a ,  ( 1o  \  b )
>. )
94, 8eqtri 2455 . 2  |-  M  =  ( a  e.  I ,  b  e.  2o  |->  <. a ,  ( 1o 
\  b ) >.
)
10 opex 4419 . 2  |-  <. A , 
( 1o  \  B
) >.  e.  _V
111, 3, 9, 10ovmpt2 6201 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 1652    e. wcel 1725    \ cdif 3309   <.cop 3809  (class class class)co 6073    e. cmpt2 6075   1oc1o 6709   2oc2o 6710
This theorem is referenced by:  efgmnvl  15338  efgval2  15348  vrgpinv  15393  frgpuptinv  15395  frgpuplem  15396  frgpnabllem1  15476
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078
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