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

Definition df-ress 13431
Description: Define a multifunction restriction operator for extensible structures, which can be used to turn statements about rings into statements about subrings, modules into submodules, etc. This definition knows nothing about individual structures and merely truncates the  Base set while leaving operators alone; individual kinds of structures will need to handle this behavior, by ignoring operators' values outside the range (like  Ring), defining a function using the base set and applying that (like  TopGrp), or explicitly truncating the slot before use (like  MetSp).

(Credit for this operator goes to Mario Carneiro).

See ressbas 13474 for the altered base set, and resslem 13477 (subrg0 15830, ressplusg 13526, subrg1 15833, ressmulr 13537) for the (un)altered other operations. (Contributed by Stefan O'Rear, 29-Nov-2014.)

Assertion
Ref Expression
df-ress  |-s  =  ( w  e.  _V ,  x  e. 
_V  |->  if ( (
Base `  w )  C_  x ,  w ,  ( w sSet  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  w ) )
>. ) ) )
Distinct variable group:    x, w

Detailed syntax breakdown of Definition df-ress
StepHypRef Expression
1 cress 13425 . 2  classs
2 vw . . 3  set  w
3 vx . . 3  set  x
4 cvv 2916 . . 3  class  _V
52cv 1648 . . . . . 6  class  w
6 cbs 13424 . . . . . 6  class  Base
75, 6cfv 5413 . . . . 5  class  ( Base `  w )
83cv 1648 . . . . 5  class  x
97, 8wss 3280 . . . 4  wff  ( Base `  w )  C_  x
10 cnx 13421 . . . . . . 7  class  ndx
1110, 6cfv 5413 . . . . . 6  class  ( Base `  ndx )
128, 7cin 3279 . . . . . 6  class  ( x  i^i  ( Base `  w
) )
1311, 12cop 3777 . . . . 5  class  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  w ) )
>.
14 csts 13422 . . . . 5  class sSet
155, 13, 14co 6040 . . . 4  class  ( w sSet  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  w
) ) >. )
169, 5, 15cif 3699 . . 3  class  if ( ( Base `  w
)  C_  x ,  w ,  ( w sSet  <.
( Base `  ndx ) ,  ( x  i^i  ( Base `  w ) )
>. ) )
172, 3, 4, 4, 16cmpt2 6042 . 2  class  ( w  e.  _V ,  x  e.  _V  |->  if ( (
Base `  w )  C_  x ,  w ,  ( w sSet  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  w ) )
>. ) ) )
181, 17wceq 1649 1  wffs  =  ( w  e.  _V ,  x  e. 
_V  |->  if ( (
Base `  w )  C_  x ,  w ,  ( w sSet  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  w ) )
>. ) ) )
Colors of variables: wff set class
This definition is referenced by:  reldmress  13470  ressval  13471
  Copyright terms: Public domain W3C validator