Theorem dvhgrp 41770

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 = (0g‘𝐷)
dvhgrp.i	⊢ 𝐼 = (invg‘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 2769	. . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈)
6	1, 2, 3, 4, 5	dvhvbase 41750	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝑈) = (𝑇 × 𝐸))
7	6	eqcomd 2775	. 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 41758	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸))) → (𝑓 + 𝑔) ∈ (𝑇 × 𝐸))
13	12	3impb 1130	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (𝑓 + 𝑔) ∈ (𝑇 × 𝐸))
14	1, 2, 3, 4, 10, 11, 8	dvhvaddass 41760	. 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 40813	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝑇)
17		eqid 2769	. . . . . . . 8 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
18	1, 17, 4, 10	dvhsca 41745	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
19	1, 17	erngdv 41656	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
20	18, 19	eqeltrd 2869	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ DivRing)
21		drnggrp 20822	. . . . . 6 ⊢ (𝐷 ∈ DivRing → 𝐷 ∈ Grp)
22	20, 21	syl 18	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp)
23		eqid 2769	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
24		dvhgrp.o	. . . . . 6 ⊢ 0 = (0g‘𝐷)
25	23, 24	grpidcl 19031	. . . . 5 ⊢ (𝐷 ∈ Grp → 0 ∈ (Base‘𝐷))
26	22, 25	syl 18	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 ∈ (Base‘𝐷))
27	1, 3, 4, 10, 23	dvhbase 41746	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸)
28	26, 27	eleqtrd 2871	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 ∈ 𝐸)
29		opelxpi 5699	. . 3 ⊢ ((( I ↾ 𝐵) ∈ 𝑇 ∧ 0 ∈ 𝐸) → ⟨( I ↾ 𝐵), 0 ⟩ ∈ (𝑇 × 𝐸))
30	16, 28, 29	syl2anc 595	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ⟨( I ↾ 𝐵), 0 ⟩ ∈ (𝑇 × 𝐸))
31		simpl 487	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
32	16	adantr 485	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ( I ↾ 𝐵) ∈ 𝑇)
33	28	adantr 485	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 0 ∈ 𝐸)
34		xp1st 8017	. . . . . 6 ⊢ (𝑓 ∈ (𝑇 × 𝐸) → (1st ‘𝑓) ∈ 𝑇)
35	34	adantl 486	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1st ‘𝑓) ∈ 𝑇)
36		xp2nd 8018	. . . . . 6 ⊢ (𝑓 ∈ (𝑇 × 𝐸) → (2nd ‘𝑓) ∈ 𝐸)
37	36	adantl 486	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (2nd ‘𝑓) ∈ 𝐸)
38	1, 2, 3, 4, 10, 8, 11	dvhopvadd 41756	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( I ↾ 𝐵) ∈ 𝑇 ∧ 0 ∈ 𝐸) ∧ ((1st ‘𝑓) ∈ 𝑇 ∧ (2nd ‘𝑓) ∈ 𝐸)) → (⟨( I ↾ 𝐵), 0 ⟩ + ⟨(1st ‘𝑓), (2nd ‘𝑓)⟩) = ⟨(( I ↾ 𝐵) ∘ (1st ‘𝑓)), ( 0 ⨣ (2nd ‘𝑓))⟩)
39	31, 32, 33, 35, 37, 38	syl122anc 1404	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0 ⟩ + ⟨(1st ‘𝑓), (2nd ‘𝑓)⟩) = ⟨(( I ↾ 𝐵) ∘ (1st ‘𝑓)), ( 0 ⨣ (2nd ‘𝑓))⟩)
