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Metamath Proof Explorer

Theorem hvsubvali

Description: Value of vector subtraction definition. (Contributed by NM, 3-Sep-1999) (New usage is discouraged.)

Ref Expression
Hypotheses hvaddcl.1 ⊢ 𝐴 ∈ ℋ
hvaddcl.2 ⊢ 𝐵 ∈ ℋ
Assertion hvsubvali ( 𝐴 −ℎ 𝐵 ) = ( 𝐴 +ℎ ( - 1 ·ℎ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 hvaddcl.1 ⊢ 𝐴 ∈ ℋ
2 hvaddcl.2 ⊢ 𝐵 ∈ ℋ
3 hvsubval ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 −ℎ 𝐵 ) = ( 𝐴 +ℎ ( - 1 ·ℎ 𝐵 ) ) )
4 1 2 3 mp2an ⊢ ( 𝐴 −ℎ 𝐵 ) = ( 𝐴 +ℎ ( - 1 ·ℎ 𝐵 ) )