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Definition df-rgspn 15859
Description: The ring-span of a set of elements in a ring is the smallest subring which contains all of them. (Contributed by Stefan O'Rear, 7-Dec-2014.)
Assertion
Ref Expression
df-rgspn  |- RingSpan  =  ( w  e.  _V  |->  ( s  e.  ~P ( Base `  w )  |->  |^|
{ t  e.  (SubRing `  w )  |  s 
C_  t } ) )
Distinct variable group:    w, s, t

Detailed syntax breakdown of Definition df-rgspn
StepHypRef Expression
1 crgspn 15857 . 2  class RingSpan
2 vw . . 3  set  w
3 cvv 2948 . . 3  class  _V
4 vs . . . 4  set  s
52cv 1651 . . . . . 6  class  w
6 cbs 13461 . . . . . 6  class  Base
75, 6cfv 5446 . . . . 5  class  ( Base `  w )
87cpw 3791 . . . 4  class  ~P ( Base `  w )
94cv 1651 . . . . . . 7  class  s
10 vt . . . . . . . 8  set  t
1110cv 1651 . . . . . . 7  class  t
129, 11wss 3312 . . . . . 6  wff  s  C_  t
13 csubrg 15856 . . . . . . 7  class SubRing
145, 13cfv 5446 . . . . . 6  class  (SubRing `  w
)
1512, 10, 14crab 2701 . . . . 5  class  { t  e.  (SubRing `  w
)  |  s  C_  t }
1615cint 4042 . . . 4  class  |^| { t  e.  (SubRing `  w
)  |  s  C_  t }
174, 8, 16cmpt 4258 . . 3  class  ( s  e.  ~P ( Base `  w )  |->  |^| { t  e.  (SubRing `  w
)  |  s  C_  t } )
182, 3, 17cmpt 4258 . 2  class  ( w  e.  _V  |->  ( s  e.  ~P ( Base `  w )  |->  |^| { t  e.  (SubRing `  w
)  |  s  C_  t } ) )
191, 18wceq 1652 1  wff RingSpan  =  ( w  e.  _V  |->  ( s  e.  ~P ( Base `  w )  |->  |^|
{ t  e.  (SubRing `  w )  |  s 
C_  t } ) )
Colors of variables: wff set class
This definition is referenced by:  rgspnval  27341
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