Theorem rngqiprngimf1 46766

Description: 𝐹 is a one-to-one function from (the base set of) a non-unital ring to the product of the (base set of the) quotient with a two-sided ideal and the (base set of the) two-sided ideal. (Contributed by AV, 7-Mar-2025.)

Hypotheses

Ref	Expression
rng2idlring.r	⊢ (𝜑 → 𝑅 ∈ Rng)
rng2idlring.i	⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))
rng2idlring.j	⊢ 𝐽 = (𝑅 ↾s 𝐼)
rng2idlring.u	⊢ (𝜑 → 𝐽 ∈ Ring)
rng2idlring.b	⊢ 𝐵 = (Base‘𝑅)
rng2idlring.t	⊢ · = (.r‘𝑅)
rng2idlring.1	⊢ 1 = (1r‘𝐽)
rngqiprngim.g	⊢ ∼ = (𝑅 ~QG 𝐼)
rngqiprngim.q	⊢ 𝑄 = (𝑅 /s ∼ )
rngqiprngim.c	⊢ 𝐶 = (Base‘𝑄)
rngqiprngim.p	⊢ 𝑃 = (𝑄 ×s 𝐽)
rngqiprngim.f	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)

Assertion

Ref	Expression
rngqiprngimf1	⊢ (𝜑 → 𝐹:𝐵–1-1→(𝐶 × 𝐼))

Distinct variable groups:   𝑥,𝐶   𝑥,𝐼   𝑥,𝐵   𝜑,𝑥   𝑥, ∼   𝑥, 1   𝑥, ·   𝑥,𝑅

Allowed substitution hints:   𝑃(𝑥)   𝑄(𝑥)   𝐹(𝑥)   𝐽(𝑥)

