Theorem drngmul0or

Description: A product is zero iff one of its factors is zero. (Contributed by NM, 8-Oct-2014)

	Ref	Expression
Hypotheses	drngmuleq0.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	drngmuleq0.o	⊢ 0 = ( 0g ‘ 𝑅 )
	drngmuleq0.t	⊢ · = ( .r ‘ 𝑅 )
	drngmuleq0.r	⊢ ( 𝜑 → 𝑅 ∈ DivRing )
	drngmuleq0.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	drngmuleq0.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	drngmul0or	⊢ ( 𝜑 → ( ( 𝑋 · 𝑌 ) = 0 ↔ ( 𝑋 = 0 ∨ 𝑌 = 0 ) ) )

Proof

Step	Hyp	Ref	Expression
1		drngmuleq0.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		drngmuleq0.o	⊢ 0 = ( 0g ‘ 𝑅 )
3		drngmuleq0.t	⊢ · = ( .r ‘ 𝑅 )
4		drngmuleq0.r	⊢ ( 𝜑 → 𝑅 ∈ DivRing )
5		drngmuleq0.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		drngmuleq0.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7		df-ne	⊢ ( 𝑋 ≠ 0 ↔ ¬ 𝑋 = 0 )
8		oveq2	⊢ ( ( 𝑋 · 𝑌 ) = 0 → ( ( ( invr ‘ 𝑅 ) ‘ 𝑋 ) · ( 𝑋 · 𝑌 ) ) = ( ( ( invr ‘ 𝑅 ) ‘ 𝑋 ) · 0 ) )
9	8	ad2antlr	⊢ ( ( ( 𝜑 ∧ ( 𝑋 · 𝑌 ) = 0 ) ∧ 𝑋 ≠ 0 ) → ( ( ( invr ‘ 𝑅 ) ‘ 𝑋 ) · ( 𝑋 · 𝑌 ) ) = ( ( ( invr ‘ 𝑅 ) ‘ 𝑋 ) · 0 ) )
10	4	adantr	⊢ ( ( 𝜑 ∧ 𝑋 ≠ 0 ) → 𝑅 ∈ DivRing )
11	5	adantr	⊢ ( ( 𝜑 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝐵 )
12		simpr	⊢ ( ( 𝜑 ∧ 𝑋 ≠ 0 ) → 𝑋 ≠ 0 )
13		eqid	⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 )
14		eqid	⊢ ( invr ‘ 𝑅 ) = ( invr ‘ 𝑅 )
15	1 2 3 13 14	drnginvrl	⊢ ( ( 𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ( ( ( invr ‘ 𝑅 ) ‘ 𝑋 ) · 𝑋 ) = ( 1r ‘ 𝑅 ) )
16	10 11 12 15	syl3anc	⊢ ( ( 𝜑 ∧ 𝑋 ≠ 0 ) → ( ( ( invr ‘ 𝑅 ) ‘ 𝑋 ) · 𝑋 ) = ( 1r ‘ 𝑅 ) )
17	16	oveq1d	⊢ ( ( 𝜑 ∧ 𝑋 ≠ 0 ) → ( ( ( ( invr ‘ 𝑅 ) ‘ 𝑋 ) · 𝑋 ) · 𝑌 ) = ( ( 1r ‘ 𝑅 ) · 𝑌 ) )
18		drngring	⊢ ( 𝑅 ∈ DivRing → 𝑅 ∈ Ring )
19	4 18	syl	⊢ ( 𝜑 → 𝑅 ∈ Ring )
20	19	adantr	⊢ ( ( 𝜑 ∧ 𝑋 ≠ 0 ) → 𝑅 ∈ Ring )
21	1 2 14	drnginvrcl	⊢ ( ( 𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ( ( invr ‘ 𝑅 ) ‘ 𝑋 ) ∈ 𝐵 )
22	10 11 12 21	syl3anc	⊢ ( ( 𝜑 ∧ 𝑋 ≠ 0 ) → ( ( invr ‘ 𝑅 ) ‘ 𝑋 ) ∈ 𝐵 )
23	6	adantr	⊢ ( ( 𝜑 ∧ 𝑋 ≠ 0 ) → 𝑌 ∈ 𝐵 )
