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 ) ) )