Theorem int-rightdistd

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

	Ref	Expression
Hypotheses	int-rightdistd.1	$$⊢\phi \to B\in ℝ$$
	int-rightdistd.2	$$⊢\phi \to C\in ℝ$$
	int-rightdistd.3	$$⊢\phi \to D\in ℝ$$
	int-rightdistd.4	$$⊢\phi \to A=B$$
Assertion	int-rightdistd	$$⊢\phi \to B\left(C+D\right)=AC+AD$$

Proof

Step	Hyp	Ref	Expression
1		int-rightdistd.1	$$⊢\phi \to B\in ℝ$$
2		int-rightdistd.2	$$⊢\phi \to C\in ℝ$$
3		int-rightdistd.3	$$⊢\phi \to D\in ℝ$$
4		int-rightdistd.4	$$⊢\phi \to A=B$$
5	1	recnd	$$⊢\phi \to B\in ℂ$$
6	2	recnd	$$⊢\phi \to C\in ℂ$$
7	3	recnd	$$⊢\phi \to D\in ℂ$$
8	6 7	addcld	$$⊢\phi \to C+D\in ℂ$$
9	5 8	mulcomd	$$⊢\phi \to B\left(C+D\right)=\left(C+D\right)B$$
10	6 5	mulcomd	$$⊢\phi \to CB=BC$$
11	4	eqcomd	$$⊢\phi \to B=A$$
12	11	oveq1d	$$⊢\phi \to BC=AC$$
13	10 12	eqtrd	$$⊢\phi \to CB=AC$$
14	7 5	mulcomd	$$⊢\phi \to DB=BD$$
15	11	oveq1d	$$⊢\phi \to BD=AD$$
16	14 15	eqtrd	$$⊢\phi \to DB=AD$$
17	13 16	oveq12d	$$⊢\phi \to CB+DB=AC+AD$$
18	6 5 7 17	joinlmuladdmuld	$$⊢\phi \to \left(C+D\right)B=AC+AD$$
19	9 18	eqtrd	$$⊢\phi \to B\left(C+D\right)=AC+AD$$