Theorem mulsubfacd

Description: Multiplication followed by the subtraction of a factor. (Contributed by Alexander van der Vekens, 28-Aug-2018)

	Ref	Expression
Hypotheses	muls1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	muls1d.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion	mulsubfacd	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) − 𝐵 ) = ( ( 𝐴 − 1 ) · 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		muls1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		muls1d.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		1cnd	⊢ ( 𝜑 → 1 ∈ ℂ )
4	1 3 2	subdird	⊢ ( 𝜑 → ( ( 𝐴 − 1 ) · 𝐵 ) = ( ( 𝐴 · 𝐵 ) − ( 1 · 𝐵 ) ) )
5	2	mulid2d	⊢ ( 𝜑 → ( 1 · 𝐵 ) = 𝐵 )
6	5	oveq2d	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) − ( 1 · 𝐵 ) ) = ( ( 𝐴 · 𝐵 ) − 𝐵 ) )
7	4 6	eqtr2d	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) − 𝐵 ) = ( ( 𝐴 − 1 ) · 𝐵 ) )