24	1 3	ringass	⊢ ( ( 𝑅 ∈ Ring ∧ ( ( ( invr ‘ 𝑅 ) ‘ 𝑋 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( ( ( invr ‘ 𝑅 ) ‘ 𝑋 ) · 𝑋 ) · 𝑌 ) = ( ( ( invr ‘ 𝑅 ) ‘ 𝑋 ) · ( 𝑋 · 𝑌 ) ) )
25	20 22 11 23 24	syl13anc	⊢ ( ( 𝜑 ∧ 𝑋 ≠ 0 ) → ( ( ( ( invr ‘ 𝑅 ) ‘ 𝑋 ) · 𝑋 ) · 𝑌 ) = ( ( ( invr ‘ 𝑅 ) ‘ 𝑋 ) · ( 𝑋 · 𝑌 ) ) )
26	1 3 13	ringlidm	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ) → ( ( 1r ‘ 𝑅 ) · 𝑌 ) = 𝑌 )
27	19 6 26	syl2anc	⊢ ( 𝜑 → ( ( 1r ‘ 𝑅 ) · 𝑌 ) = 𝑌 )
28	27	adantr	⊢ ( ( 𝜑 ∧ 𝑋 ≠ 0 ) → ( ( 1r ‘ 𝑅 ) · 𝑌 ) = 𝑌 )
29	17 25 28	3eqtr3d	⊢ ( ( 𝜑 ∧ 𝑋 ≠ 0 ) → ( ( ( invr ‘ 𝑅 ) ‘ 𝑋 ) · ( 𝑋 · 𝑌 ) ) = 𝑌 )
30	29	adantlr	⊢ ( ( ( 𝜑 ∧ ( 𝑋 · 𝑌 ) = 0 ) ∧ 𝑋 ≠ 0 ) → ( ( ( invr ‘ 𝑅 ) ‘ 𝑋 ) · ( 𝑋 · 𝑌 ) ) = 𝑌 )
31	19	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 · 𝑌 ) = 0 ) → 𝑅 ∈ Ring )
32	31	adantr	⊢ ( ( ( 𝜑 ∧ ( 𝑋 · 𝑌 ) = 0 ) ∧ 𝑋 ≠ 0 ) → 𝑅 ∈ Ring )
33	22	adantlr	⊢ ( ( ( 𝜑 ∧ ( 𝑋 · 𝑌 ) = 0 ) ∧ 𝑋 ≠ 0 ) → ( ( invr ‘ 𝑅 ) ‘ 𝑋 ) ∈ 𝐵 )
34	1 3 2	ringrz	⊢ ( ( 𝑅 ∈ Ring ∧ ( ( invr ‘ 𝑅 ) ‘ 𝑋 ) ∈ 𝐵 ) → ( ( ( invr ‘ 𝑅 ) ‘ 𝑋 ) · 0 ) = 0 )
35	32 33 34	syl2anc	⊢ ( ( ( 𝜑 ∧ ( 𝑋 · 𝑌 ) = 0 ) ∧ 𝑋 ≠ 0 ) → ( ( ( invr ‘ 𝑅 ) ‘ 𝑋 ) · 0 ) = 0 )
36	9 30 35	3eqtr3d	⊢ ( ( ( 𝜑 ∧ ( 𝑋 · 𝑌 ) = 0 ) ∧ 𝑋 ≠ 0 ) → 𝑌 = 0 )
37	36	ex	⊢ ( ( 𝜑 ∧ ( 𝑋 · 𝑌 ) = 0 ) → ( 𝑋 ≠ 0 → 𝑌 = 0 ) )
38	7 37	syl5bir	⊢ ( ( 𝜑 ∧ ( 𝑋 · 𝑌 ) = 0 ) → ( ¬ 𝑋 = 0 → 𝑌 = 0 ) )
39	38	orrd	⊢ ( ( 𝜑 ∧ ( 𝑋 · 𝑌 ) = 0 ) → ( 𝑋 = 0 ∨ 𝑌 = 0 ) )
40	39	ex	⊢ ( 𝜑 → ( ( 𝑋 · 𝑌 ) = 0 → ( 𝑋 = 0 ∨ 𝑌 = 0 ) ) )
41	1 3 2	ringlz	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ) → ( 0 · 𝑌 ) = 0 )
42	19 6 41	syl2anc	⊢ ( 𝜑 → ( 0 · 𝑌 ) = 0 )
43		oveq1	⊢ ( 𝑋 = 0 → ( 𝑋 · 𝑌 ) = ( 0 · 𝑌 ) )
44	43	eqeq1d	⊢ ( 𝑋 = 0 → ( ( 𝑋 · 𝑌 ) = 0 ↔ ( 0 · 𝑌 ) = 0 ) )
45	42 44	syl5ibrcom	⊢ ( 𝜑 → ( 𝑋 = 0 → ( 𝑋 · 𝑌 ) = 0 ) )
46	1 3 2	ringrz	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 · 0 ) = 0 )
47	19 5 46	syl2anc	⊢ ( 𝜑 → ( 𝑋 · 0 ) = 0 )
48		oveq2	⊢ ( 𝑌 = 0 → ( 𝑋 · 𝑌 ) = ( 𝑋 · 0 ) )
49	48	eqeq1d	⊢ ( 𝑌 = 0 → ( ( 𝑋 · 𝑌 ) = 0 ↔ ( 𝑋 · 0 ) = 0 ) )
50	47 49	syl5ibrcom	⊢ ( 𝜑 → ( 𝑌 = 0 → ( 𝑋 · 𝑌 ) = 0 ) )
51	45 50	jaod	⊢ ( 𝜑 → ( ( 𝑋 = 0 ∨ 𝑌 = 0 ) → ( 𝑋 · 𝑌 ) = 0 ) )
52	40 51	impbid	⊢ ( 𝜑 → ( ( 𝑋 · 𝑌 ) = 0 ↔ ( 𝑋 = 0 ∨ 𝑌 = 0 ) ) )