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Theorem pzriprnglem8 21405

Description: Lemma 8 for pzriprng 21414: 𝐼 resp. 𝐽 is a two-sided ideal of the non-unital ring 𝑅. (Contributed by AV, 21-Mar-2025.)

**Hypotheses**
Ref	Expression
pzriprng.r	⊢ 𝑅 = (ℤ_ring ×_s ℤ_ring)
pzriprng.i	⊢ 𝐼 = (ℤ × {0})
pzriprng.j	⊢ 𝐽 = (𝑅 ↾_s 𝐼)

**Assertion**
Ref	Expression
pzriprnglem8	⊢ 𝐼 ∈ (2Ideal‘𝑅)

Proof of Theorem pzriprnglem8 Dummy variables 𝑥 𝑦 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pzriprng.r	. . . . . . 7 ⊢ 𝑅 = (ℤ_ring ×_s ℤ_ring)
2	1	pzriprnglem2 21399	. . . . . 6 ⊢ (Base‘𝑅) = (ℤ × ℤ)
3	2	eleq2i 2821	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝑅) ↔ 𝑥 ∈ (ℤ × ℤ))
4		elxp2 5665	. . . . 5 ⊢ (𝑥 ∈ (ℤ × ℤ) ↔ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑥 = ⟨𝑎, 𝑏⟩)
5	3, 4	bitri 275	. . . 4 ⊢ (𝑥 ∈ (Base‘𝑅) ↔ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑥 = ⟨𝑎, 𝑏⟩)
6		pzriprng.i	. . . . 5 ⊢ 𝐼 = (ℤ × {0})
7	1, 6	pzriprnglem3 21400	. . . 4 ⊢ (𝑦 ∈ 𝐼 ↔ ∃𝑐 ∈ ℤ 𝑦 = ⟨𝑐, 0⟩)
8		simpll 766	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → 𝑎 ∈ ℤ)
9		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → 𝑐 ∈ ℤ)
10	8, 9	zmulcld 12651	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (𝑎 · 𝑐) ∈ ℤ)
11		zcn 12541	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ ℤ → 𝑏 ∈ ℂ)
12	11	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → 𝑏 ∈ ℂ)
13	12	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → 𝑏 ∈ ℂ)
14	13	mul01d 11380	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (𝑏 · 0) = 0)
15		ovex 7423	. . . . . . . . . . . . . . 15 ⊢ (𝑏 · 0) ∈ V
16	15	elsn 4607	. . . . . . . . . . . . . 14 ⊢ ((𝑏 · 0) ∈ {0} ↔ (𝑏 · 0) = 0)
17	14, 16	sylibr 234	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (𝑏 · 0) ∈ {0})
18	10, 17	opelxpd 5680	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → ⟨(𝑎 · 𝑐), (𝑏 · 0)⟩ ∈ (ℤ × {0}))
19	9, 8	zmulcld 12651	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (𝑐 · 𝑎) ∈ ℤ)
20	13	mul02d 11379	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (0 · 𝑏) = 0)
21		ovex 7423	. . . . . . . . . . . . . . 15 ⊢ (0 · 𝑏) ∈ V
22	21	elsn 4607	. . . . . . . . . . . . . 14 ⊢ ((0 · 𝑏) ∈ {0} ↔ (0 · 𝑏) = 0)
23	20, 22	sylibr 234	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (0 · 𝑏) ∈ {0})
24	19, 23	opelxpd 5680	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → ⟨(𝑐 · 𝑎), (0 · 𝑏)⟩ ∈ (ℤ × {0}))
