0prjspnrel - Mathbox for Steven Nguyen

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Theorem 0prjspnrel 41947

Description: In the zero-dimensional projective space, all vectors are equivalent to the unit vector. (Contributed by Steven Nguyen, 7-Jun-2023.)

**Hypotheses**
Ref	Expression
0prjspnrel.e	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}
0prjspnrel.b	⊢ 𝐵 = ((Base‘𝑊) ∖ {(0_g‘𝑊)})
0prjspnrel.x	⊢ · = ( ·_𝑠 ‘𝑊)
0prjspnrel.s	⊢ 𝑆 = (Base‘𝐾)
0prjspnrel.w	⊢ 𝑊 = (𝐾 freeLMod (0...0))
0prjspnrel.1	⊢ 1 = ((𝐾 unitVec (0...0))‘0)

**Assertion**
Ref	Expression
0prjspnrel	⊢ ((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) → 𝑋 ∼ 1 )

Distinct variable groups: 𝑥,𝐵,𝑦 𝑥,𝑋,𝑦,𝑙 𝑥,𝐾,𝑦,𝑙 𝑥, · ,𝑦,𝑙 𝑥, 1 ,𝑦,𝑙 𝑥,𝑆,𝑦,𝑙 Allowed substitution hints: 𝐵(𝑙) ∼ (𝑥,𝑦,𝑙) 𝑊(𝑥,𝑦,𝑙)

