dvhgrp - Mathbox for Norm Megill

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Theorem dvhgrp 36396

Description: The full vector space 𝑈 constructed from a Hilbert lattice 𝐾 (given a fiducial hyperplane 𝑊) is a group. (Contributed by NM, 19-Oct-2013.) (Revised by Mario Carneiro, 24-Jun-2014.)

**Hypotheses**
Ref	Expression
dvhgrp.b	⊢ 𝐵 = (Base‘𝐾)
dvhgrp.h	⊢ 𝐻 = (LHyp‘𝐾)
dvhgrp.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dvhgrp.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
dvhgrp.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dvhgrp.d	⊢ 𝐷 = (Scalar‘𝑈)
dvhgrp.p	⊢ ⨣ = (+_g‘𝐷)
dvhgrp.a	⊢ + = (+_g‘𝑈)
dvhgrp.o	⊢ 0 = (0_g‘𝐷)
dvhgrp.i	⊢ 𝐼 = (inv_g‘𝐷)

**Assertion**
Ref	Expression
dvhgrp	⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ Grp)

Proof of Theorem dvhgrp Dummy variables 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvhgrp.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		dvhgrp.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
3		dvhgrp.e	. . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
4		dvhgrp.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		eqid 2622	. . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈)
6	1, 2, 3, 4, 5	dvhvbase 36376	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝑈) = (𝑇 × 𝐸))
7	6	eqcomd 2628	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑇 × 𝐸) = (Base‘𝑈))
8		dvhgrp.a	. . 3 ⊢ + = (+_g‘𝑈)
9	8	a1i 11	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → + = (+_g‘𝑈))
10		dvhgrp.d	. . . 4 ⊢ 𝐷 = (Scalar‘𝑈)
11		dvhgrp.p	. . . 4 ⊢ ⨣ = (+_g‘𝐷)
12	1, 2, 3, 4, 10, 11, 8	dvhvaddcl 36384	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸))) → (𝑓 + 𝑔) ∈ (𝑇 × 𝐸))
13	12	3impb 1260	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (𝑓 + 𝑔) ∈ (𝑇 × 𝐸))
14	1, 2, 3, 4, 10, 11, 8	dvhvaddass 36386	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸) ∧ ℎ ∈ (𝑇 × 𝐸))) → ((𝑓 + 𝑔) + ℎ) = (𝑓 + (𝑔 + ℎ)))
15		dvhgrp.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
16	15, 1, 2	idltrn 35436	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝑇)
17		eqid 2622	. . . . . . . 8 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
18	1, 17, 4, 10	dvhsca 36371	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
19	1, 17	erngdv 36281	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
20	18, 19	eqeltrd 2701	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ DivRing)
21		drnggrp 18755	. . . . . 6 ⊢ (𝐷 ∈ DivRing → 𝐷 ∈ Grp)
22	20, 21	syl 17	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp)
23		eqid 2622	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
24		dvhgrp.o	. . . . . 6 ⊢ 0 = (0_g‘𝐷)
25	23, 24	grpidcl 17450	. . . . 5 ⊢ (𝐷 ∈ Grp → 0 ∈ (Base‘𝐷))
26	22, 25	syl 17	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 ∈ (Base‘𝐷))
27	1, 3, 4, 10, 23	dvhbase 36372	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸)
28	26, 27	eleqtrd 2703	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 ∈ 𝐸)
