Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df-linc Structured version   Visualization version   GIF version

Definition df-linc 47175
Description: Define the operation constructing a linear combination. Although this definition is taylored for linear combinations of vectors from left modules, it can be used for any structure having a Base, Scalar s and a scalar multiplication ·𝑠. (Contributed by AV, 29-Mar-2019.)
Assertion
Ref Expression
df-linc linC = (π‘š ∈ V ↦ (𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Baseβ€˜π‘š) ↦ (π‘š Ξ£g (π‘₯ ∈ 𝑣 ↦ ((π‘ β€˜π‘₯)( ·𝑠 β€˜π‘š)π‘₯)))))
Distinct variable group:   π‘š,𝑠,𝑣,π‘₯

Detailed syntax breakdown of Definition df-linc
StepHypRef Expression
1 clinc 47173 . 2 class linC
2 vm . . 3 setvar π‘š
3 cvv 3474 . . 3 class V
4 vs . . . 4 setvar 𝑠
5 vv . . . 4 setvar 𝑣
62cv 1540 . . . . . . 7 class π‘š
7 csca 17204 . . . . . . 7 class Scalar
86, 7cfv 6543 . . . . . 6 class (Scalarβ€˜π‘š)
9 cbs 17148 . . . . . 6 class Base
108, 9cfv 6543 . . . . 5 class (Baseβ€˜(Scalarβ€˜π‘š))
115cv 1540 . . . . 5 class 𝑣
12 cmap 8822 . . . . 5 class ↑m
1310, 11, 12co 7411 . . . 4 class ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣)
146, 9cfv 6543 . . . . 5 class (Baseβ€˜π‘š)
1514cpw 4602 . . . 4 class 𝒫 (Baseβ€˜π‘š)
16 vx . . . . . 6 setvar π‘₯
1716cv 1540 . . . . . . . 8 class π‘₯
184cv 1540 . . . . . . . 8 class 𝑠
1917, 18cfv 6543 . . . . . . 7 class (π‘ β€˜π‘₯)
20 cvsca 17205 . . . . . . . 8 class ·𝑠
216, 20cfv 6543 . . . . . . 7 class ( ·𝑠 β€˜π‘š)
2219, 17, 21co 7411 . . . . . 6 class ((π‘ β€˜π‘₯)( ·𝑠 β€˜π‘š)π‘₯)
2316, 11, 22cmpt 5231 . . . . 5 class (π‘₯ ∈ 𝑣 ↦ ((π‘ β€˜π‘₯)( ·𝑠 β€˜π‘š)π‘₯))
24 cgsu 17390 . . . . 5 class Ξ£g
256, 23, 24co 7411 . . . 4 class (π‘š Ξ£g (π‘₯ ∈ 𝑣 ↦ ((π‘ β€˜π‘₯)( ·𝑠 β€˜π‘š)π‘₯)))
264, 5, 13, 15, 25cmpo 7413 . . 3 class (𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Baseβ€˜π‘š) ↦ (π‘š Ξ£g (π‘₯ ∈ 𝑣 ↦ ((π‘ β€˜π‘₯)( ·𝑠 β€˜π‘š)π‘₯))))
272, 3, 26cmpt 5231 . 2 class (π‘š ∈ V ↦ (𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Baseβ€˜π‘š) ↦ (π‘š Ξ£g (π‘₯ ∈ 𝑣 ↦ ((π‘ β€˜π‘₯)( ·𝑠 β€˜π‘š)π‘₯)))))
281, 27wceq 1541 1 wff linC = (π‘š ∈ V ↦ (𝑠 ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Baseβ€˜π‘š) ↦ (π‘š Ξ£g (π‘₯ ∈ 𝑣 ↦ ((π‘ β€˜π‘₯)( ·𝑠 β€˜π‘š)π‘₯)))))
Colors of variables: wff setvar class
This definition is referenced by:  lincop  47177  dflinc2  47179
  Copyright terms: Public domain W3C validator