Theorem rngqiprngimf1 21490

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 20413	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Ring → 𝐽 ∈ Rng)
6	4, 5	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ Rng)
7	3, 6	eqeltrrid 2870	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)
8	1, 2, 7	rng2idlnsg 21455	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
9		nsgsubg 19268	. . . . . . . . . 10 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → 𝐼 ∈ (SubGrp‘𝑅))
10	8, 9	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (SubGrp‘𝑅))
11		rngqiprngim.q	. . . . . . . . . . 11 ⊢ 𝑄 = (𝑅 /s ∼ )
12		rngqiprngim.g	. . . . . . . . . . . 12 ⊢ ∼ = (𝑅 ~QG 𝐼)
13	12	oveq2i 7430	. . . . . . . . . . 11 ⊢ (𝑅 /s ∼ ) = (𝑅 /s (𝑅 ~QG 𝐼))
14	11, 13	eqtri 2788	. . . . . . . . . 10 ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))
15		eqid 2765	. . . . . . . . . 10 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
16	14, 15	qus2idrng 21462	. . . . . . . . 9 ⊢ ((𝑅 ∈ Rng ∧ 𝐼 ∈ (2Ideal‘𝑅) ∧ 𝐼 ∈ (SubGrp‘𝑅)) → 𝑄 ∈ Rng)
17	1, 2, 10, 16	syl3anc 1398	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ Rng)
18		rnggrp 20280	. . . . . . . . 9 ⊢ (𝑄 ∈ Rng → 𝑄 ∈ Grp)
19	18	grpmndd 19057	. . . . . . . 8 ⊢ (𝑄 ∈ Rng → 𝑄 ∈ Mnd)
20	17, 19	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ Mnd)
21		ringmnd 20369	. . . . . . . 8 ⊢ (𝐽 ∈ Ring → 𝐽 ∈ Mnd)
22	4, 21	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Mnd)
23		rngqiprngim.p	. . . . . . . 8 ⊢ 𝑃 = (𝑄 ×s 𝐽)
24	23	xpsmnd0 18873	. . . . . . 7 ⊢ ((𝑄 ∈ Mnd ∧ 𝐽 ∈ Mnd) → (0g‘𝑃) = ⟨(0g‘𝑄), (0g‘𝐽)⟩)
25	20, 22, 24	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (0g‘𝑃) = ⟨(0g‘𝑄), (0g‘𝐽)⟩)
26	25	sneqd 4603	. . . . 5 ⊢ (𝜑 → {(0g‘𝑃)} = {⟨(0g‘𝑄), (0g‘𝐽)⟩})
27	26	imaeq2d 6064	. . . 4 ⊢ (𝜑 → (◡𝐹 “ {(0g‘𝑃)}) = (◡𝐹 “ {⟨(0g‘𝑄), (0g‘𝐽)⟩}))
28		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑥𝜑
29		opex 5447	. . . . . . 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 6680	. . . . 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 18	. . . 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 21488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐹‘𝑎) = ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩)
40	39	eleq1d 2850	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝐹‘𝑎) ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩} ↔ ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩}))
41	40	rabbidva 3424	. . . . 5 ⊢ (𝜑 → {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩}} = {𝑎 ∈ 𝐵 ∣ ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩}})
42		eceq1 8740	. . . . . . . 8 ⊢ (𝑎 = (0g‘𝑅) → [𝑎] ∼ = [(0g‘𝑅)] ∼ )
43		oveq2 7427	. . . . . . . 8 ⊢ (𝑎 = (0g‘𝑅) → ( 1 · 𝑎) = ( 1 · (0g‘𝑅)))
44	42, 43	opeq12d 4848	. . . . . . 7 ⊢ (𝑎 = (0g‘𝑅) → ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ = ⟨[(0g‘𝑅)] ∼ , ( 1 · (0g‘𝑅))⟩)
45	44	eleq1d 2850	. . . . . 6 ⊢ (𝑎 = (0g‘𝑅) → (⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩} ↔ ⟨[(0g‘𝑅)] ∼ , ( 1 · (0g‘𝑅))⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩}))
46		rnggrp 20280	. . . . . . . . 9 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