Proof of Theorem rngqiprngimf1

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rng2idlring.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Rng)
2		rng2idlring.i	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))
3		rng2idlring.j	. . . . . . . . . . . 12 ⊢ 𝐽 = (𝑅 ↾s 𝐼)
4		rng2idlring.u	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ Ring)
5		ringrng 46642	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Ring → 𝐽 ∈ Rng)
6	4, 5	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ Rng)
7	3, 6	eqeltrrid 2839	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)
8	1, 2, 7	rng2idlnsg 46743	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
9		nsgsubg 19033	. . . . . . . . . 10 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → 𝐼 ∈ (SubGrp‘𝑅))
10	8, 9	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (SubGrp‘𝑅))
11		rngqiprngim.q	. . . . . . . . . . 11 ⊢ 𝑄 = (𝑅 /s ∼ )
12		rngqiprngim.g	. . . . . . . . . . . 12 ⊢ ∼ = (𝑅 ~QG 𝐼)
13	12	oveq2i 7417	. . . . . . . . . . 11 ⊢ (𝑅 /s ∼ ) = (𝑅 /s (𝑅 ~QG 𝐼))
14	11, 13	eqtri 2761	. . . . . . . . . 10 ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))
15		eqid 2733	. . . . . . . . . 10 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
16	14, 15	qus2idrng 46749	. . . . . . . . 9 ⊢ ((𝑅 ∈ Rng ∧ 𝐼 ∈ (2Ideal‘𝑅) ∧ 𝐼 ∈ (SubGrp‘𝑅)) → 𝑄 ∈ Rng)
17	1, 2, 10, 16	syl3anc 1372	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ Rng)
18		rnggrp 46641	. . . . . . . . 9 ⊢ (𝑄 ∈ Rng → 𝑄 ∈ Grp)
19	18	grpmndd 18829	. . . . . . . 8 ⊢ (𝑄 ∈ Rng → 𝑄 ∈ Mnd)
20	17, 19	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ Mnd)
21		ringmnd 20060	. . . . . . . 8 ⊢ (𝐽 ∈ Ring → 𝐽 ∈ Mnd)
22	4, 21	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Mnd)
23		rngqiprngim.p	. . . . . . . 8 ⊢ 𝑃 = (𝑄 ×s 𝐽)
24	23	xpsmnd0 18663	. . . . . . 7 ⊢ ((𝑄 ∈ Mnd ∧ 𝐽 ∈ Mnd) → (0g‘𝑃) = ⟨(0g‘𝑄), (0g‘𝐽)⟩)
25	20, 22, 24	syl2anc 585	. . . . . 6 ⊢ (𝜑 → (0g‘𝑃) = ⟨(0g‘𝑄), (0g‘𝐽)⟩)
26	25	sneqd 4640	. . . . 5 ⊢ (𝜑 → {(0g‘𝑃)} = {⟨(0g‘𝑄), (0g‘𝐽)⟩})
27	26	imaeq2d 6058	. . . 4 ⊢ (𝜑 → (◡𝐹 “ {(0g‘𝑃)}) = (◡𝐹 “ {⟨(0g‘𝑄), (0g‘𝐽)⟩}))
28		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑥𝜑
29		opex 5464	. . . . . . 7 ⊢ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩ ∈ V
30	29	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩ ∈ V)
31		rngqiprngim.f	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)
32	28, 30, 31	fnmptd 6689	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐵)
33		fncnvima2 7060	. . . . 5 ⊢ (𝐹 Fn 𝐵 → (◡𝐹 “ {⟨(0g‘𝑄), (0g‘𝐽)⟩}) = {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩}})
34	32, 33	syl 17	. . . 4 ⊢ (𝜑 → (◡𝐹 “ {⟨(0g‘𝑄), (0g‘𝐽)⟩}) = {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩}})
35		rng2idlring.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅)
36		rng2idlring.t	. . . . . . . 8 ⊢ · = (.r‘𝑅)
37		rng2idlring.1	. . . . . . . 8 ⊢ 1 = (1r‘𝐽)
38		rngqiprngim.c	. . . . . . . 8 ⊢ 𝐶 = (Base‘𝑄)
39	1, 2, 3, 4, 35, 36, 37, 12, 11, 38, 23, 31	rngqiprngimfv 46764	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐹‘𝑎) = ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩)
40	39	eleq1d 2819	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝐹‘𝑎) ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩} ↔ ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩}))
41	40	rabbidva 3440	. . . . 5 ⊢ (𝜑 → {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩}} = {𝑎 ∈ 𝐵 ∣ ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩}})