29		opelxpi 5148	. . 3 ⊢ ((( I ↾ 𝐵) ∈ 𝑇 ∧ 0 ∈ 𝐸) → ⟨( I ↾ 𝐵), 0 ⟩ ∈ (𝑇 × 𝐸))
30	16, 28, 29	syl2anc 693	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ⟨( I ↾ 𝐵), 0 ⟩ ∈ (𝑇 × 𝐸))
31		simpl 473	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
32	16	adantr 481	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ( I ↾ 𝐵) ∈ 𝑇)
33	28	adantr 481	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 0 ∈ 𝐸)
34		xp1st 7198	. . . . . 6 ⊢ (𝑓 ∈ (𝑇 × 𝐸) → (1^st ‘𝑓) ∈ 𝑇)
35	34	adantl 482	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1^st ‘𝑓) ∈ 𝑇)
36		xp2nd 7199	. . . . . 6 ⊢ (𝑓 ∈ (𝑇 × 𝐸) → (2^nd ‘𝑓) ∈ 𝐸)
37	36	adantl 482	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (2^nd ‘𝑓) ∈ 𝐸)
38	1, 2, 3, 4, 10, 8, 11	dvhopvadd 36382	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( I ↾ 𝐵) ∈ 𝑇 ∧ 0 ∈ 𝐸) ∧ ((1^st ‘𝑓) ∈ 𝑇 ∧ (2^nd ‘𝑓) ∈ 𝐸)) → (⟨( I ↾ 𝐵), 0 ⟩ + ⟨(1^st ‘𝑓), (2^nd ‘𝑓)⟩) = ⟨(( I ↾ 𝐵) ∘ (1^st ‘𝑓)), ( 0 ⨣ (2^nd ‘𝑓))⟩)
39	31, 32, 33, 35, 37, 38	syl122anc 1335	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0 ⟩ + ⟨(1^st ‘𝑓), (2^nd ‘𝑓)⟩) = ⟨(( I ↾ 𝐵) ∘ (1^st ‘𝑓)), ( 0 ⨣ (2^nd ‘𝑓))⟩)
40	15, 1, 2	ltrn1o 35410	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (1^st ‘𝑓) ∈ 𝑇) → (1^st ‘𝑓):𝐵–1-1-onto→𝐵)
41	35, 40	syldan 487	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1^st ‘𝑓):𝐵–1-1-onto→𝐵)
42		f1of 6137	. . . . . 6 ⊢ ((1^st ‘𝑓):𝐵–1-1-onto→𝐵 → (1^st ‘𝑓):𝐵⟶𝐵)
43		fcoi2 6079	. . . . . 6 ⊢ ((1^st ‘𝑓):𝐵⟶𝐵 → (( I ↾ 𝐵) ∘ (1^st ‘𝑓)) = (1^st ‘𝑓))
44	41, 42, 43	3syl 18	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝐵) ∘ (1^st ‘𝑓)) = (1^st ‘𝑓))
45	22	adantr 481	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 𝐷 ∈ Grp)
46	27	adantr 481	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (Base‘𝐷) = 𝐸)
47	37, 46	eleqtrrd 2704	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (2^nd ‘𝑓) ∈ (Base‘𝐷))
48	23, 11, 24	grplid 17452	. . . . . 6 ⊢ ((𝐷 ∈ Grp ∧ (2^nd ‘𝑓) ∈ (Base‘𝐷)) → ( 0 ⨣ (2^nd ‘𝑓)) = (2^nd ‘𝑓))
49	45, 47, 48	syl2anc 693	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ( 0 ⨣ (2^nd ‘𝑓)) = (2^nd ‘𝑓))
50	44, 49	opeq12d 4410	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨(( I ↾ 𝐵) ∘ (1^st ‘𝑓)), ( 0 ⨣ (2^nd ‘𝑓))⟩ = ⟨(1^st ‘𝑓), (2^nd ‘𝑓)⟩)
51	39, 50	eqtrd 2656	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0 ⟩ + ⟨(1^st ‘𝑓), (2^nd ‘𝑓)⟩) = ⟨(1^st ‘𝑓), (2^nd ‘𝑓)⟩)
52		1st2nd2 7205	. . . . 5 ⊢ (𝑓 ∈ (𝑇 × 𝐸) → 𝑓 = ⟨(1^st ‘𝑓), (2^nd ‘𝑓)⟩)
53	52	adantl 482	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 𝑓 = ⟨(1^st ‘𝑓), (2^nd ‘𝑓)⟩)
54	53	oveq2d 6666	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0 ⟩ + 𝑓) = (⟨( I ↾ 𝐵), 0 ⟩ + ⟨(1^st ‘𝑓), (2^nd ‘𝑓)⟩))
55	51, 54, 53	3eqtr4d 2666	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0 ⟩ + 𝑓) = 𝑓)
56	1, 2	ltrncnv 35432	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (1^st ‘𝑓) ∈ 𝑇) → ^◡(1^st ‘𝑓) ∈ 𝑇)
57	35, 56	syldan 487	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ^◡(1^st ‘𝑓) ∈ 𝑇)
