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Definition df-asp 16070
Description: Define the algebraic span of a set of vectors in an algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
Assertion
Ref Expression
df-asp  |- AlgSpan  =  ( w  e. AssAlg  |->  ( s  e.  ~P ( Base `  w )  |->  |^| { t  e.  ( (SubRing `  w
)  i^i  ( LSubSp `  w ) )  |  s  C_  t }
) )
Distinct variable group:    t, s, w

Detailed syntax breakdown of Definition df-asp
StepHypRef Expression
1 casp 16067 . 2  class AlgSpan
2 vw . . 3  set  w
3 casa 16066 . . 3  class AssAlg
4 vs . . . 4  set  s
52cv 1631 . . . . . 6  class  w
6 cbs 13164 . . . . . 6  class  Base
75, 6cfv 5271 . . . . 5  class  ( Base `  w )
87cpw 3638 . . . 4  class  ~P ( Base `  w )
94cv 1631 . . . . . . 7  class  s
10 vt . . . . . . . 8  set  t
1110cv 1631 . . . . . . 7  class  t
129, 11wss 3165 . . . . . 6  wff  s  C_  t
13 csubrg 15557 . . . . . . . 8  class SubRing
145, 13cfv 5271 . . . . . . 7  class  (SubRing `  w
)
15 clss 15705 . . . . . . . 8  class  LSubSp
165, 15cfv 5271 . . . . . . 7  class  ( LSubSp `  w )
1714, 16cin 3164 . . . . . 6  class  ( (SubRing `  w )  i^i  ( LSubSp `
 w ) )
1812, 10, 17crab 2560 . . . . 5  class  { t  e.  ( (SubRing `  w
)  i^i  ( LSubSp `  w ) )  |  s  C_  t }
1918cint 3878 . . . 4  class  |^| { t  e.  ( (SubRing `  w
)  i^i  ( LSubSp `  w ) )  |  s  C_  t }
204, 8, 19cmpt 4093 . . 3  class  ( s  e.  ~P ( Base `  w )  |->  |^| { t  e.  ( (SubRing `  w
)  i^i  ( LSubSp `  w ) )  |  s  C_  t }
)
212, 3, 20cmpt 4093 . 2  class  ( w  e. AssAlg  |->  ( s  e. 
~P ( Base `  w
)  |->  |^| { t  e.  ( (SubRing `  w
)  i^i  ( LSubSp `  w ) )  |  s  C_  t }
) )
221, 21wceq 1632 1  wff AlgSpan  =  ( w  e. AssAlg  |->  ( s  e.  ~P ( Base `  w )  |->  |^| { t  e.  ( (SubRing `  w
)  i^i  ( LSubSp `  w ) )  |  s  C_  t }
) )
Colors of variables: wff set class
This definition is referenced by:  aspval  16084
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