42		eceq1 8738	. . . . . . . 8 ⊢ (𝑎 = (0g‘𝑅) → [𝑎] ∼ = [(0g‘𝑅)] ∼ )
43		oveq2 7414	. . . . . . . 8 ⊢ (𝑎 = (0g‘𝑅) → ( 1 · 𝑎) = ( 1 · (0g‘𝑅)))
44	42, 43	opeq12d 4881	. . . . . . 7 ⊢ (𝑎 = (0g‘𝑅) → ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ = ⟨[(0g‘𝑅)] ∼ , ( 1 · (0g‘𝑅))⟩)
45	44	eleq1d 2819	. . . . . 6 ⊢ (𝑎 = (0g‘𝑅) → (⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩} ↔ ⟨[(0g‘𝑅)] ∼ , ( 1 · (0g‘𝑅))⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩}))
46		rnggrp 46641	. . . . . . . . 9 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
47	1, 46	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Grp)
48	47	grpmndd 18829	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Mnd)
49		eqid 2733	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅)
50	35, 49	mndidcl 18637	. . . . . . 7 ⊢ (𝑅 ∈ Mnd → (0g‘𝑅) ∈ 𝐵)
51	48, 50	syl 17	. . . . . 6 ⊢ (𝜑 → (0g‘𝑅) ∈ 𝐵)
52	12	eceq2i 8741	. . . . . . . . 9 ⊢ [(0g‘𝑅)] ∼ = [(0g‘𝑅)](𝑅 ~QG 𝐼)
53	14, 49	qus0 19063	. . . . . . . . . 10 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → [(0g‘𝑅)](𝑅 ~QG 𝐼) = (0g‘𝑄))
54	8, 53	syl 17	. . . . . . . . 9 ⊢ (𝜑 → [(0g‘𝑅)](𝑅 ~QG 𝐼) = (0g‘𝑄))
55	52, 54	eqtrid 2785	. . . . . . . 8 ⊢ (𝜑 → [(0g‘𝑅)] ∼ = (0g‘𝑄))
56	1, 2, 7	rng2idl0 46744	. . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝑅) ∈ 𝐼)
57	35, 15	2idlss 20861	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ (2Ideal‘𝑅) → 𝐼 ⊆ 𝐵)
58	2, 57	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ⊆ 𝐵)
59	3, 35, 49	ress0g 18650	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Mnd ∧ (0g‘𝑅) ∈ 𝐼 ∧ 𝐼 ⊆ 𝐵) → (0g‘𝑅) = (0g‘𝐽))
60	48, 56, 58, 59	syl3anc 1372	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝑅) = (0g‘𝐽))
61	60	oveq2d 7422	. . . . . . . . 9 ⊢ (𝜑 → ( 1 · (0g‘𝑅)) = ( 1 · (0g‘𝐽)))
62	3, 36	ressmulr 17249	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (2Ideal‘𝑅) → · = (.r‘𝐽))
63	2, 62	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → · = (.r‘𝐽))
64	63	oveqd 7423	. . . . . . . . 9 ⊢ (𝜑 → ( 1 · (0g‘𝐽)) = ( 1 (.r‘𝐽)(0g‘𝐽)))
65		eqid 2733	. . . . . . . . . . 11 ⊢ (Base‘𝐽) = (Base‘𝐽)
66	65, 37	ringidcl 20077	. . . . . . . . . 10 ⊢ (𝐽 ∈ Ring → 1 ∈ (Base‘𝐽))
67		eqid 2733	. . . . . . . . . . 11 ⊢ (.r‘𝐽) = (.r‘𝐽)
68		eqid 2733	. . . . . . . . . . 11 ⊢ (0g‘𝐽) = (0g‘𝐽)
69	65, 67, 68	ringrz 20102	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Ring ∧ 1 ∈ (Base‘𝐽)) → ( 1 (.r‘𝐽)(0g‘𝐽)) = (0g‘𝐽))
70	4, 66, 69	syl2anc2 586	. . . . . . . . 9 ⊢ (𝜑 → ( 1 (.r‘𝐽)(0g‘𝐽)) = (0g‘𝐽))
71	61, 64, 70	3eqtrd 2777	. . . . . . . 8 ⊢ (𝜑 → ( 1 · (0g‘𝑅)) = (0g‘𝐽))
72	55, 71	opeq12d 4881	. . . . . . 7 ⊢ (𝜑 → ⟨[(0g‘𝑅)] ∼ , ( 1 · (0g‘𝑅))⟩ = ⟨(0g‘𝑄), (0g‘𝐽)⟩)
73		opex 5464	. . . . . . . 8 ⊢ ⟨[(0g‘𝑅)] ∼ , ( 1 · (0g‘𝑅))⟩ ∈ V
74	73	elsn 4643	. . . . . . 7 ⊢ (⟨[(0g‘𝑅)] ∼ , ( 1 · (0g‘𝑅))⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩} ↔ ⟨[(0g‘𝑅)] ∼ , ( 1 · (0g‘𝑅))⟩ = ⟨(0g‘𝑄), (0g‘𝐽)⟩)
75	72, 74	sylibr 233	. . . . . 6 ⊢ (𝜑 → ⟨[(0g‘𝑅)] ∼ , ( 1 · (0g‘𝑅))⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩})
76		opex 5464	. . . . . . . . . 10 ⊢ ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ V
77	76	elsn 4643	. . . . . . . . 9 ⊢ (⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩} ↔ ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ = ⟨(0g‘𝑄), (0g‘𝐽)⟩)
78	12	ovexi 7440	. . . . . . . . . . 11 ⊢ ∼ ∈ V
79		ecexg 8704	. . . . . . . . . . 11 ⊢ ( ∼ ∈ V → [𝑎] ∼ ∈ V)
