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Theorem pzriprnglem8 21516

Description: Lemma 8 for pzriprng 21525: 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 21510	. . . . . 6 ⊢ (Base‘𝑅) = (ℤ × ℤ)
3	2	eleq2i 2830	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝑅) ↔ 𝑥 ∈ (ℤ × ℤ))
4		elxp2 5712	. . . . 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 21511	. . . 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 12725	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (𝑎 · 𝑐) ∈ ℤ)
11		zcn 12615	. . . . . . . . . . . . . . . . 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 11457	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (𝑏 · 0) = 0)
15		ovex 7463	. . . . . . . . . . . . . . 15 ⊢ (𝑏 · 0) ∈ V
16	15	elsn 4645	. . . . . . . . . . . . . 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 5727	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → ⟨(𝑎 · 𝑐), (𝑏 · 0)⟩ ∈ (ℤ × {0}))
19	9, 8	zmulcld 12725	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (𝑐 · 𝑎) ∈ ℤ)
20	13	mul02d 11456	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (0 · 𝑏) = 0)
21		ovex 7463	. . . . . . . . . . . . . . 15 ⊢ (0 · 𝑏) ∈ V
22	21	elsn 4645	. . . . . . . . . . . . . 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 5727	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → ⟨(𝑐 · 𝑎), (0 · 𝑏)⟩ ∈ (ℤ × {0}))
25		zringbas 21481	. . . . . . . . . . . . . . 15 ⊢ ℤ = (Base‘ℤ_ring)
26		zringring 21477	. . . . . . . . . . . . . . . 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 12622	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → 0 ∈ ℤ)
30	28, 29	zmulcld 12725	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (𝑏 · 0) ∈ ℤ)
31		zringmulr 21485	. . . . . . . . . . . . . . 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 17621	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (⟨𝑎, 𝑏⟩(._r‘𝑅)⟨𝑐, 0⟩) = ⟨(𝑎 · 𝑐), (𝑏 · 0)⟩)
34	33	eleq1d 2823	. . . . . . . . . . . . 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 12725	. . . . . . . . . . . . . . . 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 12622	. . . . . . . . . . . . . . . . 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 12725	. . . . . . . . . . . . . . . 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 17621	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ 𝑐 ∈ ℤ) → (⟨𝑐, 0⟩(._r‘𝑅)⟨𝑎, 𝑏⟩) = ⟨(𝑐 · 𝑎), (0 · 𝑏)⟩)
44	43	eleq1d 2823	. . . . . . . . . . . . 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 7439	. . . . . . . . . . . . . 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 2832	. . . . . . . . . . 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 7439	. . . . . . . . . . . . 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 2832	. . . . . . . . . . 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 3152	. . . . . . 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 3198	. . . . 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 3196	. 2 ⊢ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ 𝐼 ((𝑥(._r‘𝑅)𝑦) ∈ 𝐼 ∧ (𝑦(._r‘𝑅)𝑥) ∈ 𝐼)
65	1	pzriprnglem1 21509	. . 3 ⊢ 𝑅 ∈ Rng
66	1, 6	pzriprnglem4 21512	. . 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 21283	. . 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 1536 ∈ wcel 2105 ∀wral 3058 ∃wrex 3067 {csn 4630 ⟨cop 4636 × cxp 5686 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 0cc0 11152 · cmul 11157 ℤcz 12610 Basecbs 17244 ↾_s cress 17273 ._rcmulr 17298 ×_s cxps 17552 SubGrpcsubg 19150 Rngcrng 20169 Ringcrg 20250 2Idealc2idl 21276 ℤ_ringczring 21474

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-addf 11231 ax-mulf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17487 df-prds 17493 df-imas 17554 df-xps 17556 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-subg 19153 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-oppr 20350 df-subrng 20562 df-subrg 20586 df-lss 20947 df-sra 21189 df-rgmod 21190 df-lidl 21235 df-2idl 21277 df-cnfld 21382 df-zring 21475

This theorem is referenced by: pzriprnglem12 21520 pzriprnglem13 21521 pzriprngALT 21523 pzriprng1ALT 21524

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