MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-efg Unicode version

Definition df-efg 15018
Description: Define the free group equivalence relation, which is the smallest equivalence relation  ~~ such that for any words 
A ,  B and formal symbol  x with inverse  inv g x,  A B  ~~  A x ( inv g
x ) B. (Contributed by Mario Carneiro, 1-Oct-2015.)
Assertion
Ref Expression
df-efg  |- ~FG  =  ( i  e.  _V  |->  |^| { r  |  ( r  Er Word  (
i  X.  2o )  /\  A. x  e. Word 
( i  X.  2o ) A. n  e.  ( 0 ... ( # `  x ) ) A. y  e.  i  A. z  e.  2o  x
r ( x splice  <. n ,  n ,  <" <. y ,  z >. <. y ,  ( 1o  \ 
z ) >. "> >.
) ) } )
Distinct variable group:    i, n, r, x, y, z

Detailed syntax breakdown of Definition df-efg
StepHypRef Expression
1 cefg 15015 . 2  class ~FG
2 vi . . 3  set  i
3 cvv 2788 . . 3  class  _V
42cv 1622 . . . . . . . . 9  class  i
5 c2o 6473 . . . . . . . . 9  class  2o
64, 5cxp 4687 . . . . . . . 8  class  ( i  X.  2o )
76cword 11403 . . . . . . 7  class Word  ( i  X.  2o )
8 vr . . . . . . . 8  set  r
98cv 1622 . . . . . . 7  class  r
107, 9wer 6657 . . . . . 6  wff  r  Er Word 
( i  X.  2o )
11 vx . . . . . . . . . . . 12  set  x
1211cv 1622 . . . . . . . . . . 11  class  x
13 vn . . . . . . . . . . . . . 14  set  n
1413cv 1622 . . . . . . . . . . . . 13  class  n
15 vy . . . . . . . . . . . . . . . 16  set  y
1615cv 1622 . . . . . . . . . . . . . . 15  class  y
17 vz . . . . . . . . . . . . . . . 16  set  z
1817cv 1622 . . . . . . . . . . . . . . 15  class  z
1916, 18cop 3643 . . . . . . . . . . . . . 14  class  <. y ,  z >.
20 c1o 6472 . . . . . . . . . . . . . . . 16  class  1o
2120, 18cdif 3149 . . . . . . . . . . . . . . 15  class  ( 1o 
\  z )
2216, 21cop 3643 . . . . . . . . . . . . . 14  class  <. y ,  ( 1o  \ 
z ) >.
2319, 22cs2 11491 . . . . . . . . . . . . 13  class  <" <. y ,  z >. <. y ,  ( 1o  \ 
z ) >. ">
2414, 14, 23cotp 3644 . . . . . . . . . . . 12  class  <. n ,  n ,  <" <. y ,  z >. <. y ,  ( 1o  \ 
z ) >. "> >.
25 csplice 11407 . . . . . . . . . . . 12  class splice
2612, 24, 25co 5858 . . . . . . . . . . 11  class  ( x splice  <. n ,  n , 
<" <. y ,  z
>. <. y ,  ( 1o  \  z )
>. "> >. )
2712, 26, 9wbr 4023 . . . . . . . . . 10  wff  x r ( x splice  <. n ,  n ,  <" <. y ,  z >. <. y ,  ( 1o  \ 
z ) >. "> >.
)
2827, 17, 5wral 2543 . . . . . . . . 9  wff  A. z  e.  2o  x r ( x splice  <. n ,  n ,  <" <. y ,  z >. <. y ,  ( 1o  \ 
z ) >. "> >.
)
2928, 15, 4wral 2543 . . . . . . . 8  wff  A. y  e.  i  A. z  e.  2o  x r ( x splice  <. n ,  n ,  <" <. y ,  z >. <. y ,  ( 1o  \ 
z ) >. "> >.
)
30 cc0 8737 . . . . . . . . 9  class  0
31 chash 11337 . . . . . . . . . 10  class  #
3212, 31cfv 5255 . . . . . . . . 9  class  ( # `  x )
33 cfz 10782 . . . . . . . . 9  class  ...
3430, 32, 33co 5858 . . . . . . . 8  class  ( 0 ... ( # `  x
) )
3529, 13, 34wral 2543 . . . . . . 7  wff  A. n  e.  ( 0 ... ( # `
 x ) ) A. y  e.  i 
A. z  e.  2o  x r ( x splice  <. n ,  n , 
<" <. y ,  z
>. <. y ,  ( 1o  \  z )
>. "> >. )
3635, 11, 7wral 2543 . . . . . 6  wff  A. x  e. Word  ( i  X.  2o ) A. n  e.  ( 0 ... ( # `  x ) ) A. y  e.  i  A. z  e.  2o  x
r ( x splice  <. n ,  n ,  <" <. y ,  z >. <. y ,  ( 1o  \ 
z ) >. "> >.
)
3710, 36wa 358 . . . . 5  wff  ( r  Er Word  ( i  X.  2o )  /\  A. x  e. Word  ( i  X.  2o ) A. n  e.  ( 0 ... ( # `
 x ) ) A. y  e.  i 
A. z  e.  2o  x r ( x splice  <. n ,  n , 
<" <. y ,  z
>. <. y ,  ( 1o  \  z )
>. "> >. )
)
3837, 8cab 2269 . . . 4  class  { r  |  ( r  Er Word 
( i  X.  2o )  /\  A. x  e. Word 
( i  X.  2o ) A. n  e.  ( 0 ... ( # `  x ) ) A. y  e.  i  A. z  e.  2o  x
r ( x splice  <. n ,  n ,  <" <. y ,  z >. <. y ,  ( 1o  \ 
z ) >. "> >.
) ) }
3938cint 3862 . . 3  class  |^| { r  |  ( r  Er Word 
( i  X.  2o )  /\  A. x  e. Word 
( i  X.  2o ) A. n  e.  ( 0 ... ( # `  x ) ) A. y  e.  i  A. z  e.  2o  x
r ( x splice  <. n ,  n ,  <" <. y ,  z >. <. y ,  ( 1o  \ 
z ) >. "> >.
) ) }
402, 3, 39cmpt 4077 . 2  class  ( i  e.  _V  |->  |^| { r  |  ( r  Er Word 
( i  X.  2o )  /\  A. x  e. Word 
( i  X.  2o ) A. n  e.  ( 0 ... ( # `  x ) ) A. y  e.  i  A. z  e.  2o  x
r ( x splice  <. n ,  n ,  <" <. y ,  z >. <. y ,  ( 1o  \ 
z ) >. "> >.
) ) } )
411, 40wceq 1623 1  wff ~FG  =  ( i  e.  _V  |->  |^| { r  |  ( r  Er Word  (
i  X.  2o )  /\  A. x  e. Word 
( i  X.  2o ) A. n  e.  ( 0 ... ( # `  x ) ) A. y  e.  i  A. z  e.  2o  x
r ( x splice  <. n ,  n ,  <" <. y ,  z >. <. y ,  ( 1o  \ 
z ) >. "> >.
) ) } )
Colors of variables: wff set class
This definition is referenced by:  efgval  15026
  Copyright terms: Public domain W3C validator