Metamath Proof Explorer

Theorem hvsubcl

Description: Closure of vector subtraction. (Contributed by NM, 17-Aug-1999) (New usage is discouraged.)

Ref Expression
Assertion hvsubcl ⊢A∈ℋ∧B∈ℋ→A-ℎB∈ℋ

Proof

Step Hyp Ref Expression
1 hvsubval ⊢A∈ℋ∧B∈ℋ→A-ℎB=A+ℎ-1⋅ℎB
2 neg1cn ⊢−1∈ℂ
3 hvmulcl ⊢−1∈ℂ∧B∈ℋ→-1⋅ℎB∈ℋ
4 2 3 mpan ⊢B∈ℋ→-1⋅ℎB∈ℋ
5 hvaddcl ⊢A∈ℋ∧-1⋅ℎB∈ℋ→A+ℎ-1⋅ℎB∈ℋ
6 4 5 sylan2 ⊢A∈ℋ∧B∈ℋ→A+ℎ-1⋅ℎB∈ℋ
7 1 6 eqeltrd ⊢A∈ℋ∧B∈ℋ→A-ℎB∈ℋ