40	15, 1, 2	ltrn1o 40787	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (1st ‘𝑓) ∈ 𝑇) → (1st ‘𝑓):𝐵–1-1-onto→𝐵)
41	35, 40	syldan 602	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1st ‘𝑓):𝐵–1-1-onto→𝐵)
42		f1of 6821	. . . . . 6 ⊢ ((1st ‘𝑓):𝐵–1-1-onto→𝐵 → (1st ‘𝑓):𝐵⟶𝐵)
43		fcoi2 6754	. . . . . 6 ⊢ ((1st ‘𝑓):𝐵⟶𝐵 → (( I ↾ 𝐵) ∘ (1st ‘𝑓)) = (1st ‘𝑓))
44	41, 42, 43	3syl 19	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝐵) ∘ (1st ‘𝑓)) = (1st ‘𝑓))
45	22	adantr 485	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 𝐷 ∈ Grp)
46	27	adantr 485	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (Base‘𝐷) = 𝐸)
47	37, 46	eleqtrrd 2872	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (2nd ‘𝑓) ∈ (Base‘𝐷))
48	23, 11, 24	grplid 19033	. . . . . 6 ⊢ ((𝐷 ∈ Grp ∧ (2nd ‘𝑓) ∈ (Base‘𝐷)) → ( 0 ⨣ (2nd ‘𝑓)) = (2nd ‘𝑓))
49	45, 47, 48	syl2anc 595	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ( 0 ⨣ (2nd ‘𝑓)) = (2nd ‘𝑓))
50	44, 49	opeq12d 4850	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨(( I ↾ 𝐵) ∘ (1st ‘𝑓)), ( 0 ⨣ (2nd ‘𝑓))⟩ = ⟨(1st ‘𝑓), (2nd ‘𝑓)⟩)
51	39, 50	eqtrd 2804	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0 ⟩ + ⟨(1st ‘𝑓), (2nd ‘𝑓)⟩) = ⟨(1st ‘𝑓), (2nd ‘𝑓)⟩)
52		1st2nd2 8024	. . . . 5 ⊢ (𝑓 ∈ (𝑇 × 𝐸) → 𝑓 = ⟨(1st ‘𝑓), (2nd ‘𝑓)⟩)
53	52	adantl 486	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 𝑓 = ⟨(1st ‘𝑓), (2nd ‘𝑓)⟩)
54	53	oveq2d 7427	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0 ⟩ + 𝑓) = (⟨( I ↾ 𝐵), 0 ⟩ + ⟨(1st ‘𝑓), (2nd ‘𝑓)⟩))
55	51, 54, 53	3eqtr4d 2814	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0 ⟩ + 𝑓) = 𝑓)
56	1, 2	ltrncnv 40809	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (1st ‘𝑓) ∈ 𝑇) → ◡(1st ‘𝑓) ∈ 𝑇)
57	35, 56	syldan 602	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ◡(1st ‘𝑓) ∈ 𝑇)
58		dvhgrp.i	. . . . . 6 ⊢ 𝐼 = (invg‘𝐷)
59	23, 58	grpinvcl 19053	. . . . 5 ⊢ ((𝐷 ∈ Grp ∧ (2nd ‘𝑓) ∈ (Base‘𝐷)) → (𝐼‘(2nd ‘𝑓)) ∈ (Base‘𝐷))
60	45, 47, 59	syl2anc 595	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐼‘(2nd ‘𝑓)) ∈ (Base‘𝐷))
61	60, 46	eleqtrd 2871	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐼‘(2nd ‘𝑓)) ∈ 𝐸)
62		opelxpi 5699	. . 3 ⊢ ((◡(1st ‘𝑓) ∈ 𝑇 ∧ (𝐼‘(2nd ‘𝑓)) ∈ 𝐸) → ⟨◡(1st ‘𝑓), (𝐼‘(2nd ‘𝑓))⟩ ∈ (𝑇 × 𝐸))
63	57, 61, 62	syl2anc 595	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨◡(1st ‘𝑓), (𝐼‘(2nd ‘𝑓))⟩ ∈ (𝑇 × 𝐸))
64	53	oveq2d 7427	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨◡(1st ‘𝑓), (𝐼‘(2nd ‘𝑓))⟩ + 𝑓) = (⟨◡(1st ‘𝑓), (𝐼‘(2nd ‘𝑓))⟩ + ⟨(1st ‘𝑓), (2nd ‘𝑓)⟩))
65	1, 2, 3, 4, 10, 8, 11	dvhopvadd 41756	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (◡(1st ‘𝑓) ∈ 𝑇 ∧ (𝐼‘(2nd ‘𝑓)) ∈ 𝐸) ∧ ((1st ‘𝑓) ∈ 𝑇 ∧ (2nd ‘𝑓) ∈ 𝐸)) → (⟨◡(1st ‘𝑓), (𝐼‘(2nd ‘𝑓))⟩ + ⟨(1st ‘𝑓), (2nd ‘𝑓)⟩) = ⟨(◡(1st ‘𝑓) ∘ (1st ‘𝑓)), ((𝐼‘(2nd ‘𝑓)) ⨣ (2nd ‘𝑓))⟩)
