Theorem rngqiprngimf1 21255

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 20220	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Ring → 𝐽 ∈ Rng)
6	4, 5	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ Rng)
7	3, 6	eqeltrrid 2841	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)
8	1, 2, 7	rng2idlnsg 21221	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
9		nsgsubg 19087	. . . . . . . . . 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 7369	. . . . . . . . . . 11 ⊢ (𝑅 /s ∼ ) = (𝑅 /s (𝑅 ~QG 𝐼))
14	11, 13	eqtri 2759	. . . . . . . . . 10 ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))
15		eqid 2736	. . . . . . . . . 10 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
16	14, 15	qus2idrng 21228	. . . . . . . . 9 ⊢ ((𝑅 ∈ Rng ∧ 𝐼 ∈ (2Ideal‘𝑅) ∧ 𝐼 ∈ (SubGrp‘𝑅)) → 𝑄 ∈ Rng)
17	1, 2, 10, 16	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ Rng)
18		rnggrp 20093	. . . . . . . . 9 ⊢ (𝑄 ∈ Rng → 𝑄 ∈ Grp)
19	18	grpmndd 18876	. . . . . . . 8 ⊢ (𝑄 ∈ Rng → 𝑄 ∈ Mnd)
20	17, 19	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ Mnd)
21		ringmnd 20178	. . . . . . . 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 18703	. . . . . . 7 ⊢ ((𝑄 ∈ Mnd ∧ 𝐽 ∈ Mnd) → (0g‘𝑃) = ⟨(0g‘𝑄), (0g‘𝐽)⟩)
25	20, 22, 24	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (0g‘𝑃) = ⟨(0g‘𝑄), (0g‘𝐽)⟩)
26	25	sneqd 4592	. . . . 5 ⊢ (𝜑 → {(0g‘𝑃)} = {⟨(0g‘𝑄), (0g‘𝐽)⟩})
27	26	imaeq2d 6019	. . . 4 ⊢ (𝜑 → (◡𝐹 “ {(0g‘𝑃)}) = (◡𝐹 “ {⟨(0g‘𝑄), (0g‘𝐽)⟩}))
28		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑥𝜑
29		opex 5412	. . . . . . 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 6633	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐵)
33		fncnvima2 7006	. . . . 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 21253	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐹‘𝑎) = ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩)
40	39	eleq1d 2821	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝐹‘𝑎) ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩} ↔ ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩}))
41	40	rabbidva 3405	. . . . 5 ⊢ (𝜑 → {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩}} = {𝑎 ∈ 𝐵 ∣ ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩}})
42		eceq1 8674	. . . . . . . 8 ⊢ (𝑎 = (0g‘𝑅) → [𝑎] ∼ = [(0g‘𝑅)] ∼ )
43		oveq2 7366	. . . . . . . 8 ⊢ (𝑎 = (0g‘𝑅) → ( 1 · 𝑎) = ( 1 · (0g‘𝑅)))
44	42, 43	opeq12d 4837	. . . . . . 7 ⊢ (𝑎 = (0g‘𝑅) → ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ = ⟨[(0g‘𝑅)] ∼ , ( 1 · (0g‘𝑅))⟩)
45	44	eleq1d 2821	. . . . . 6 ⊢ (𝑎 = (0g‘𝑅) → (⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩} ↔ ⟨[(0g‘𝑅)] ∼ , ( 1 · (0g‘𝑅))⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩}))
46		rnggrp 20093	. . . . . . . . 9 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