80	78, 79	ax-mp 5	. . . . . . . . . 10 ⊢ [𝑎] ∼ ∈ V
81		ovex 7439	. . . . . . . . . 10 ⊢ ( 1 · 𝑎) ∈ V
82	80, 81	opth 5476	. . . . . . . . 9 ⊢ (⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ = ⟨(0g‘𝑄), (0g‘𝐽)⟩ ↔ ([𝑎] ∼ = (0g‘𝑄) ∧ ( 1 · 𝑎) = (0g‘𝐽)))
83	77, 82	bitri 275	. . . . . . . 8 ⊢ (⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩} ↔ ([𝑎] ∼ = (0g‘𝑄) ∧ ( 1 · 𝑎) = (0g‘𝐽)))
84	1, 2, 3, 4, 35, 36, 37, 12, 11	rngqiprngimf1lem 46760	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (([𝑎] ∼ = (0g‘𝑄) ∧ ( 1 · 𝑎) = (0g‘𝐽)) → 𝑎 = (0g‘𝑅)))
85	83, 84	biimtrid 241	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩} → 𝑎 = (0g‘𝑅)))
86	85	imp 408	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩}) → 𝑎 = (0g‘𝑅))
87	45, 51, 75, 86	rabeqsnd 4671	. . . . 5 ⊢ (𝜑 → {𝑎 ∈ 𝐵 ∣ ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩}} = {(0g‘𝑅)})
88	41, 87	eqtrd 2773	. . . 4 ⊢ (𝜑 → {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩}} = {(0g‘𝑅)})
89	27, 34, 88	3eqtrd 2777	. . 3 ⊢ (𝜑 → (◡𝐹 “ {(0g‘𝑃)}) = {(0g‘𝑅)})
90	1, 2, 3, 4, 35, 36, 37, 12, 11, 38, 23, 31	rngqiprngghm 46765	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑃))
91		eqid 2733	. . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃)
92		eqid 2733	. . . . 5 ⊢ (0g‘𝑃) = (0g‘𝑃)
93	35, 91, 49, 92	kerf1ghm 20275	. . . 4 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑃) → (𝐹:𝐵–1-1→(Base‘𝑃) ↔ (◡𝐹 “ {(0g‘𝑃)}) = {(0g‘𝑅)}))
94	90, 93	syl 17	. . 3 ⊢ (𝜑 → (𝐹:𝐵–1-1→(Base‘𝑃) ↔ (◡𝐹 “ {(0g‘𝑃)}) = {(0g‘𝑅)}))
95	89, 94	mpbird 257	. 2 ⊢ (𝜑 → 𝐹:𝐵–1-1→(Base‘𝑃))
96		eqidd 2734	. . 3 ⊢ (𝜑 → 𝐹 = 𝐹)
97		eqidd 2734	. . 3 ⊢ (𝜑 → 𝐵 = 𝐵)
98	1, 2, 3, 4, 35, 36, 37, 12, 11, 38, 23	rngqipbas 46761	. . 3 ⊢ (𝜑 → (Base‘𝑃) = (𝐶 × 𝐼))
99	96, 97, 98	f1eq123d 6823	. 2 ⊢ (𝜑 → (𝐹:𝐵–1-1→(Base‘𝑃) ↔ 𝐹:𝐵–1-1→(𝐶 × 𝐼)))
100	95, 99	mpbid 231	1 ⊢ (𝜑 → 𝐹:𝐵–1-1→(𝐶 × 𝐼))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {crab 3433  Vcvv 3475   ⊆ wss 3948  {csn 4628  ⟨cop 4634   ↦ cmpt 5231   × cxp 5674  ^◡ccnv 5675   “ cima 5679   Fn wfn 6536  –1-1→wf1 6538  ‘cfv 6541  (class class class)co 7406  [cec 8698  Basecbs 17141   ↾_s cress 17170  ._rcmulr 17195  0_gc0g 17382   /_s cqus 17448   ×_s cxps 17449  Mndcmnd 18622  Grpcgrp 18816  SubGrpcsubg 18995  NrmSGrpcnsg 18996   ~_QG cqg 18997   GrpHom cghm 19084  1_rcur 19999  Ringcrg 20050  2Idealc2idl 20849  Rngcrng 46635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-tpos 8208  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-2o 8464  df-er 8700  df-ec 8702  df-qs 8706  df-map 8819  df-ixp 8889  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-sup 9434  df-inf 9435  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-z 12556  df-dec 12675  df-uz 12820  df-fz 13482  df-struct 17077  df-sets 17094  df-slot 17112  df-ndx 17124  df-base 17142  df-ress 17171  df-plusg 17207  df-mulr 17208  df-sca 17210  df-vsca 17211  df-ip 17212  df-tset 17213  df-ple 17214  df-ds 17216  df-hom 17218  df-cco 17219  df-0g 17384  df-prds 17390  df-imas 17451  df-qus 17452  df-xps 17453  df-mgm 18558  df-sgrp 18607  df-mnd 18623  df-grp 18819  df-minusg 18820  df-sbg 18821  df-subg 18998  df-nsg 18999  df-eqg 19000  df-ghm 19085  df-cmn 19645  df-abl 19646  df-mgp 19983  df-ur 20000  df-ring 20052  df-oppr 20143  df-dvdsr 20164  df-unit 20165  df-invr 20195  df-lss 20536  df-sra 20778  df-rgmod 20779  df-lidl 20780  df-2idl 20850  df-rng 46636  df-subrng 46710

This theorem is referenced by:  rngqiprngim  46770