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Definition df-scaf 16951
Description: Define the functionalization of the  .s operator. This restricts the value of  .s to the stated domain, which is necessary when working with restricted structures, whose operations may be defined on a larger set than the true base. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
df-scaf  |-  .sf 
=  ( g  e. 
_V  |->  ( x  e.  ( Base `  (Scalar `  g ) ) ,  y  e.  ( Base `  g )  |->  ( x ( .s `  g
) y ) ) )
Distinct variable group:    x, g, y

Detailed syntax breakdown of Definition df-scaf
StepHypRef Expression
1 cscaf 16949 . 2  class  .sf
2 vg . . 3  setvar  g
3 cvv 2972 . . 3  class  _V
4 vx . . . 4  setvar  x
5 vy . . . 4  setvar  y
62cv 1368 . . . . . 6  class  g
7 csca 14241 . . . . . 6  class Scalar
86, 7cfv 5418 . . . . 5  class  (Scalar `  g )
9 cbs 14174 . . . . 5  class  Base
108, 9cfv 5418 . . . 4  class  ( Base `  (Scalar `  g )
)
116, 9cfv 5418 . . . 4  class  ( Base `  g )
124cv 1368 . . . . 5  class  x
135cv 1368 . . . . 5  class  y
14 cvsca 14242 . . . . . 6  class  .s
156, 14cfv 5418 . . . . 5  class  ( .s
`  g )
1612, 13, 15co 6091 . . . 4  class  ( x ( .s `  g
) y )
174, 5, 10, 11, 16cmpt2 6093 . . 3  class  ( x  e.  ( Base `  (Scalar `  g ) ) ,  y  e.  ( Base `  g )  |->  ( x ( .s `  g
) y ) )
182, 3, 17cmpt 4350 . 2  class  ( g  e.  _V  |->  ( x  e.  ( Base `  (Scalar `  g ) ) ,  y  e.  ( Base `  g )  |->  ( x ( .s `  g
) y ) ) )
191, 18wceq 1369 1  wff  .sf 
=  ( g  e. 
_V  |->  ( x  e.  ( Base `  (Scalar `  g ) ) ,  y  e.  ( Base `  g )  |->  ( x ( .s `  g
) y ) ) )
Colors of variables: wff setvar class
This definition is referenced by:  scaffval  16966
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