dvhgrp - Mathbox for Norm Megill

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Theorem dvhgrp 39048

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 2738	. . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈)
6	1, 2, 3, 4, 5	dvhvbase 39028	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝑈) = (𝑇 × 𝐸))
7	6	eqcomd 2744	. 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 39036	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸))) → (𝑓 + 𝑔) ∈ (𝑇 × 𝐸))
13	12	3impb 1113	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (𝑓 + 𝑔) ∈ (𝑇 × 𝐸))
14	1, 2, 3, 4, 10, 11, 8	dvhvaddass 39038	. 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 38091	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝑇)
17		eqid 2738	. . . . . . . 8 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
18	1, 17, 4, 10	dvhsca 39023	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
19	1, 17	erngdv 38934	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
20	18, 19	eqeltrd 2839	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ DivRing)
21		drnggrp 19914	. . . . . 6 ⊢ (𝐷 ∈ DivRing → 𝐷 ∈ Grp)
22	20, 21	syl 17	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp)
23		eqid 2738	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
24		dvhgrp.o	. . . . . 6 ⊢ 0 = (0_g‘𝐷)
25	23, 24	grpidcl 18522	. . . . 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 39024	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸)
28	26, 27	eleqtrd 2841	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 ∈ 𝐸)
29		opelxpi 5617	. . 3 ⊢ ((( I ↾ 𝐵) ∈ 𝑇 ∧ 0 ∈ 𝐸) → ⟨( I ↾ 𝐵), 0 ⟩ ∈ (𝑇 × 𝐸))
30	16, 28, 29	syl2anc 583	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ⟨( I ↾ 𝐵), 0 ⟩ ∈ (𝑇 × 𝐸))
31		simpl 482	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
32	16	adantr 480	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ( I ↾ 𝐵) ∈ 𝑇)
33	28	adantr 480	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 0 ∈ 𝐸)
34		xp1st 7836	. . . . . 6 ⊢ (𝑓 ∈ (𝑇 × 𝐸) → (1^st ‘𝑓) ∈ 𝑇)
35	34	adantl 481	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1^st ‘𝑓) ∈ 𝑇)
36		xp2nd 7837	. . . . . 6 ⊢ (𝑓 ∈ (𝑇 × 𝐸) → (2^nd ‘𝑓) ∈ 𝐸)
37	36	adantl 481	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (2^nd ‘𝑓) ∈ 𝐸)
38	1, 2, 3, 4, 10, 8, 11	dvhopvadd 39034	. . . . 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 1377	. . . 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 38065	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (1^st ‘𝑓) ∈ 𝑇) → (1^st ‘𝑓):𝐵–1-1-onto→𝐵)
41	35, 40	syldan 590	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1^st ‘𝑓):𝐵–1-1-onto→𝐵)
42		f1of 6700	. . . . . 6 ⊢ ((1^st ‘𝑓):𝐵–1-1-onto→𝐵 → (1^st ‘𝑓):𝐵⟶𝐵)
43		fcoi2 6633	. . . . . 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 480	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 𝐷 ∈ Grp)
46	27	adantr 480	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (Base‘𝐷) = 𝐸)
47	37, 46	eleqtrrd 2842	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (2^nd ‘𝑓) ∈ (Base‘𝐷))
48	23, 11, 24	grplid 18524	. . . . . 6 ⊢ ((𝐷 ∈ Grp ∧ (2^nd ‘𝑓) ∈ (Base‘𝐷)) → ( 0 ⨣ (2^nd ‘𝑓)) = (2^nd ‘𝑓))
49	45, 47, 48	syl2anc 583	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ( 0 ⨣ (2^nd ‘𝑓)) = (2^nd ‘𝑓))
