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Theorem pzriprnglem8 21461

Description: Lemma 8 for pzriprng 21470: 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 21455	. . . . . 6 ⊢ (Base‘𝑅) = (ℤ × ℤ)
3	2	eleq2i 2825	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝑅) ↔ 𝑥 ∈ (ℤ × ℤ))
4		elxp2 5689	. . . . 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 21456	. . . 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 12711	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (𝑎 · 𝑐) ∈ ℤ)
11		zcn 12601	. . . . . . . . . . . . . . . . 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 11442	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (𝑏 · 0) = 0)
15		ovex 7446	. . . . . . . . . . . . . . 15 ⊢ (𝑏 · 0) ∈ V
16	15	elsn 4621	. . . . . . . . . . . . . 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 5704	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → ⟨(𝑎 · 𝑐), (𝑏 · 0)⟩ ∈ (ℤ × {0}))
19	9, 8	zmulcld 12711	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (𝑐 · 𝑎) ∈ ℤ)
20	13	mul02d 11441	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (0 · 𝑏) = 0)
21		ovex 7446	. . . . . . . . . . . . . . 15 ⊢ (0 · 𝑏) ∈ V
22	21	elsn 4621	. . . . . . . . . . . . . 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 5704	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → ⟨(𝑐 · 𝑎), (0 · 𝑏)⟩ ∈ (ℤ × {0}))
25		zringbas 21426	. . . . . . . . . . . . . . 15 ⊢ ℤ = (Base‘ℤ_ring)
26		zringring 21422	. . . . . . . . . . . . . . . 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 12608	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → 0 ∈ ℤ)
30	28, 29	zmulcld 12711	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (𝑏 · 0) ∈ ℤ)
31		zringmulr 21430	. . . . . . . . . . . . . . 15 ⊢ · = (._r‘ℤ_ring)
32		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ (._r‘𝑅) = (._r‘𝑅)
33	1, 25, 25, 27, 27, 8, 28, 9, 29, 10, 30, 31, 31, 32	xpsmul 17591	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (⟨𝑎, 𝑏⟩(._r‘𝑅)⟨𝑐, 0⟩) = ⟨(𝑎 · 𝑐), (𝑏 · 0)⟩)
34	33	eleq1d 2818	. . . . . . . . . . . . 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 12711	. . . . . . . . . . . . . . . 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 12608	. . . . . . . . . . . . . . . . 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 12711	. . . . . . . . . . . . . . . 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 17591	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (⟨𝑐, 0⟩(._r‘𝑅)⟨𝑎, 𝑏⟩) = ⟨(𝑐 · 𝑎), (0 · 𝑏)⟩)
44	43	eleq1d 2818	. . . . . . . . . . . . 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 7422	. . . . . . . . . . . . . 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 2827	. . . . . . . . . . 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 7422	. . . . . . . . . . . . 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 2827	. . . . . . . . . . 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 3142	. . . . . . 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 3188	. . . . 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 3186	. 2 ⊢ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ 𝐼 ((𝑥(._r‘𝑅)𝑦) ∈ 𝐼 ∧ (𝑦(._r‘𝑅)𝑥) ∈ 𝐼)
65	1	pzriprnglem1 21454	. . 3 ⊢ 𝑅 ∈ Rng
66	1, 6	pzriprnglem4 21457	. . 3 ⊢ 𝐼 ∈ (SubGrp‘𝑅)
67		eqid 2734	. . . 4 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
68		eqid 2734	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
69	67, 68, 32	df2idl2rng 21228	. . 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 1539 ∈ wcel 2107 ∀wral 3050 ∃wrex 3059 {csn 4606 ⟨cop 4612 × cxp 5663 ‘cfv 6541 (class class class)co 7413 ℂcc 11135 0cc0 11137 · cmul 11142 ℤcz 12596 Basecbs 17229 ↾_s cress 17252 ._rcmulr 17274 ×_s cxps 17522 SubGrpcsubg 19107 Rngcrng 20117 Ringcrg 20198 2Idealc2idl 21221 ℤ_ringczring 21419

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214 ax-addf 11216 ax-mulf 11217

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4888 df-iun 4973 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7870 df-1st 7996 df-2nd 7997 df-tpos 8233 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8727 df-map 8850 df-ixp 8920 df-en 8968 df-dom 8969 df-sdom 8970 df-fin 8971 df-sup 9464 df-inf 9465 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12510 df-z 12597 df-dec 12717 df-uz 12861 df-fz 13530 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17230 df-ress 17253 df-plusg 17286 df-mulr 17287 df-starv 17288 df-sca 17289 df-vsca 17290 df-ip 17291 df-tset 17292 df-ple 17293 df-ds 17295 df-unif 17296 df-hom 17297 df-cco 17298 df-0g 17457 df-prds 17463 df-imas 17524 df-xps 17526 df-mgm 18622 df-sgrp 18701 df-mnd 18717 df-grp 18923 df-minusg 18924 df-subg 19110 df-cmn 19768 df-abl 19769 df-mgp 20106 df-rng 20118 df-ur 20147 df-ring 20200 df-cring 20201 df-oppr 20302 df-subrng 20514 df-subrg 20538 df-lss 20898 df-sra 21140 df-rgmod 21141 df-lidl 21180 df-2idl 21222 df-cnfld 21327 df-zring 21420

This theorem is referenced by: pzriprnglem12 21465 pzriprnglem13 21466 pzriprngALT 21468 pzriprng1ALT 21469

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