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Definition df-scaf 17710
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 17708 . 2  class  .sf
2 vg . . 3  setvar  g
3 cvv 3106 . . 3  class  _V
4 vx . . . 4  setvar  x
5 vy . . . 4  setvar  y
62cv 1397 . . . . . 6  class  g
7 csca 14787 . . . . . 6  class Scalar
86, 7cfv 5570 . . . . 5  class  (Scalar `  g )
9 cbs 14716 . . . . 5  class  Base
108, 9cfv 5570 . . . 4  class  ( Base `  (Scalar `  g )
)
116, 9cfv 5570 . . . 4  class  ( Base `  g )
124cv 1397 . . . . 5  class  x
135cv 1397 . . . . 5  class  y
14 cvsca 14788 . . . . . 6  class  .s
156, 14cfv 5570 . . . . 5  class  ( .s
`  g )
1612, 13, 15co 6270 . . . 4  class  ( x ( .s `  g
) y )
174, 5, 10, 11, 16cmpt2 6272 . . 3  class  ( x  e.  ( Base `  (Scalar `  g ) ) ,  y  e.  ( Base `  g )  |->  ( x ( .s `  g
) y ) )
182, 3, 17cmpt 4497 . 2  class  ( g  e.  _V  |->  ( x  e.  ( Base `  (Scalar `  g ) ) ,  y  e.  ( Base `  g )  |->  ( x ( .s `  g
) y ) ) )
191, 18wceq 1398 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  17725
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