47	1, 46	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Grp)
48	47	grpmndd 18876	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Mnd)
49		eqid 2736	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅)
50	35, 49	mndidcl 18674	. . . . . . 7 ⊢ (𝑅 ∈ Mnd → (0g‘𝑅) ∈ 𝐵)
51	48, 50	syl 17	. . . . . 6 ⊢ (𝜑 → (0g‘𝑅) ∈ 𝐵)
52	12	eceq2i 8677	. . . . . . . . 9 ⊢ [(0g‘𝑅)] ∼ = [(0g‘𝑅)](𝑅 ~QG 𝐼)
53	14, 49	qus0 19118	. . . . . . . . . 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 2783	. . . . . . . 8 ⊢ (𝜑 → [(0g‘𝑅)] ∼ = (0g‘𝑄))
56	1, 2, 7	rng2idl0 21222	. . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝑅) ∈ 𝐼)
57	35, 15	2idlss 21217	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ (2Ideal‘𝑅) → 𝐼 ⊆ 𝐵)
58	2, 57	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ⊆ 𝐵)
59	3, 35, 49	ress0g 18687	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Mnd ∧ (0g‘𝑅) ∈ 𝐼 ∧ 𝐼 ⊆ 𝐵) → (0g‘𝑅) = (0g‘𝐽))
60	48, 56, 58, 59	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝑅) = (0g‘𝐽))
61	60	oveq2d 7374	. . . . . . . . 9 ⊢ (𝜑 → ( 1 · (0g‘𝑅)) = ( 1 · (0g‘𝐽)))
62	3, 36	ressmulr 17227	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (2Ideal‘𝑅) → · = (.r‘𝐽))
63	2, 62	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → · = (.r‘𝐽))
64	63	oveqd 7375	. . . . . . . . 9 ⊢ (𝜑 → ( 1 · (0g‘𝐽)) = ( 1 (.r‘𝐽)(0g‘𝐽)))
65		eqid 2736	. . . . . . . . . . 11 ⊢ (Base‘𝐽) = (Base‘𝐽)
66	65, 37	ringidcl 20200	. . . . . . . . . 10 ⊢ (𝐽 ∈ Ring → 1 ∈ (Base‘𝐽))
67		eqid 2736	. . . . . . . . . . 11 ⊢ (.r‘𝐽) = (.r‘𝐽)
68		eqid 2736	. . . . . . . . . . 11 ⊢ (0g‘𝐽) = (0g‘𝐽)
69	65, 67, 68	ringrz 20229	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Ring ∧ 1 ∈ (Base‘𝐽)) → ( 1 (.r‘𝐽)(0g‘𝐽)) = (0g‘𝐽))
70	4, 66, 69	syl2anc2 585	. . . . . . . . 9 ⊢ (𝜑 → ( 1 (.r‘𝐽)(0g‘𝐽)) = (0g‘𝐽))
71	61, 64, 70	3eqtrd 2775	. . . . . . . 8 ⊢ (𝜑 → ( 1 · (0g‘𝑅)) = (0g‘𝐽))
72	55, 71	opeq12d 4837	. . . . . . 7 ⊢ (𝜑 → ⟨[(0g‘𝑅)] ∼ , ( 1 · (0g‘𝑅))⟩ = ⟨(0g‘𝑄), (0g‘𝐽)⟩)
73		opex 5412	. . . . . . . 8 ⊢ ⟨[(0g‘𝑅)] ∼ , ( 1 · (0g‘𝑅))⟩ ∈ V
74	73	elsn 4595	. . . . . . 7 ⊢ (⟨[(0g‘𝑅)] ∼ , ( 1 · (0g‘𝑅))⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩} ↔ ⟨[(0g‘𝑅)] ∼ , ( 1 · (0g‘𝑅))⟩ = ⟨(0g‘𝑄), (0g‘𝐽)⟩)
75	72, 74	sylibr 234	. . . . . 6 ⊢ (𝜑 → ⟨[(0g‘𝑅)] ∼ , ( 1 · (0g‘𝑅))⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩})
76		opex 5412	. . . . . . . . . 10 ⊢ ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ V
77	76	elsn 4595	. . . . . . . . 9 ⊢ (⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩} ↔ ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ = ⟨(0g‘𝑄), (0g‘𝐽)⟩)
78	12	ovexi 7392	. . . . . . . . . . 11 ⊢ ∼ ∈ V
79		ecexg 8639	. . . . . . . . . . 11 ⊢ ( ∼ ∈ V → [𝑎] ∼ ∈ V)
80	78, 79	ax-mp 5	. . . . . . . . . 10 ⊢ [𝑎] ∼ ∈ V