Proof of Theorem 0prjspnrel Dummy variables 𝑚 𝑛 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. 2 ⊢ ((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
2		0prjspnrel.b	. . . 4 ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0_g‘𝑊)})
3		0prjspnrel.w	. . . 4 ⊢ 𝑊 = (𝐾 freeLMod (0...0))
4		0prjspnrel.1	. . . 4 ⊢ 1 = ((𝐾 unitVec (0...0))‘0)
5	2, 3, 4	0prjspnlem 41943	. . 3 ⊢ (𝐾 ∈ DivRing → 1 ∈ 𝐵)
6	5	adantr 480	. 2 ⊢ ((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) → 1 ∈ 𝐵)
7		ovexd 7440	. . . . . 6 ⊢ ((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) → (0...0) ∈ V)
8		difss 4126	. . . . . . . . 9 ⊢ ((Base‘𝑊) ∖ {(0_g‘𝑊)}) ⊆ (Base‘𝑊)
9	2, 8	eqsstri 4011	. . . . . . . 8 ⊢ 𝐵 ⊆ (Base‘𝑊)
10	9	sseli 3973	. . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ (Base‘𝑊))
11	10	adantl 481	. . . . . 6 ⊢ ((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑊))
12		eqid 2726	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
13		eqid 2726	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
14	3, 12, 13	frlmbasf 21655	. . . . . 6 ⊢ (((0...0) ∈ V ∧ 𝑋 ∈ (Base‘𝑊)) → 𝑋:(0...0)⟶(Base‘𝐾))
15	7, 11, 14	syl2anc 583	. . . . 5 ⊢ ((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) → 𝑋:(0...0)⟶(Base‘𝐾))
16		c0ex 11212	. . . . . . . 8 ⊢ 0 ∈ V
17	16	snid 4659	. . . . . . 7 ⊢ 0 ∈ {0}
18		fz0sn 13607	. . . . . . 7 ⊢ (0...0) = {0}
19	17, 18	eleqtrri 2826	. . . . . 6 ⊢ 0 ∈ (0...0)
20	19	a1i 11	. . . . 5 ⊢ ((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) → 0 ∈ (0...0))
21	15, 20	ffvelcdmd 7081	. . . 4 ⊢ ((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) → (𝑋‘0) ∈ (Base‘𝐾))
22		sneq 4633	. . . . . . 7 ⊢ (𝑛 = (𝑋‘0) → {𝑛} = {(𝑋‘0)})
23	22	xpeq2d 5699	. . . . . 6 ⊢ (𝑛 = (𝑋‘0) → ((0...0) × {𝑛}) = ((0...0) × {(𝑋‘0)}))
24	23	eqeq2d 2737	. . . . 5 ⊢ (𝑛 = (𝑋‘0) → (𝑋 = ((0...0) × {𝑛}) ↔ 𝑋 = ((0...0) × {(𝑋‘0)})))
25	24	adantl 481	. . . 4 ⊢ (((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 = (𝑋‘0)) → (𝑋 = ((0...0) × {𝑛}) ↔ 𝑋 = ((0...0) × {(𝑋‘0)})))
26	3, 12, 13	frlmbasmap 21654	. . . . . 6 ⊢ (((0...0) ∈ V ∧ 𝑋 ∈ (Base‘𝑊)) → 𝑋 ∈ ((Base‘𝐾) ↑_m (0...0)))
27	7, 11, 26	syl2anc 583	. . . . 5 ⊢ ((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ ((Base‘𝐾) ↑_m (0...0)))
28		fvex 6898	. . . . . 6 ⊢ (Base‘𝐾) ∈ V
29	18, 28, 16	mapsnconst 8888	. . . . 5 ⊢ (𝑋 ∈ ((Base‘𝐾) ↑_m (0...0)) → 𝑋 = ((0...0) × {(𝑋‘0)}))
30	27, 29	syl 17	. . . 4 ⊢ ((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) → 𝑋 = ((0...0) × {(𝑋‘0)}))
31	21, 25, 30	rspcedvd 3608	. . 3 ⊢ ((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) → ∃𝑛 ∈ (Base‘𝐾)𝑋 = ((0...0) × {𝑛}))
32		simprl 768	. . . . 5 ⊢ (((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ (𝑛 ∈ (Base‘𝐾) ∧ 𝑋 = ((0...0) × {𝑛}))) → 𝑛 ∈ (Base‘𝐾))
33		0prjspnrel.s	. . . . 5 ⊢ 𝑆 = (Base‘𝐾)
34	32, 33	eleqtrrdi 2838	. . . 4 ⊢ (((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ (𝑛 ∈ (Base‘𝐾) ∧ 𝑋 = ((0...0) × {𝑛}))) → 𝑛 ∈ 𝑆)
35		oveq1 7412	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑚 · 1 ) = (𝑛 · 1 ))
36	35	eqeq2d 2737	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝑋 = (𝑚 · 1 ) ↔ 𝑋 = (𝑛 · 1 )))