66	31, 57, 61, 35, 37, 65	syl122anc 1404	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨◡(1st ‘𝑓), (𝐼‘(2nd ‘𝑓))⟩ + ⟨(1st ‘𝑓), (2nd ‘𝑓)⟩) = ⟨(◡(1st ‘𝑓) ∘ (1st ‘𝑓)), ((𝐼‘(2nd ‘𝑓)) ⨣ (2nd ‘𝑓))⟩)
67		f1ococnv1 6851	. . . . . 6 ⊢ ((1st ‘𝑓):𝐵–1-1-onto→𝐵 → (◡(1st ‘𝑓) ∘ (1st ‘𝑓)) = ( I ↾ 𝐵))
68	41, 67	syl 18	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (◡(1st ‘𝑓) ∘ (1st ‘𝑓)) = ( I ↾ 𝐵))
69	23, 11, 24, 58	grplinv 19055	. . . . . 6 ⊢ ((𝐷 ∈ Grp ∧ (2nd ‘𝑓) ∈ (Base‘𝐷)) → ((𝐼‘(2nd ‘𝑓)) ⨣ (2nd ‘𝑓)) = 0 )
70	45, 47, 69	syl2anc 595	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ((𝐼‘(2nd ‘𝑓)) ⨣ (2nd ‘𝑓)) = 0 )
71	68, 70	opeq12d 4850	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨(◡(1st ‘𝑓) ∘ (1st ‘𝑓)), ((𝐼‘(2nd ‘𝑓)) ⨣ (2nd ‘𝑓))⟩ = ⟨( I ↾ 𝐵), 0 ⟩)
72	66, 71	eqtrd 2804	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨◡(1st ‘𝑓), (𝐼‘(2nd ‘𝑓))⟩ + ⟨(1st ‘𝑓), (2nd ‘𝑓)⟩) = ⟨( I ↾ 𝐵), 0 ⟩)
73	64, 72	eqtrd 2804	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨◡(1st ‘𝑓), (𝐼‘(2nd ‘𝑓))⟩ + 𝑓) = ⟨( I ↾ 𝐵), 0 ⟩)
74	7, 9, 13, 14, 30, 55, 63, 73	isgrpd 19024	1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ⟨cop 4600   I cid 5556   × cxp 5660  ^◡ccnv 5661   ↾ cres 5664   ∘ ccom 5666  ⟶wf 6533  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7411  1^st c1st 7983  2^nd c2nd 7984  Basecbs 17268  +_gcplusg 17309  Scalarcsca 17312  0_gc0g 17491  Grpcgrp 18999  inv_gcminusg 19000  DivRingcdr 20812  HLchlt 40013  LHypclh 40647  LTrncltrn 40764  TEndoctendo 41415  EDRingcedring 41416  DVecHcdvh 41741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-riotaBAD 39616

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-1st 7985  df-2nd 7986  df-tpos 8221  df-undef 8268  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-5 12305  df-6 12306  df-n0 12504  df-z 12591  df-uz 12862  df-fz 13535  df-struct 17206  df-sets 17223  df-slot 17241  df-ndx 17253  df-base 17269  df-ress 17290  df-plusg 17322  df-mulr 17323  df-sca 17325  df-vsca 17326  df-0g 17493  df-proset 18349  df-poset 18368  df-plt 18383  df-lub 18399  df-glb 18400  df-join 18401  df-meet 18402  df-p0 18478  df-p1 18479  df-lat 18487  df-clat 18554  df-mgm 18697  df-sgrp 18776  df-mnd 18792  df-grp 19002  df-minusg 19003  df-cmn 19851  df-abl 19852  df-mgp 20216  df-rng 20230  df-ur 20263  df-ring 20316  df-oppr 20418  df-dvdsr 20438  df-unit 20439  df-invr 20469  df-dvr 20482  df-drng 20814  df-oposet 39839  df-ol 39841  df-oml 39842  df-covers 39929  df-ats 39930  df-atl 39961  df-cvlat 39985  df-hlat 40014  df-llines 40161  df-lplanes 40162  df-lvols 40163  df-lines 40164  df-psubsp 40166  df-pmap 40167  df-padd 40459  df-lhyp 40651  df-laut 40652  df-ldil 40767  df-ltrn 40768  df-trl 40822  df-tendo 41418  df-edring 41420  df-dvech 41742

This theorem is referenced by:  dvhlveclem  41771