81		ovex 7391	. . . . . . . . . 10 ⊢ ( 1 · 𝑎) ∈ V
82	80, 81	opth 5424	. . . . . . . . 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 21249	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (([𝑎] ∼ = (0g‘𝑄) ∧ ( 1 · 𝑎) = (0g‘𝐽)) → 𝑎 = (0g‘𝑅)))
85	83, 84	biimtrid 242	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩} → 𝑎 = (0g‘𝑅)))
86	85	imp 406	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩}) → 𝑎 = (0g‘𝑅))
87	45, 51, 75, 86	rabeqsnd 4626	. . . . 5 ⊢ (𝜑 → {𝑎 ∈ 𝐵 ∣ ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩}} = {(0g‘𝑅)})
88	41, 87	eqtrd 2771	. . . 4 ⊢ (𝜑 → {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩}} = {(0g‘𝑅)})
89	27, 34, 88	3eqtrd 2775	. . 3 ⊢ (𝜑 → (◡𝐹 “ {(0g‘𝑃)}) = {(0g‘𝑅)})
90	1, 2, 3, 4, 35, 36, 37, 12, 11, 38, 23, 31	rngqiprngghm 21254	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑃))
91		eqid 2736	. . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃)
92		eqid 2736	. . . . 5 ⊢ (0g‘𝑃) = (0g‘𝑃)
93	35, 91, 49, 92	kerf1ghm 19176	. . . 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 2737	. . 3 ⊢ (𝜑 → 𝐹 = 𝐹)
97		eqidd 2737	. . 3 ⊢ (𝜑 → 𝐵 = 𝐵)
98	1, 2, 3, 4, 35, 36, 37, 12, 11, 38, 23	rngqipbas 21250	. . 3 ⊢ (𝜑 → (Base‘𝑃) = (𝐶 × 𝐼))
99	96, 97, 98	f1eq123d 6766	. 2 ⊢ (𝜑 → (𝐹:𝐵–1-1→(Base‘𝑃) ↔ 𝐹:𝐵–1-1→(𝐶 × 𝐼)))
100	95, 99	mpbid 232	1 ⊢ (𝜑 → 𝐹:𝐵–1-1→(𝐶 × 𝐼))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {crab 3399  Vcvv 3440   ⊆ wss 3901  {csn 4580  ⟨cop 4586   ↦ cmpt 5179   × cxp 5622  ^◡ccnv 5623   “ cima 5627   Fn wfn 6487  –1-1→wf1 6489  ‘cfv 6492  (class class class)co 7358  [cec 8633  Basecbs 17136   ↾_s cress 17157  ._rcmulr 17178  0_gc0g 17359   /_s cqus 17426   ×_s cxps 17427  Mndcmnd 18659  Grpcgrp 18863  SubGrpcsubg 19050  NrmSGrpcnsg 19051   ~_QG cqg 19052   GrpHom cghm 19141  Rngcrng 20087  1_rcur 20116  Ringcrg 20168  2Idealc2idl 21204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-tpos 8168  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-ec 8637  df-qs 8641  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-inf 9346  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-fz 13424  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-mulr 17191  df-sca 17193  df-vsca 17194  df-ip 17195  df-tset 17196  df-ple 17197  df-ds 17199  df-hom 17201  df-cco 17202  df-0g 17361  df-prds 17367  df-imas 17429  df-qus 17430  df-xps 17431  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18866  df-minusg 18867  df-sbg 18868  df-subg 19053  df-nsg 19054  df-eqg 19055  df-ghm 19142  df-cmn 19711  df-abl 19712  df-mgp 20076  df-rng 20088  df-ur 20117  df-ring 20170  df-oppr 20273  df-dvdsr 20293  df-unit 20294  df-invr 20324  df-subrng 20479  df-lss 20883  df-sra 21125  df-rgmod 21126  df-lidl 21163  df-2idl 21205

This theorem is referenced by:  rngqiprngim  21259