37	36	adantl 481	. . . 4 ⊢ ((((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ (𝑛 ∈ (Base‘𝐾) ∧ 𝑋 = ((0...0) × {𝑛}))) ∧ 𝑚 = 𝑛) → (𝑋 = (𝑚 · 1 ) ↔ 𝑋 = (𝑛 · 1 )))
38		ovexd 7440	. . . . . . . . 9 ⊢ (((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) → (0...0) ∈ V)
39		simpr 484	. . . . . . . . 9 ⊢ (((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) → 𝑛 ∈ (Base‘𝐾))
40	5	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) → 1 ∈ 𝐵)
41	9, 40	sselid 3975	. . . . . . . . 9 ⊢ (((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) → 1 ∈ (Base‘𝑊))
42		0prjspnrel.x	. . . . . . . . 9 ⊢ · = ( ·_𝑠 ‘𝑊)
43		eqid 2726	. . . . . . . . 9 ⊢ (._r‘𝐾) = (._r‘𝐾)
44	3, 13, 12, 38, 39, 41, 42, 43	frlmvscafval 21661	. . . . . . . 8 ⊢ (((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) → (𝑛 · 1 ) = (((0...0) × {𝑛}) ∘_f (._r‘𝐾) 1 ))
45	3, 12, 13	frlmbasf 21655	. . . . . . . . . . 11 ⊢ (((0...0) ∈ V ∧ 1 ∈ (Base‘𝑊)) → 1 :(0...0)⟶(Base‘𝐾))
46	38, 41, 45	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) → 1 :(0...0)⟶(Base‘𝐾))
47		drngring 20594	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ DivRing → 𝐾 ∈ Ring)
48		eqid 2726	. . . . . . . . . . . . . . 15 ⊢ (1_r‘𝐾) = (1_r‘𝐾)
49	12, 48	ringidcl 20165	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ Ring → (1_r‘𝐾) ∈ (Base‘𝐾))
50	47, 49	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ DivRing → (1_r‘𝐾) ∈ (Base‘𝐾))
51	50	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) → (1_r‘𝐾) ∈ (Base‘𝐾))
52	51	snssd 4807	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) → {(1_r‘𝐾)} ⊆ (Base‘𝐾))
53	4	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ (0...0) → 1 = ((𝐾 unitVec (0...0))‘0))
54		elfz1eq 13518	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ (0...0) → 𝑑 = 0)
55	53, 54	fveq12d 6892	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ (0...0) → ( 1 ‘𝑑) = (((𝐾 unitVec (0...0))‘0)‘0))
56	55	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) ∧ 𝑑 ∈ (0...0)) → ( 1 ‘𝑑) = (((𝐾 unitVec (0...0))‘0)‘0))
57		eqid 2726	. . . . . . . . . . . . . . 15 ⊢ (𝐾 unitVec (0...0)) = (𝐾 unitVec (0...0))
58		simplll 772	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) ∧ 𝑑 ∈ (0...0)) → 𝐾 ∈ DivRing)
59		ovexd 7440	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) ∧ 𝑑 ∈ (0...0)) → (0...0) ∈ V)
60	19	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) ∧ 𝑑 ∈ (0...0)) → 0 ∈ (0...0))
61	57, 58, 59, 60, 48	uvcvv1 21684	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) ∧ 𝑑 ∈ (0...0)) → (((𝐾 unitVec (0...0))‘0)‘0) = (1_r‘𝐾))
62		fvex 6898	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 unitVec (0...0))‘0)‘0) ∈ V
63	62	elsn 4638	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 unitVec (0...0))‘0)‘0) ∈ {(1_r‘𝐾)} ↔ (((𝐾 unitVec (0...0))‘0)‘0) = (1_r‘𝐾))
64	61, 63	sylibr 233	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) ∧ 𝑑 ∈ (0...0)) → (((𝐾 unitVec (0...0))‘0)‘0) ∈ {(1_r‘𝐾)})
65	56, 64	eqeltrd 2827	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) ∧ 𝑑 ∈ (0...0)) → ( 1 ‘𝑑) ∈ {(1_r‘𝐾)})
66	65	ralrimiva 3140	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) → ∀𝑑 ∈ (0...0)( 1 ‘𝑑) ∈ {(1_r‘𝐾)})
67		fcdmssb 7117	. . . . . . . . . . 11 ⊢ (({(1_r‘𝐾)} ⊆ (Base‘𝐾) ∧ ∀𝑑 ∈ (0...0)( 1 ‘𝑑) ∈ {(1_r‘𝐾)}) → ( 1 :(0...0)⟶(Base‘𝐾) ↔ 1 :(0...0)⟶{(1_r‘𝐾)}))