47	1, 46	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Grp)
48	47	grpmndd 19057	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Mnd)
49		eqid 2765	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅)
50	35, 49	mndidcl 18840	. . . . . . 7 ⊢ (𝑅 ∈ Mnd → (0g‘𝑅) ∈ 𝐵)
51	48, 50	syl 18	. . . . . 6 ⊢ (𝜑 → (0g‘𝑅) ∈ 𝐵)
52	12	eceq2i 8743	. . . . . . . . 9 ⊢ [(0g‘𝑅)] ∼ = [(0g‘𝑅)](𝑅 ~QG 𝐼)
53	14, 49	qus0 19304	. . . . . . . . . 10 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → [(0g‘𝑅)](𝑅 ~QG 𝐼) = (0g‘𝑄))
54	8, 53	syl 18	. . . . . . . . 9 ⊢ (𝜑 → [(0g‘𝑅)](𝑅 ~QG 𝐼) = (0g‘𝑄))
55	52, 54	eqtrid 2812	. . . . . . . 8 ⊢ (𝜑 → [(0g‘𝑅)] ∼ = (0g‘𝑄))
56	1, 2, 7	rng2idl0 21456	. . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝑅) ∈ 𝐼)
57	35, 15	2idlss 21451	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ (2Ideal‘𝑅) → 𝐼 ⊆ 𝐵)
58	2, 57	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ⊆ 𝐵)
59	3, 35, 49	ress0g 18855	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Mnd ∧ (0g‘𝑅) ∈ 𝐼 ∧ 𝐼 ⊆ 𝐵) → (0g‘𝑅) = (0g‘𝐽))
60	48, 56, 58, 59	syl3anc 1398	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝑅) = (0g‘𝐽))
61	60	oveq2d 7435	. . . . . . . . 9 ⊢ (𝜑 → ( 1 · (0g‘𝑅)) = ( 1 · (0g‘𝐽)))
62	3, 36	ressmulr 17382	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (2Ideal‘𝑅) → · = (.r‘𝐽))
63	2, 62	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → · = (.r‘𝐽))
64	63	oveqd 7436	. . . . . . . . 9 ⊢ (𝜑 → ( 1 · (0g‘𝐽)) = ( 1 (.r‘𝐽)(0g‘𝐽)))
65		eqid 2765	. . . . . . . . . . 11 ⊢ (Base‘𝐽) = (Base‘𝐽)
66	65, 37	ringidcl 20393	. . . . . . . . . 10 ⊢ (𝐽 ∈ Ring → 1 ∈ (Base‘𝐽))
67		eqid 2765	. . . . . . . . . . 11 ⊢ (.r‘𝐽) = (.r‘𝐽)
68		eqid 2765	. . . . . . . . . . 11 ⊢ (0g‘𝐽) = (0g‘𝐽)
69	65, 67, 68	ringrz 20423	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Ring ∧ 1 ∈ (Base‘𝐽)) → ( 1 (.r‘𝐽)(0g‘𝐽)) = (0g‘𝐽))
70	4, 66, 69	syl2anc2 597	. . . . . . . . 9 ⊢ (𝜑 → ( 1 (.r‘𝐽)(0g‘𝐽)) = (0g‘𝐽))
71	61, 64, 70	3eqtrd 2804	. . . . . . . 8 ⊢ (𝜑 → ( 1 · (0g‘𝑅)) = (0g‘𝐽))
72	55, 71	opeq12d 4848	. . . . . . 7 ⊢ (𝜑 → ⟨[(0g‘𝑅)] ∼ , ( 1 · (0g‘𝑅))⟩ = ⟨(0g‘𝑄), (0g‘𝐽)⟩)
73		opex 5447	. . . . . . . 8 ⊢ ⟨[(0g‘𝑅)] ∼ , ( 1 · (0g‘𝑅))⟩ ∈ V
74	73	elsn 4606	. . . . . . 7 ⊢ (⟨[(0g‘𝑅)] ∼ , ( 1 · (0g‘𝑅))⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩} ↔ ⟨[(0g‘𝑅)] ∼ , ( 1 · (0g‘𝑅))⟩ = ⟨(0g‘𝑄), (0g‘𝐽)⟩)
75	72, 74	sylibr 237	. . . . . 6 ⊢ (𝜑 → ⟨[(0g‘𝑅)] ∼ , ( 1 · (0g‘𝑅))⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩})
76		opex 5447	. . . . . . . . . 10 ⊢ ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ V
77	76	elsn 4606	. . . . . . . . 9 ⊢ (⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩} ↔ ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ = ⟨(0g‘𝑄), (0g‘𝐽)⟩)
78	12	ovexi 7453	. . . . . . . . . . 11 ⊢ ∼ ∈ V
79		ecexg 8704	. . . . . . . . . . 11 ⊢ ( ∼ ∈ V → [𝑎] ∼ ∈ V)
80	78, 79	ax-mp 5	. . . . . . . . . 10 ⊢ [𝑎] ∼ ∈ V
81		ovex 7452	. . . . . . . . . 10 ⊢ ( 1 · 𝑎) ∈ V
82	80, 81	opth 5460	. . . . . . . . 9 ⊢ (⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ = ⟨(0g‘𝑄), (0g‘𝐽)⟩ ↔ ([𝑎] ∼ = (0g‘𝑄) ∧ ( 1 · 𝑎) = (0g‘𝐽)))
