Theorem redivdird

Description: Distribution of division over addition. (Contributed by SN, 9-Apr-2026)

	Ref	Expression
Hypotheses	rediv23d.a	$$⊢\phi \to A\in ℝ$$
	rediv23d.b	$$⊢\phi \to B\in ℝ$$
	rediv23d.c	$$⊢\phi \to C\in ℝ$$
	rediv23d.z	$$⊢\phi \to C\ne 0$$
Assertion	redivdird	Could not format assertion : No typesetting found for |- ( ph -> ( ( A + B ) /R C ) = ( ( A /R C ) + ( B /R C ) ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		rediv23d.a	$$⊢\phi \to A\in ℝ$$
2		rediv23d.b	$$⊢\phi \to B\in ℝ$$
3		rediv23d.c	$$⊢\phi \to C\in ℝ$$
4		rediv23d.z	$$⊢\phi \to C\ne 0$$
5	3	recnd	$$⊢\phi \to C\in ℂ$$
6	1 3 4	sn-redivcld	Could not format ( ph -> ( A /R C ) e. RR ) : No typesetting found for |- ( ph -> ( A /R C ) e. RR ) with typecode |-
7	6	recnd	Could not format ( ph -> ( A /R C ) e. CC ) : No typesetting found for |- ( ph -> ( A /R C ) e. CC ) with typecode |-
8	2 3 4	sn-redivcld	Could not format ( ph -> ( B /R C ) e. RR ) : No typesetting found for |- ( ph -> ( B /R C ) e. RR ) with typecode |-
9	8	recnd	Could not format ( ph -> ( B /R C ) e. CC ) : No typesetting found for |- ( ph -> ( B /R C ) e. CC ) with typecode |-
10	5 7 9	adddid	Could not format ( ph -> ( C x. ( ( A /R C ) + ( B /R C ) ) ) = ( ( C x. ( A /R C ) ) + ( C x. ( B /R C ) ) ) ) : No typesetting found for |- ( ph -> ( C x. ( ( A /R C ) + ( B /R C ) ) ) = ( ( C x. ( A /R C ) ) + ( C x. ( B /R C ) ) ) ) with typecode |-
11	1 3 4	redivcan2d	Could not format ( ph -> ( C x. ( A /R C ) ) = A ) : No typesetting found for |- ( ph -> ( C x. ( A /R C ) ) = A ) with typecode |-
12	2 3 4	redivcan2d	Could not format ( ph -> ( C x. ( B /R C ) ) = B ) : No typesetting found for |- ( ph -> ( C x. ( B /R C ) ) = B ) with typecode |-
13	11 12	oveq12d	Could not format ( ph -> ( ( C x. ( A /R C ) ) + ( C x. ( B /R C ) ) ) = ( A + B ) ) : No typesetting found for |- ( ph -> ( ( C x. ( A /R C ) ) + ( C x. ( B /R C ) ) ) = ( A + B ) ) with typecode |-
14	10 13	eqtrd	Could not format ( ph -> ( C x. ( ( A /R C ) + ( B /R C ) ) ) = ( A + B ) ) : No typesetting found for |- ( ph -> ( C x. ( ( A /R C ) + ( B /R C ) ) ) = ( A + B ) ) with typecode |-
15	1 2	readdcld	$$⊢\phi \to A+B\in ℝ$$
16	6 8	readdcld	Could not format ( ph -> ( ( A /R C ) + ( B /R C ) ) e. RR ) : No typesetting found for |- ( ph -> ( ( A /R C ) + ( B /R C ) ) e. RR ) with typecode |-
17	15 16 3 4	redivmuld	Could not format ( ph -> ( ( ( A + B ) /R C ) = ( ( A /R C ) + ( B /R C ) ) <-> ( C x. ( ( A /R C ) + ( B /R C ) ) ) = ( A + B ) ) ) : No typesetting found for |- ( ph -> ( ( ( A + B ) /R C ) = ( ( A /R C ) + ( B /R C ) ) <-> ( C x. ( ( A /R C ) + ( B /R C ) ) ) = ( A + B ) ) ) with typecode |-
18	14 17	mpbird	Could not format ( ph -> ( ( A + B ) /R C ) = ( ( A /R C ) + ( B /R C ) ) ) : No typesetting found for |- ( ph -> ( ( A + B ) /R C ) = ( ( A /R C ) + ( B /R C ) ) ) with typecode |-