68	52, 66, 67	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) → ( 1 :(0...0)⟶(Base‘𝐾) ↔ 1 :(0...0)⟶{(1_r‘𝐾)}))
69	46, 68	mpbid 231	. . . . . . . . 9 ⊢ (((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) → 1 :(0...0)⟶{(1_r‘𝐾)})
70		vex 3472	. . . . . . . . . 10 ⊢ 𝑛 ∈ V
71	70	a1i 11	. . . . . . . . 9 ⊢ (((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) → 𝑛 ∈ V)
72		elsni 4640	. . . . . . . . . . 11 ⊢ (𝑐 ∈ {(1_r‘𝐾)} → 𝑐 = (1_r‘𝐾))
73	72	oveq2d 7421	. . . . . . . . . 10 ⊢ (𝑐 ∈ {(1_r‘𝐾)} → (𝑛(._r‘𝐾)𝑐) = (𝑛(._r‘𝐾)(1_r‘𝐾)))
74	47	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) → 𝐾 ∈ Ring)
75	12, 43, 48	ringridm 20169	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Ring ∧ 𝑛 ∈ (Base‘𝐾)) → (𝑛(._r‘𝐾)(1_r‘𝐾)) = 𝑛)
76	74, 39, 75	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) → (𝑛(._r‘𝐾)(1_r‘𝐾)) = 𝑛)
77	73, 76	sylan9eqr 2788	. . . . . . . . 9 ⊢ ((((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) ∧ 𝑐 ∈ {(1_r‘𝐾)}) → (𝑛(._r‘𝐾)𝑐) = 𝑛)
78	38, 69, 71, 71, 77	caofid2 7701	. . . . . . . 8 ⊢ (((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) → (((0...0) × {𝑛}) ∘_f (._r‘𝐾) 1 ) = ((0...0) × {𝑛}))
79	44, 78	eqtrd 2766	. . . . . . 7 ⊢ (((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) → (𝑛 · 1 ) = ((0...0) × {𝑛}))
80	79	eqeq2d 2737	. . . . . 6 ⊢ (((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) → (𝑋 = (𝑛 · 1 ) ↔ 𝑋 = ((0...0) × {𝑛})))
81	80	biimprd 247	. . . . 5 ⊢ (((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ 𝑛 ∈ (Base‘𝐾)) → (𝑋 = ((0...0) × {𝑛}) → 𝑋 = (𝑛 · 1 )))
82	81	impr 454	. . . 4 ⊢ (((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ (𝑛 ∈ (Base‘𝐾) ∧ 𝑋 = ((0...0) × {𝑛}))) → 𝑋 = (𝑛 · 1 ))
83	34, 37, 82	rspcedvd 3608	. . 3 ⊢ (((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) ∧ (𝑛 ∈ (Base‘𝐾) ∧ 𝑋 = ((0...0) × {𝑛}))) → ∃𝑚 ∈ 𝑆 𝑋 = (𝑚 · 1 ))
84	31, 83	rexlimddv 3155	. 2 ⊢ ((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) → ∃𝑚 ∈ 𝑆 𝑋 = (𝑚 · 1 ))
85		0prjspnrel.e	. . 3 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}
86	85	prjsprel 41924	. 2 ⊢ (𝑋 ∼ 1 ↔ ((𝑋 ∈ 𝐵 ∧ 1 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝑆 𝑋 = (𝑚 · 1 )))
87	1, 6, 84, 86	syl21anbrc 1341	1 ⊢ ((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) → 𝑋 ∼ 1 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ∃wrex 3064 Vcvv 3468 ∖ cdif 3940 ⊆ wss 3943 {csn 4623 class class class wbr 5141 {copab 5203 × cxp 5667 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 ∘_f cof 7665 ↑_m cmap 8822 0cc0 11112 ...cfz 13490 Basecbs 17153 ._rcmulr 17207 ·_𝑠 cvsca 17210 0_gc0g 17394 1_rcur 20086 Ringcrg 20138 DivRingcdr 20587 freeLMod cfrlm 21641 unitVec cuvc 21677

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-hom 17230 df-cco 17231 df-0g 17396 df-prds 17402 df-pws 17404 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19050 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-nzr 20415 df-subrg 20471 df-drng 20589 df-lmod 20708 df-lss 20779 df-sra 21021 df-rgmod 21022 df-dsmm 21627 df-frlm 21642 df-uvc 21678

This theorem is referenced by: 0prjspn 41948

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