Theorem int-leftdistd

Description: AdditionMultiplicationLeftDistribution generator rule. (Contributed by Stanislas Polu, 7-Apr-2020)

	Ref	Expression
Hypotheses	int-leftdistd.1	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	int-leftdistd.2	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
	int-leftdistd.3	⊢ ( 𝜑 → 𝐷 ∈ ℝ )
	int-leftdistd.4	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	int-leftdistd	⊢ ( 𝜑 → ( ( 𝐶 + 𝐷 ) · 𝐵 ) = ( ( 𝐶 · 𝐴 ) + ( 𝐷 · 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		int-leftdistd.1	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
2		int-leftdistd.2	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
3		int-leftdistd.3	⊢ ( 𝜑 → 𝐷 ∈ ℝ )
4		int-leftdistd.4	⊢ ( 𝜑 → 𝐴 = 𝐵 )
5	2	recnd	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
6	3	recnd	⊢ ( 𝜑 → 𝐷 ∈ ℂ )
7	1	recnd	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
8	5 6 7	adddird	⊢ ( 𝜑 → ( ( 𝐶 + 𝐷 ) · 𝐵 ) = ( ( 𝐶 · 𝐵 ) + ( 𝐷 · 𝐵 ) ) )
9	5 7	mulcld	⊢ ( 𝜑 → ( 𝐶 · 𝐵 ) ∈ ℂ )
10	6 7	mulcld	⊢ ( 𝜑 → ( 𝐷 · 𝐵 ) ∈ ℂ )
11	9 10	addcomd	⊢ ( 𝜑 → ( ( 𝐶 · 𝐵 ) + ( 𝐷 · 𝐵 ) ) = ( ( 𝐷 · 𝐵 ) + ( 𝐶 · 𝐵 ) ) )
12	10 9	addcomd	⊢ ( 𝜑 → ( ( 𝐷 · 𝐵 ) + ( 𝐶 · 𝐵 ) ) = ( ( 𝐶 · 𝐵 ) + ( 𝐷 · 𝐵 ) ) )
13	4	eqcomd	⊢ ( 𝜑 → 𝐵 = 𝐴 )
14	13	oveq2d	⊢ ( 𝜑 → ( 𝐶 · 𝐵 ) = ( 𝐶 · 𝐴 ) )
15	13	oveq2d	⊢ ( 𝜑 → ( 𝐷 · 𝐵 ) = ( 𝐷 · 𝐴 ) )
16	14 15	oveq12d	⊢ ( 𝜑 → ( ( 𝐶 · 𝐵 ) + ( 𝐷 · 𝐵 ) ) = ( ( 𝐶 · 𝐴 ) + ( 𝐷 · 𝐴 ) ) )
17	12 16	eqtrd	⊢ ( 𝜑 → ( ( 𝐷 · 𝐵 ) + ( 𝐶 · 𝐵 ) ) = ( ( 𝐶 · 𝐴 ) + ( 𝐷 · 𝐴 ) ) )
18	8 11 17	3eqtrd	⊢ ( 𝜑 → ( ( 𝐶 + 𝐷 ) · 𝐵 ) = ( ( 𝐶 · 𝐴 ) + ( 𝐷 · 𝐴 ) ) )