50	44, 49	opeq12d 4809	. . . 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 2778	. . 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 7843	. . . . 5 ⊢ (𝑓 ∈ (𝑇 × 𝐸) → 𝑓 = ⟨(1^st ‘𝑓), (2^nd ‘𝑓)⟩)
53	52	adantl 481	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 𝑓 = ⟨(1^st ‘𝑓), (2^nd ‘𝑓)⟩)
54	53	oveq2d 7271	. . 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 2788	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0 ⟩ + 𝑓) = 𝑓)
56	1, 2	ltrncnv 38087	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (1^st ‘𝑓) ∈ 𝑇) → ^◡(1^st ‘𝑓) ∈ 𝑇)
57	35, 56	syldan 590	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ^◡(1^st ‘𝑓) ∈ 𝑇)
58		dvhgrp.i	. . . . . 6 ⊢ 𝐼 = (inv_g‘𝐷)
59	23, 58	grpinvcl 18542	. . . . 5 ⊢ ((𝐷 ∈ Grp ∧ (2^nd ‘𝑓) ∈ (Base‘𝐷)) → (𝐼‘(2^nd ‘𝑓)) ∈ (Base‘𝐷))
60	45, 47, 59	syl2anc 583	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐼‘(2^nd ‘𝑓)) ∈ (Base‘𝐷))
61	60, 46	eleqtrd 2841	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐼‘(2^nd ‘𝑓)) ∈ 𝐸)
62		opelxpi 5617	. . 3 ⊢ ((^◡(1^st ‘𝑓) ∈ 𝑇 ∧ (𝐼‘(2^nd ‘𝑓)) ∈ 𝐸) → ⟨^◡(1^st ‘𝑓), (𝐼‘(2^nd ‘𝑓))⟩ ∈ (𝑇 × 𝐸))
63	57, 61, 62	syl2anc 583	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨^◡(1^st ‘𝑓), (𝐼‘(2^nd ‘𝑓))⟩ ∈ (𝑇 × 𝐸))
64	53	oveq2d 7271	. . 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 39034	. . . . 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 1377	. . . 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 6728	. . . . . 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 18543	. . . . . 6 ⊢ ((𝐷 ∈ Grp ∧ (2^nd ‘𝑓) ∈ (Base‘𝐷)) → ((𝐼‘(2^nd ‘𝑓)) ⨣ (2^nd ‘𝑓)) = 0 )
70	45, 47, 69	syl2anc 583	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ((𝐼‘(2^nd ‘𝑓)) ⨣ (2^nd ‘𝑓)) = 0 )
71	68, 70	opeq12d 4809	. . . 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 2778	. . 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 2778	. 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 18516	1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ Grp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ⟨cop 4564 I cid 5479 × cxp 5578 ^◡ccnv 5579 ↾ cres 5582 ∘ ccom 5584 ⟶wf 6414 –1-1-onto→wf1o 6417 ‘cfv 6418 (class class class)co 7255 1^st c1st 7802 2^nd c2nd 7803 Basecbs 16840 +_gcplusg 16888 Scalarcsca 16891 0_gc0g 17067 Grpcgrp 18492 inv_gcminusg 18493 DivRingcdr 19906 HLchlt 37291 LHypclh 37925 LTrncltrn 38042 TEndoctendo 38693 EDRingcedring 38694 DVecHcdvh 39019

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-riotaBAD 36894

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-undef 8060 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-0g 17069 df-proset 17928 df-poset 17946 df-plt 17963 df-lub 17979 df-glb 17980 df-join 17981 df-meet 17982 df-p0 18058 df-p1 18059 df-lat 18065 df-clat 18132 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-grp 18495 df-minusg 18496 df-mgp 19636 df-ur 19653 df-ring 19700 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-drng 19908 df-oposet 37117 df-ol 37119 df-oml 37120 df-covers 37207 df-ats 37208 df-atl 37239 df-cvlat 37263 df-hlat 37292 df-llines 37439 df-lplanes 37440 df-lvols 37441 df-lines 37442 df-psubsp 37444 df-pmap 37445 df-padd 37737 df-lhyp 37929 df-laut 37930 df-ldil 38045 df-ltrn 38046 df-trl 38100 df-tendo 38696 df-edring 38698 df-dvech 39020

This theorem is referenced by: dvhlveclem 39049

W3C validator