25		zringbas 21370	. . . . . . . . . . . . . . 15 ⊢ ℤ = (Base‘ℤ_ring)
26		zringring 21366	. . . . . . . . . . . . . . . 16 ⊢ ℤ_ring ∈ Ring
27	26	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → ℤ_ring ∈ Ring)
28		simplr 768	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → 𝑏 ∈ ℤ)
29		0zd 12548	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → 0 ∈ ℤ)
30	28, 29	zmulcld 12651	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (𝑏 · 0) ∈ ℤ)
31		zringmulr 21374	. . . . . . . . . . . . . . 15 ⊢ · = (._r‘ℤ_ring)
32		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ (._r‘𝑅) = (._r‘𝑅)
33	1, 25, 25, 27, 27, 8, 28, 9, 29, 10, 30, 31, 31, 32	xpsmul 17545	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (⟨𝑎, 𝑏⟩(._r‘𝑅)⟨𝑐, 0⟩) = ⟨(𝑎 · 𝑐), (𝑏 · 0)⟩)
34	33	eleq1d 2814	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → ((⟨𝑎, 𝑏⟩(._r‘𝑅)⟨𝑐, 0⟩) ∈ (ℤ × {0}) ↔ ⟨(𝑎 · 𝑐), (𝑏 · 0)⟩ ∈ (ℤ × {0})))
35		simpl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ ℤ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → 𝑐 ∈ ℤ)
36		simprl 770	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ ℤ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → 𝑎 ∈ ℤ)
37	35, 36	zmulcld 12651	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ ℤ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → (𝑐 · 𝑎) ∈ ℤ)
38	37	ancoms 458	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (𝑐 · 𝑎) ∈ ℤ)
39		0zd 12548	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ ℤ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → 0 ∈ ℤ)
40		simprr 772	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ ℤ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → 𝑏 ∈ ℤ)
41	39, 40	zmulcld 12651	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ ℤ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → (0 · 𝑏) ∈ ℤ)
42	41	ancoms 458	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (0 · 𝑏) ∈ ℤ)
43	1, 25, 25, 27, 27, 9, 29, 8, 28, 38, 42, 31, 31, 32	xpsmul 17545	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (⟨𝑐, 0⟩(._r‘𝑅)⟨𝑎, 𝑏⟩) = ⟨(𝑐 · 𝑎), (0 · 𝑏)⟩)
44	43	eleq1d 2814	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → ((⟨𝑐, 0⟩(._r‘𝑅)⟨𝑎, 𝑏⟩) ∈ (ℤ × {0}) ↔ ⟨(𝑐 · 𝑎), (0 · 𝑏)⟩ ∈ (ℤ × {0})))
