Metamath Proof Explorer


Theorem dflinc2

Description: Alternative definition of linear combinations using the function operation. (Contributed by AV, 1-Apr-2019)

Ref Expression
Assertion dflinc2 linC = m V s Base Scalar m v , v 𝒫 Base m m s m f I v

Proof

Step Hyp Ref Expression
1 df-linc linC = m V s Base Scalar m v , v 𝒫 Base m m i v s i m i
2 elmapfn s Base Scalar m v s Fn v
3 2 adantr s Base Scalar m v v 𝒫 Base m s Fn v
4 fnresi I v Fn v
5 4 a1i s Base Scalar m v v 𝒫 Base m I v Fn v
6 vex v V
7 6 a1i s Base Scalar m v v 𝒫 Base m v V
8 inidm v v = v
9 eqidd s Base Scalar m v v 𝒫 Base m i v s i = s i
10 fvresi i v I v i = i
11 10 adantl s Base Scalar m v v 𝒫 Base m i v I v i = i
12 3 5 7 7 8 9 11 offval s Base Scalar m v v 𝒫 Base m s m f I v = i v s i m i
13 12 eqcomd s Base Scalar m v v 𝒫 Base m i v s i m i = s m f I v
14 13 oveq2d s Base Scalar m v v 𝒫 Base m m i v s i m i = m s m f I v
15 14 mpoeq3ia s Base Scalar m v , v 𝒫 Base m m i v s i m i = s Base Scalar m v , v 𝒫 Base m m s m f I v
16 15 mpteq2i m V s Base Scalar m v , v 𝒫 Base m m i v s i m i = m V s Base Scalar m v , v 𝒫 Base m m s m f I v
17 1 16 eqtri linC = m V s Base Scalar m v , v 𝒫 Base m m s m f I v