Theorem trimul0or 16845

Description: Real number trichotomy implies that if a product is zero, one of its factors must be zero. (Contributed by Jim Kingdon, 27-May-2026.)

Assertion

Ref	Expression
trimul0or	⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → ∀𝑢 ∈ ℂ ∀𝑣 ∈ ℂ ((𝑢 · 𝑣) = 0 → (𝑢 = 0 ∨ 𝑣 = 0)))

Distinct variable group:   𝑣,𝑢,𝑥,𝑦

Proof of Theorem trimul0or

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 531	. . . . . . . 8 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → 𝑢 ∈ ℂ)
2	1	ad3antrrr 492	. . . . . . 7 ⊢ (((((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑢 # 0) ∧ 𝑣 # 0) ∧ (𝑢 · 𝑣) = 0) → 𝑢 ∈ ℂ)
3		simplrr 538	. . . . . . . 8 ⊢ (((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑢 # 0) → 𝑣 ∈ ℂ)
4	3	ad2antrr 488	. . . . . . 7 ⊢ (((((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑢 # 0) ∧ 𝑣 # 0) ∧ (𝑢 · 𝑣) = 0) → 𝑣 ∈ ℂ)
5		simpllr 536	. . . . . . 7 ⊢ (((((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑢 # 0) ∧ 𝑣 # 0) ∧ (𝑢 · 𝑣) = 0) → 𝑢 # 0)
6		simplr 529	. . . . . . 7 ⊢ (((((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑢 # 0) ∧ 𝑣 # 0) ∧ (𝑢 · 𝑣) = 0) → 𝑣 # 0)
7	2, 4, 5, 6	mulap0d 8932	. . . . . 6 ⊢ (((((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑢 # 0) ∧ 𝑣 # 0) ∧ (𝑢 · 𝑣) = 0) → (𝑢 · 𝑣) # 0)
8		simpr 110	. . . . . . 7 ⊢ (((((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑢 # 0) ∧ 𝑣 # 0) ∧ (𝑢 · 𝑣) = 0) → (𝑢 · 𝑣) = 0)
9	2, 4	mulcld 8294	. . . . . . . 8 ⊢ (((((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑢 # 0) ∧ 𝑣 # 0) ∧ (𝑢 · 𝑣) = 0) → (𝑢 · 𝑣) ∈ ℂ)
10		0cn 8266	. . . . . . . 8 ⊢ 0 ∈ ℂ
11		apti 8896	. . . . . . . 8 ⊢ (((𝑢 · 𝑣) ∈ ℂ ∧ 0 ∈ ℂ) → ((𝑢 · 𝑣) = 0 ↔ ¬ (𝑢 · 𝑣) # 0))
12	9, 10, 11	sylancl 413	. . . . . . 7 ⊢ (((((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑢 # 0) ∧ 𝑣 # 0) ∧ (𝑢 · 𝑣) = 0) → ((𝑢 · 𝑣) = 0 ↔ ¬ (𝑢 · 𝑣) # 0))
13	8, 12	mpbid 147	. . . . . 6 ⊢ (((((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑢 # 0) ∧ 𝑣 # 0) ∧ (𝑢 · 𝑣) = 0) → ¬ (𝑢 · 𝑣) # 0)
14	7, 13	pm2.21dd 625	. . . . 5 ⊢ (((((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑢 # 0) ∧ 𝑣 # 0) ∧ (𝑢 · 𝑣) = 0) → (𝑢 = 0 ∨ 𝑣 = 0))
15	14	ex 115	. . . 4 ⊢ ((((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑢 # 0) ∧ 𝑣 # 0) → ((𝑢 · 𝑣) = 0 → (𝑢 = 0 ∨ 𝑣 = 0)))
16		simpr 110	. . . . . . 7 ⊢ ((((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑢 # 0) ∧ ¬ 𝑣 # 0) → ¬ 𝑣 # 0)
17	3	adantr 276	. . . . . . . 8 ⊢ ((((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑢 # 0) ∧ ¬ 𝑣 # 0) → 𝑣 ∈ ℂ)
18		apti 8896	. . . . . . . 8 ⊢ ((𝑣 ∈ ℂ ∧ 0 ∈ ℂ) → (𝑣 = 0 ↔ ¬ 𝑣 # 0))
19	17, 10, 18	sylancl 413	. . . . . . 7 ⊢ ((((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑢 # 0) ∧ ¬ 𝑣 # 0) → (𝑣 = 0 ↔ ¬ 𝑣 # 0))
20	16, 19	mpbird 167	. . . . . 6 ⊢ ((((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑢 # 0) ∧ ¬ 𝑣 # 0) → 𝑣 = 0)
21	20	olcd 742	. . . . 5 ⊢ ((((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑢 # 0) ∧ ¬ 𝑣 # 0) → (𝑢 = 0 ∨ 𝑣 = 0))
22	21	a1d 22	. . . 4 ⊢ ((((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑢 # 0) ∧ ¬ 𝑣 # 0) → ((𝑢 · 𝑣) = 0 → (𝑢 = 0 ∨ 𝑣 = 0)))
23		breq1 4112	. . . . . . 7 ⊢ (𝑧 = 𝑣 → (𝑧 # 0 ↔ 𝑣 # 0))
24	23	dcbid 846	. . . . . 6 ⊢ (𝑧 = 𝑣 → (DECID 𝑧 # 0 ↔ DECID 𝑣 # 0))