45	34, 44	anbi12d 632	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (((⟨𝑎, 𝑏⟩(._r‘𝑅)⟨𝑐, 0⟩) ∈ (ℤ × {0}) ∧ (⟨𝑐, 0⟩(._r‘𝑅)⟨𝑎, 𝑏⟩) ∈ (ℤ × {0})) ↔ (⟨(𝑎 · 𝑐), (𝑏 · 0)⟩ ∈ (ℤ × {0}) ∧ ⟨(𝑐 · 𝑎), (0 · 𝑏)⟩ ∈ (ℤ × {0}))))
46	18, 24, 45	mpbir2and 713	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → ((⟨𝑎, 𝑏⟩(._r‘𝑅)⟨𝑐, 0⟩) ∈ (ℤ × {0}) ∧ (⟨𝑐, 0⟩(._r‘𝑅)⟨𝑎, 𝑏⟩) ∈ (ℤ × {0})))
47	46	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) ∧ (𝑦 = ⟨𝑐, 0⟩ ∧ 𝑥 = ⟨𝑎, 𝑏⟩)) → ((⟨𝑎, 𝑏⟩(._r‘𝑅)⟨𝑐, 0⟩) ∈ (ℤ × {0}) ∧ (⟨𝑐, 0⟩(._r‘𝑅)⟨𝑎, 𝑏⟩) ∈ (ℤ × {0})))
48		oveq12 7399	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑐, 0⟩) → (𝑥(._r‘𝑅)𝑦) = (⟨𝑎, 𝑏⟩(._r‘𝑅)⟨𝑐, 0⟩))
49	48	ancoms 458	. . . . . . . . . . . . 13 ⊢ ((𝑦 = ⟨𝑐, 0⟩ ∧ 𝑥 = ⟨𝑎, 𝑏⟩) → (𝑥(._r‘𝑅)𝑦) = (⟨𝑎, 𝑏⟩(._r‘𝑅)⟨𝑐, 0⟩))
50	49	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) ∧ (𝑦 = ⟨𝑐, 0⟩ ∧ 𝑥 = ⟨𝑎, 𝑏⟩)) → (𝑥(._r‘𝑅)𝑦) = (⟨𝑎, 𝑏⟩(._r‘𝑅)⟨𝑐, 0⟩))
51	6	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) ∧ (𝑦 = ⟨𝑐, 0⟩ ∧ 𝑥 = ⟨𝑎, 𝑏⟩)) → 𝐼 = (ℤ × {0}))
52	50, 51	eleq12d 2823	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) ∧ (𝑦 = ⟨𝑐, 0⟩ ∧ 𝑥 = ⟨𝑎, 𝑏⟩)) → ((𝑥(._r‘𝑅)𝑦) ∈ 𝐼 ↔ (⟨𝑎, 𝑏⟩(._r‘𝑅)⟨𝑐, 0⟩) ∈ (ℤ × {0})))
53		oveq12 7399	. . . . . . . . . . . . 13 ⊢ ((𝑦 = ⟨𝑐, 0⟩ ∧ 𝑥 = ⟨𝑎, 𝑏⟩) → (𝑦(._r‘𝑅)𝑥) = (⟨𝑐, 0⟩(._r‘𝑅)⟨𝑎, 𝑏⟩))
54	53	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) ∧ (𝑦 = ⟨𝑐, 0⟩ ∧ 𝑥 = ⟨𝑎, 𝑏⟩)) → (𝑦(._r‘𝑅)𝑥) = (⟨𝑐, 0⟩(._r‘𝑅)⟨𝑎, 𝑏⟩))
55	54, 51	eleq12d 2823	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) ∧ (𝑦 = ⟨𝑐, 0⟩ ∧ 𝑥 = ⟨𝑎, 𝑏⟩)) → ((𝑦(._r‘𝑅)𝑥) ∈ 𝐼 ↔ (⟨𝑐, 0⟩(._r‘𝑅)⟨𝑎, 𝑏⟩) ∈ (ℤ × {0})))
56	52, 55	anbi12d 632	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) ∧ (𝑦 = ⟨𝑐, 0⟩ ∧ 𝑥 = ⟨𝑎, 𝑏⟩)) → (((𝑥(._r‘𝑅)𝑦) ∈ 𝐼 ∧ (𝑦(._r‘𝑅)𝑥) ∈ 𝐼) ↔ ((⟨𝑎, 𝑏⟩(._r‘𝑅)⟨𝑐, 0⟩) ∈ (ℤ × {0}) ∧ (⟨𝑐, 0⟩(._r‘𝑅)⟨𝑎, 𝑏⟩) ∈ (ℤ × {0}))))
57	47, 56	mpbird 257	. . . . . . . . 9 ⊢ ((((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) ∧ (𝑦 = ⟨𝑐, 0⟩ ∧ 𝑥 = ⟨𝑎, 𝑏⟩)) → ((𝑥(._r‘𝑅)𝑦) ∈ 𝐼 ∧ (𝑦(._r‘𝑅)𝑥) ∈ 𝐼))
58	57	exp32 420	. . . . . . . 8 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (𝑦 = ⟨𝑐, 0⟩ → (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝑥(._r‘𝑅)𝑦) ∈ 𝐼 ∧ (𝑦(._r‘𝑅)𝑥) ∈ 𝐼))))
59	58	rexlimdva 3135	. . . . . . 7 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → (∃𝑐 ∈ ℤ 𝑦 = ⟨𝑐, 0⟩ → (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝑥(._r‘𝑅)𝑦) ∈ 𝐼 ∧ (𝑦(._r‘𝑅)𝑥) ∈ 𝐼))))
60	59	com23 86	. . . . . 6 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → (𝑥 = ⟨𝑎, 𝑏⟩ → (∃𝑐 ∈ ℤ 𝑦 = ⟨𝑐, 0⟩ → ((𝑥(._r‘𝑅)𝑦) ∈ 𝐼 ∧ (𝑦(._r‘𝑅)𝑥) ∈ 𝐼))))
