Theorem int-rightdistd

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

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

Proof

Step	Hyp	Ref	Expression
1		int-rightdistd.1	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
2		int-rightdistd.2	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
3		int-rightdistd.3	⊢ ( 𝜑 → 𝐷 ∈ ℝ )
4		int-rightdistd.4	⊢ ( 𝜑 → 𝐴 = 𝐵 )
5	1	recnd	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
6	2	recnd	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
7	3	recnd	⊢ ( 𝜑 → 𝐷 ∈ ℂ )
8	6 7	addcld	⊢ ( 𝜑 → ( 𝐶 + 𝐷 ) ∈ ℂ )
9	5 8	mulcomd	⊢ ( 𝜑 → ( 𝐵 · ( 𝐶 + 𝐷 ) ) = ( ( 𝐶 + 𝐷 ) · 𝐵 ) )
10	6 5	mulcomd	⊢ ( 𝜑 → ( 𝐶 · 𝐵 ) = ( 𝐵 · 𝐶 ) )
11	4	eqcomd	⊢ ( 𝜑 → 𝐵 = 𝐴 )
12	11	oveq1d	⊢ ( 𝜑 → ( 𝐵 · 𝐶 ) = ( 𝐴 · 𝐶 ) )
13	10 12	eqtrd	⊢ ( 𝜑 → ( 𝐶 · 𝐵 ) = ( 𝐴 · 𝐶 ) )
14	7 5	mulcomd	⊢ ( 𝜑 → ( 𝐷 · 𝐵 ) = ( 𝐵 · 𝐷 ) )
15	11	oveq1d	⊢ ( 𝜑 → ( 𝐵 · 𝐷 ) = ( 𝐴 · 𝐷 ) )
16	14 15	eqtrd	⊢ ( 𝜑 → ( 𝐷 · 𝐵 ) = ( 𝐴 · 𝐷 ) )
17	13 16	oveq12d	⊢ ( 𝜑 → ( ( 𝐶 · 𝐵 ) + ( 𝐷 · 𝐵 ) ) = ( ( 𝐴 · 𝐶 ) + ( 𝐴 · 𝐷 ) ) )
18	6 5 7 17	joinlmuladdmuld	⊢ ( 𝜑 → ( ( 𝐶 + 𝐷 ) · 𝐵 ) = ( ( 𝐴 · 𝐶 ) + ( 𝐴 · 𝐷 ) ) )
19	9 18	eqtrd	⊢ ( 𝜑 → ( 𝐵 · ( 𝐶 + 𝐷 ) ) = ( ( 𝐴 · 𝐶 ) + ( 𝐴 · 𝐷 ) ) )