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Definition df-lco 33262
Description: Define the operation constructing the set of all linear combinations for a set of vectors. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 28-Jul-2019.)
Assertion
Ref Expression
df-lco  |- LinCo  =  ( m  e.  _V , 
v  e.  ~P ( Base `  m )  |->  { c  e.  ( Base `  m )  |  E. s  e.  ( ( Base `  (Scalar `  m
) )  ^m  v
) ( s finSupp  ( 0g `  (Scalar `  m
) )  /\  c  =  ( s ( linC  `  m ) v ) ) } )
Distinct variable group:    m, c, s, v

Detailed syntax breakdown of Definition df-lco
StepHypRef Expression
1 clinco 33260 . 2  class LinCo
2 vm . . 3  setvar  m
3 vv . . 3  setvar  v
4 cvv 3106 . . 3  class  _V
52cv 1397 . . . . 5  class  m
6 cbs 14716 . . . . 5  class  Base
75, 6cfv 5570 . . . 4  class  ( Base `  m )
87cpw 3999 . . 3  class  ~P ( Base `  m )
9 vs . . . . . . . 8  setvar  s
109cv 1397 . . . . . . 7  class  s
11 csca 14787 . . . . . . . . 9  class Scalar
125, 11cfv 5570 . . . . . . . 8  class  (Scalar `  m )
13 c0g 14929 . . . . . . . 8  class  0g
1412, 13cfv 5570 . . . . . . 7  class  ( 0g
`  (Scalar `  m )
)
15 cfsupp 7821 . . . . . . 7  class finSupp
1610, 14, 15wbr 4439 . . . . . 6  wff  s finSupp  ( 0g `  (Scalar `  m
) )
17 vc . . . . . . . 8  setvar  c
1817cv 1397 . . . . . . 7  class  c
193cv 1397 . . . . . . . 8  class  v
20 clinc 33259 . . . . . . . . 9  class linC
215, 20cfv 5570 . . . . . . . 8  class  ( linC  `  m )
2210, 19, 21co 6270 . . . . . . 7  class  ( s ( linC  `  m )
v )
2318, 22wceq 1398 . . . . . 6  wff  c  =  ( s ( linC  `  m ) v )
2416, 23wa 367 . . . . 5  wff  ( s finSupp 
( 0g `  (Scalar `  m ) )  /\  c  =  ( s
( linC  `  m )
v ) )
2512, 6cfv 5570 . . . . . 6  class  ( Base `  (Scalar `  m )
)
26 cmap 7412 . . . . . 6  class  ^m
2725, 19, 26co 6270 . . . . 5  class  ( (
Base `  (Scalar `  m
) )  ^m  v
)
2824, 9, 27wrex 2805 . . . 4  wff  E. s  e.  ( ( Base `  (Scalar `  m ) )  ^m  v ) ( s finSupp 
( 0g `  (Scalar `  m ) )  /\  c  =  ( s
( linC  `  m )
v ) )
2928, 17, 7crab 2808 . . 3  class  { c  e.  ( Base `  m
)  |  E. s  e.  ( ( Base `  (Scalar `  m ) )  ^m  v ) ( s finSupp 
( 0g `  (Scalar `  m ) )  /\  c  =  ( s
( linC  `  m )
v ) ) }
302, 3, 4, 8, 29cmpt2 6272 . 2  class  ( m  e.  _V ,  v  e.  ~P ( Base `  m )  |->  { c  e.  ( Base `  m
)  |  E. s  e.  ( ( Base `  (Scalar `  m ) )  ^m  v ) ( s finSupp 
( 0g `  (Scalar `  m ) )  /\  c  =  ( s
( linC  `  m )
v ) ) } )
311, 30wceq 1398 1  wff LinCo  =  ( m  e.  _V , 
v  e.  ~P ( Base `  m )  |->  { c  e.  ( Base `  m )  |  E. s  e.  ( ( Base `  (Scalar `  m
) )  ^m  v
) ( s finSupp  ( 0g `  (Scalar `  m
) )  /\  c  =  ( s ( linC  `  m ) v ) ) } )
Colors of variables: wff setvar class
This definition is referenced by:  lcoop  33266
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