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Theorem pzriprnglem8 21460

Description: Lemma 8 for pzriprng 21469: 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 21454	. . . . . 6 ⊢ (Base‘𝑅) = (ℤ × ℤ)
3	2	eleq2i 2829	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝑅) ↔ 𝑥 ∈ (ℤ × ℤ))
4		elxp2 5658	. . . . 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 21455	. . . 4 ⊢ (𝑦 ∈ 𝐼 ↔ ∃𝑐 ∈ ℤ 𝑦 = ⟨𝑐, 0⟩)
8		simpll 767	. . . . . . . . . . . . . 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 12616	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (𝑎 · 𝑐) ∈ ℤ)
11		zcn 12507	. . . . . . . . . . . . . . . . 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 11346	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (𝑏 · 0) = 0)
15		ovex 7403	. . . . . . . . . . . . . . 15 ⊢ (𝑏 · 0) ∈ V
16	15	elsn 4597	. . . . . . . . . . . . . 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 5673	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → ⟨(𝑎 · 𝑐), (𝑏 · 0)⟩ ∈ (ℤ × {0}))
19	9, 8	zmulcld 12616	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (𝑐 · 𝑎) ∈ ℤ)
20	13	mul02d 11345	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (0 · 𝑏) = 0)
21		ovex 7403	. . . . . . . . . . . . . . 15 ⊢ (0 · 𝑏) ∈ V
22	21	elsn 4597	. . . . . . . . . . . . . 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 5673	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → ⟨(𝑐 · 𝑎), (0 · 𝑏)⟩ ∈ (ℤ × {0}))
25		zringbas 21425	. . . . . . . . . . . . . . 15 ⊢ ℤ = (Base‘ℤ_ring)
26		zringring 21421	. . . . . . . . . . . . . . . 16 ⊢ ℤ_ring ∈ Ring
27	26	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → ℤ_ring ∈ Ring)
28		simplr 769	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → 𝑏 ∈ ℤ)
29		0zd 12514	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → 0 ∈ ℤ)
30	28, 29	zmulcld 12616	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (𝑏 · 0) ∈ ℤ)
31		zringmulr 21429	. . . . . . . . . . . . . . 15 ⊢ · = (._r‘ℤ_ring)
32		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (._r‘𝑅) = (._r‘𝑅)
33	1, 25, 25, 27, 27, 8, 28, 9, 29, 10, 30, 31, 31, 32	xpsmul 17510	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (⟨𝑎, 𝑏⟩(._r‘𝑅)⟨𝑐, 0⟩) = ⟨(𝑎 · 𝑐), (𝑏 · 0)⟩)
34	33	eleq1d 2822	. . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ ℤ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → 𝑎 ∈ ℤ)
37	35, 36	zmulcld 12616	. . . . . . . . . . . . . . . 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 12514	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ ℤ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → 0 ∈ ℤ)
40		simprr 773	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ ℤ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → 𝑏 ∈ ℤ)
41	39, 40	zmulcld 12616	. . . . . . . . . . . . . . . 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 17510	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (⟨𝑐, 0⟩(._r‘𝑅)⟨𝑎, 𝑏⟩) = ⟨(𝑐 · 𝑎), (0 · 𝑏)⟩)
44	43	eleq1d 2822	. . . . . . . . . . . . 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 633	. . . . . . . . . . . 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 714	. . . . . . . . . . 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 7379	. . . . . . . . . . . . . 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 2831	. . . . . . . . . . 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 7379	. . . . . . . . . . . . 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 2831	. . . . . . . . . . 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 633	. . . . . . . . . 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 3139	. . . . . . 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 599	. . 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 21453	. . 3 ⊢ 𝑅 ∈ Rng
66	1, 6	pzriprnglem4 21456	. . 3 ⊢ 𝐼 ∈ (SubGrp‘𝑅)
67		eqid 2737	. . . 4 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
68		eqid 2737	. . . 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 693	. 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 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 {csn 4582 ⟨cop 4588 × cxp 5632 ‘cfv 6502 (class class class)co 7370 ℂcc 11038 0cc0 11040 · cmul 11045 ℤcz 12502 Basecbs 17150 ↾_s cress 17171 ._rcmulr 17192 ×_s cxps 17441 SubGrpcsubg 19067 Rngcrng 20104 Ringcrg 20185 2Idealc2idl 21221 ℤ_ringczring 21418

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117 ax-addf 11119 ax-mulf 11120

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-om 7821 df-1st 7945 df-2nd 7946 df-tpos 8180 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-1o 8409 df-2o 8410 df-er 8647 df-map 8779 df-ixp 8850 df-en 8898 df-dom 8899 df-sdom 8900 df-fin 8901 df-sup 9359 df-inf 9360 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-nn 12160 df-2 12222 df-3 12223 df-4 12224 df-5 12225 df-6 12226 df-7 12227 df-8 12228 df-9 12229 df-n0 12416 df-z 12503 df-dec 12622 df-uz 12766 df-fz 13438 df-struct 17088 df-sets 17105 df-slot 17123 df-ndx 17135 df-base 17151 df-ress 17172 df-plusg 17204 df-mulr 17205 df-starv 17206 df-sca 17207 df-vsca 17208 df-ip 17209 df-tset 17210 df-ple 17211 df-ds 17213 df-unif 17214 df-hom 17215 df-cco 17216 df-0g 17375 df-prds 17381 df-imas 17443 df-xps 17445 df-mgm 18579 df-sgrp 18658 df-mnd 18674 df-grp 18883 df-minusg 18884 df-subg 19070 df-cmn 19728 df-abl 19729 df-mgp 20093 df-rng 20105 df-ur 20134 df-ring 20187 df-cring 20188 df-oppr 20290 df-subrng 20496 df-subrg 20520 df-lss 20900 df-sra 21142 df-rgmod 21143 df-lidl 21180 df-2idl 21222 df-cnfld 21327 df-zring 21419

This theorem is referenced by: pzriprnglem12 21464 pzriprnglem13 21465 pzriprngALT 21467 pzriprng1ALT 21468

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