58		dvhgrp.i	. . . . . 6 ⊢ 𝐼 = (inv_g‘𝐷)
59	23, 58	grpinvcl 17467	. . . . 5 ⊢ ((𝐷 ∈ Grp ∧ (2^nd ‘𝑓) ∈ (Base‘𝐷)) → (𝐼‘(2^nd ‘𝑓)) ∈ (Base‘𝐷))
60	45, 47, 59	syl2anc 693	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐼‘(2^nd ‘𝑓)) ∈ (Base‘𝐷))
61	60, 46	eleqtrd 2703	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐼‘(2^nd ‘𝑓)) ∈ 𝐸)
62		opelxpi 5148	. . 3 ⊢ ((^◡(1^st ‘𝑓) ∈ 𝑇 ∧ (𝐼‘(2^nd ‘𝑓)) ∈ 𝐸) → ⟨^◡(1^st ‘𝑓), (𝐼‘(2^nd ‘𝑓))⟩ ∈ (𝑇 × 𝐸))
63	57, 61, 62	syl2anc 693	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨^◡(1^st ‘𝑓), (𝐼‘(2^nd ‘𝑓))⟩ ∈ (𝑇 × 𝐸))
64	53	oveq2d 6666	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨^◡(1^st ‘𝑓), (𝐼‘(2^nd ‘𝑓))⟩ + 𝑓) = (⟨^◡(1^st ‘𝑓), (𝐼‘(2^nd ‘𝑓))⟩ + ⟨(1^st ‘𝑓), (2^nd ‘𝑓)⟩))
65	1, 2, 3, 4, 10, 8, 11	dvhopvadd 36382	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (^◡(1^st ‘𝑓) ∈ 𝑇 ∧ (𝐼‘(2^nd ‘𝑓)) ∈ 𝐸) ∧ ((1^st ‘𝑓) ∈ 𝑇 ∧ (2^nd ‘𝑓) ∈ 𝐸)) → (⟨^◡(1^st ‘𝑓), (𝐼‘(2^nd ‘𝑓))⟩ + ⟨(1^st ‘𝑓), (2^nd ‘𝑓)⟩) = ⟨(^◡(1^st ‘𝑓) ∘ (1^st ‘𝑓)), ((𝐼‘(2^nd ‘𝑓)) ⨣ (2^nd ‘𝑓))⟩)
66	31, 57, 61, 35, 37, 65	syl122anc 1335	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨^◡(1^st ‘𝑓), (𝐼‘(2^nd ‘𝑓))⟩ + ⟨(1^st ‘𝑓), (2^nd ‘𝑓)⟩) = ⟨(^◡(1^st ‘𝑓) ∘ (1^st ‘𝑓)), ((𝐼‘(2^nd ‘𝑓)) ⨣ (2^nd ‘𝑓))⟩)
67		f1ococnv1 6165	. . . . . 6 ⊢ ((1^st ‘𝑓):𝐵–1-1-onto→𝐵 → (^◡(1^st ‘𝑓) ∘ (1^st ‘𝑓)) = ( I ↾ 𝐵))
68	41, 67	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (^◡(1^st ‘𝑓) ∘ (1^st ‘𝑓)) = ( I ↾ 𝐵))
69	23, 11, 24, 58	grplinv 17468	. . . . . 6 ⊢ ((𝐷 ∈ Grp ∧ (2^nd ‘𝑓) ∈ (Base‘𝐷)) → ((𝐼‘(2^nd ‘𝑓)) ⨣ (2^nd ‘𝑓)) = 0 )
70	45, 47, 69	syl2anc 693	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ((𝐼‘(2^nd ‘𝑓)) ⨣ (2^nd ‘𝑓)) = 0 )
71	68, 70	opeq12d 4410	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨(^◡(1^st ‘𝑓) ∘ (1^st ‘𝑓)), ((𝐼‘(2^nd ‘𝑓)) ⨣ (2^nd ‘𝑓))⟩ = ⟨( I ↾ 𝐵), 0 ⟩)
72	66, 71	eqtrd 2656	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨^◡(1^st ‘𝑓), (𝐼‘(2^nd ‘𝑓))⟩ + ⟨(1^st ‘𝑓), (2^nd ‘𝑓)⟩) = ⟨( I ↾ 𝐵), 0 ⟩)
73	64, 72	eqtrd 2656	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨^◡(1^st ‘𝑓), (𝐼‘(2^nd ‘𝑓))⟩ + 𝑓) = ⟨( I ↾ 𝐵), 0 ⟩)
74	7, 9, 13, 14, 30, 55, 63, 73	isgrpd 17444	1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ Grp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ⟨cop 4183 I cid 5023 × cxp 5112 ^◡ccnv 5113 ↾ cres 5116 ∘ ccom 5118 ⟶wf 5884 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 1^st c1st 7166 2^nd c2nd 7167 Basecbs 15857 +_gcplusg 15941 Scalarcsca 15944 0_gc0g 16100 Grpcgrp 17422 inv_gcminusg 17423 DivRingcdr 18747 HLchlt 34637 LHypclh 35270 LTrncltrn 35387 TEndoctendo 36040 EDRingcedring 36041 DVecHcdvh 36367

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-riotaBAD 34239

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-undef 7399 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-0g 16102 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 df-tendo 36043 df-edring 36045 df-dvech 36368

This theorem is referenced by: dvhlveclem 36397

W3C validator