61	60	rexlimivv 3180	. . . . 5 ⊢ (∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑥 = ⟨𝑎, 𝑏⟩ → (∃𝑐 ∈ ℤ 𝑦 = ⟨𝑐, 0⟩ → ((𝑥(._r‘𝑅)𝑦) ∈ 𝐼 ∧ (𝑦(._r‘𝑅)𝑥) ∈ 𝐼)))
62	61	imp 406	. . . 4 ⊢ ((∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ 𝑥 = ⟨𝑎, 𝑏⟩ ∧ ∃𝑐 ∈ ℤ 𝑦 = ⟨𝑐, 0⟩) → ((𝑥(._r‘𝑅)𝑦) ∈ 𝐼 ∧ (𝑦(._r‘𝑅)𝑥) ∈ 𝐼))
63	5, 7, 62	syl2anb 598	. . 3 ⊢ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝐼) → ((𝑥(._r‘𝑅)𝑦) ∈ 𝐼 ∧ (𝑦(._r‘𝑅)𝑥) ∈ 𝐼))
64	63	rgen2 3178	. 2 ⊢ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ 𝐼 ((𝑥(._r‘𝑅)𝑦) ∈ 𝐼 ∧ (𝑦(._r‘𝑅)𝑥) ∈ 𝐼)
65	1	pzriprnglem1 21398	. . 3 ⊢ 𝑅 ∈ Rng
66	1, 6	pzriprnglem4 21401	. . 3 ⊢ 𝐼 ∈ (SubGrp‘𝑅)
67		eqid 2730	. . . 4 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
68		eqid 2730	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
69	67, 68, 32	df2idl2rng 21173	. . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) → (𝐼 ∈ (2Ideal‘𝑅) ↔ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ 𝐼 ((𝑥(._r‘𝑅)𝑦) ∈ 𝐼 ∧ (𝑦(._r‘𝑅)𝑥) ∈ 𝐼)))
70	65, 66, 69	mp2an 692	. 2 ⊢ (𝐼 ∈ (2Ideal‘𝑅) ↔ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ 𝐼 ((𝑥(._r‘𝑅)𝑦) ∈ 𝐼 ∧ (𝑦(._r‘𝑅)𝑥) ∈ 𝐼))
71	64, 70	mpbir 231	1 ⊢ 𝐼 ∈ (2Ideal‘𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3045 ∃wrex 3054 {csn 4592 ⟨cop 4598 × cxp 5639 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 0cc0 11075 · cmul 11080 ℤcz 12536 Basecbs 17186 ↾_s cress 17207 ._rcmulr 17228 ×_s cxps 17476 SubGrpcsubg 19059 Rngcrng 20068 Ringcrg 20149 2Idealc2idl 21166 ℤ_ringczring 21363

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-addf 11154 ax-mulf 11155

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9400 df-inf 9401 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-fz 13476 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-0g 17411 df-prds 17417 df-imas 17478 df-xps 17480 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-grp 18875 df-minusg 18876 df-subg 19062 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-cring 20152 df-oppr 20253 df-subrng 20462 df-subrg 20486 df-lss 20845 df-sra 21087 df-rgmod 21088 df-lidl 21125 df-2idl 21167 df-cnfld 21272 df-zring 21364

This theorem is referenced by: pzriprnglem12 21409 pzriprnglem13 21410 pzriprngALT 21412 pzriprng1ALT 21413

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