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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 ) ) )