Metamath Proof Explorer

Theorem submul2

Description: Convert a subtraction to addition using multiplication by a negative. (Contributed by NM, 2-Feb-2007)

Ref Expression
Assertion submul2 ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A − B ⁢ C = A + B ⁢ − C

Proof

Step Hyp Ref Expression
1 mulneg2 ⊢ B ∈ ℂ ∧ C ∈ ℂ → B ⁢ − C = − B ⁢ C
2 1 adantl ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → B ⁢ − C = − B ⁢ C
3 2 oveq2d ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A + B ⁢ − C = A + − B ⁢ C
4 mulcl ⊢ B ∈ ℂ ∧ C ∈ ℂ → B ⁢ C ∈ ℂ
5 negsub ⊢ A ∈ ℂ ∧ B ⁢ C ∈ ℂ → A + − B ⁢ C = A − B ⁢ C
6 4 5 sylan2 ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A + − B ⁢ C = A − B ⁢ C
7 3 6 eqtr2d ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A − B ⁢ C = A + B ⁢ − C
8 7 3impb ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A − B ⁢ C = A + B ⁢ − C