83	77, 82	bitri 278	. . . . . . . 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 21484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (([𝑎] ∼ = (0g‘𝑄) ∧ ( 1 · 𝑎) = (0g‘𝐽)) → 𝑎 = (0g‘𝑅)))
85	83, 84	biimtrid 245	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩} → 𝑎 = (0g‘𝑅)))
86	85	imp 412	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩}) → 𝑎 = (0g‘𝑅))
87	45, 51, 75, 86	rabeqsnd 4637	. . . . 5 ⊢ (𝜑 → {𝑎 ∈ 𝐵 ∣ ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩ ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩}} = {(0g‘𝑅)})
88	41, 87	eqtrd 2800	. . . 4 ⊢ (𝜑 → {𝑎 ∈ 𝐵 ∣ (𝐹‘𝑎) ∈ {⟨(0g‘𝑄), (0g‘𝐽)⟩}} = {(0g‘𝑅)})
89	27, 34, 88	3eqtrd 2804	. . 3 ⊢ (𝜑 → (◡𝐹 “ {(0g‘𝑃)}) = {(0g‘𝑅)})
90	1, 2, 3, 4, 35, 36, 37, 12, 11, 38, 23, 31	rngqiprngghm 21489	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑃))
91		eqid 2765	. . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃)
92		eqid 2765	. . . . 5 ⊢ (0g‘𝑃) = (0g‘𝑃)
93	35, 91, 49, 92	kerf1ghm 19361	. . . 4 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑃) → (𝐹:𝐵–1-1→(Base‘𝑃) ↔ (◡𝐹 “ {(0g‘𝑃)}) = {(0g‘𝑅)}))
94	90, 93	syl 18	. . 3 ⊢ (𝜑 → (𝐹:𝐵–1-1→(Base‘𝑃) ↔ (◡𝐹 “ {(0g‘𝑃)}) = {(0g‘𝑅)}))
95	89, 94	mpbird 260	. 2 ⊢ (𝜑 → 𝐹:𝐵–1-1→(Base‘𝑃))
96		eqidd 2766	. . 3 ⊢ (𝜑 → 𝐹 = 𝐹)
97		eqidd 2766	. . 3 ⊢ (𝜑 → 𝐵 = 𝐵)
98	1, 2, 3, 4, 35, 36, 37, 12, 11, 38, 23	rngqipbas 21485	. . 3 ⊢ (𝜑 → (Base‘𝑃) = (𝐶 × 𝐼))
99	96, 97, 98	f1eq123d 6816	. 2 ⊢ (𝜑 → (𝐹:𝐵–1-1→(Base‘𝑃) ↔ 𝐹:𝐵–1-1→(𝐶 × 𝐼)))
100	95, 99	mpbid 235	1 ⊢ (𝜑 → 𝐹:𝐵–1-1→(𝐶 × 𝐼))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {crab 3418  Vcvv 3457   ⊆ wss 3906  {csn 4591  ⟨cop 4597   ↦ cmpt 5194   × cxp 5661  ^◡ccnv 5662   “ cima 5666   Fn wfn 6535  –1-1→wf1 6537  ‘cfv 6540  (class class class)co 7419  [cec 8698  Basecbs 17291   ↾_s cress 17312  ._rcmulr 17333  0_gc0g 17514   /_s cqus 17581   ×_s cxps 17582  Mndcmnd 18824  Grpcgrp 19044  SubGrpcsubg 19230  NrmSGrpcnsg 19231   ~_QG cqg 19232   GrpHom cghm 19327  Rngcrng 20274  1_rcur 20307  Ringcrg 20359  2Idealc2idl 21438

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-tpos 8228  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-er 8700  df-ec 8702  df-qs 8706  df-map 8832  df-ixp 8902  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-sup 9409  df-inf 9410  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-nn 12249  df-2 12318  df-3 12319  df-4 12320  df-5 12321  df-6 12322  df-7 12323  df-8 12324  df-9 12325  df-n0 12520  df-z 12607  df-dec 12728  df-uz 12879  df-fz 13552  df-struct 17229  df-sets 17246  df-slot 17264  df-ndx 17276  df-base 17292  df-ress 17313  df-plusg 17345  df-mulr 17346  df-sca 17348  df-vsca 17349  df-ip 17350  df-tset 17351  df-ple 17352  df-ds 17354  df-hom 17356  df-cco 17357  df-0g 17516  df-prds 17522  df-imas 17584  df-qus 17585  df-xps 17586  df-mgm 18720  df-sgrp 18809  df-mnd 18825  df-grp 19047  df-minusg 19048  df-sbg 19049  df-subg 19233  df-nsg 19234  df-eqg 19235  df-ghm 19328  df-cmn 19896  df-abl 19897  df-mgp 20261  df-rng 20275  df-ur 20308  df-ring 20361  df-oppr 20465  df-dvdsr 20485  df-unit 20486  df-invr 20516  df-subrng 20695  df-lss 21103  df-sra 21344  df-rgmod 21345  df-lidl 21382  df-2idl 21439

This theorem is used by:  rngqiprngim  21494