25		breq2 4113	. . . . . . . . . 10 ⊢ (𝑤 = 0 → (𝑧 # 𝑤 ↔ 𝑧 # 0))
26	25	dcbid 846	. . . . . . . . 9 ⊢ (𝑤 = 0 → (DECID 𝑧 # 𝑤 ↔ DECID 𝑧 # 0))
27		cndcap 16844	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ∀𝑧 ∈ ℂ ∀𝑤 ∈ ℂ DECID 𝑧 # 𝑤)
28	27	biimpi 120	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → ∀𝑧 ∈ ℂ ∀𝑤 ∈ ℂ DECID 𝑧 # 𝑤)
29	28	adantr 276	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → ∀𝑧 ∈ ℂ ∀𝑤 ∈ ℂ DECID 𝑧 # 𝑤)
30	29	r19.21bi 2630	. . . . . . . . 9 ⊢ (((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑧 ∈ ℂ) → ∀𝑤 ∈ ℂ DECID 𝑧 # 𝑤)
31		0cnd 8267	. . . . . . . . 9 ⊢ (((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑧 ∈ ℂ) → 0 ∈ ℂ)
32	26, 30, 31	rspcdva 2926	. . . . . . . 8 ⊢ (((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑧 ∈ ℂ) → DECID 𝑧 # 0)
33	32	ralrimiva 2615	. . . . . . 7 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → ∀𝑧 ∈ ℂ DECID 𝑧 # 0)
34	33	adantr 276	. . . . . 6 ⊢ (((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑢 # 0) → ∀𝑧 ∈ ℂ DECID 𝑧 # 0)
35	24, 34, 3	rspcdva 2926	. . . . 5 ⊢ (((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑢 # 0) → DECID 𝑣 # 0)
36		exmiddc 844	. . . . 5 ⊢ (DECID 𝑣 # 0 → (𝑣 # 0 ∨ ¬ 𝑣 # 0))
37	35, 36	syl 14	. . . 4 ⊢ (((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑢 # 0) → (𝑣 # 0 ∨ ¬ 𝑣 # 0))
38	15, 22, 37	mpjaodan 806	. . 3 ⊢ (((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ 𝑢 # 0) → ((𝑢 · 𝑣) = 0 → (𝑢 = 0 ∨ 𝑣 = 0)))
39		simpr 110	. . . . . 6 ⊢ (((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ ¬ 𝑢 # 0) → ¬ 𝑢 # 0)
40	1	adantr 276	. . . . . . 7 ⊢ (((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ ¬ 𝑢 # 0) → 𝑢 ∈ ℂ)
41		apti 8896	. . . . . . 7 ⊢ ((𝑢 ∈ ℂ ∧ 0 ∈ ℂ) → (𝑢 = 0 ↔ ¬ 𝑢 # 0))
42	40, 10, 41	sylancl 413	. . . . . 6 ⊢ (((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ ¬ 𝑢 # 0) → (𝑢 = 0 ↔ ¬ 𝑢 # 0))
43	39, 42	mpbird 167	. . . . 5 ⊢ (((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ ¬ 𝑢 # 0) → 𝑢 = 0)
44	43	orcd 741	. . . 4 ⊢ (((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ ¬ 𝑢 # 0) → (𝑢 = 0 ∨ 𝑣 = 0))
45	44	a1d 22	. . 3 ⊢ (((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) ∧ ¬ 𝑢 # 0) → ((𝑢 · 𝑣) = 0 → (𝑢 = 0 ∨ 𝑣 = 0)))
46		breq1 4112	. . . . . 6 ⊢ (𝑧 = 𝑢 → (𝑧 # 0 ↔ 𝑢 # 0))
47	46	dcbid 846	. . . . 5 ⊢ (𝑧 = 𝑢 → (DECID 𝑧 # 0 ↔ DECID 𝑢 # 0))
48	47, 33, 1	rspcdva 2926	. . . 4 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → DECID 𝑢 # 0)
49		exmiddc 844	. . . 4 ⊢ (DECID 𝑢 # 0 → (𝑢 # 0 ∨ ¬ 𝑢 # 0))
50	48, 49	syl 14	. . 3 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑢 # 0 ∨ ¬ 𝑢 # 0))
51	38, 45, 50	mpjaodan 806	. 2 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → ((𝑢 · 𝑣) = 0 → (𝑢 = 0 ∨ 𝑣 = 0)))
52	51	ralrimivva 2624	1 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → ∀𝑢 ∈ ℂ ∀𝑣 ∈ ℂ ((𝑢 · 𝑣) = 0 → (𝑢 = 0 ∨ 𝑣 = 0)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716  DECID wdc 842   ∨ w3o 1004   = wceq 1398   ∈ wcel 2203  ∀wral 2520   class class class wbr 4109  (class class class)co 6050  ℂcc 8125  ℝcr 8126  0cc0 8127   · cmul 8132   < clt 8308   # cap 8855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-po 4417  df-iso 4418  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-2 9296  df-cj 11527  df-re 11528  df-im 11529

This theorem is referenced by: (None)