dvhgrp - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dvhgrp		Structured version Visualization version GIF version

Theorem dvhgrp 41731

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 2762	. . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈)
6	1, 2, 3, 4, 5	dvhvbase 41711	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝑈) = (𝑇 × 𝐸))
7	6	eqcomd 2768	. 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 41719	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸))) → (𝑓 + 𝑔) ∈ (𝑇 × 𝐸))
13	12	3impb 1127	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (𝑓 + 𝑔) ∈ (𝑇 × 𝐸))
14	1, 2, 3, 4, 10, 11, 8	dvhvaddass 41721	. 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 40774	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝑇)
17		eqid 2762	. . . . . . . 8 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
18	1, 17, 4, 10	dvhsca 41706	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
19	1, 17	erngdv 41617	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
20	18, 19	eqeltrd 2862	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ DivRing)
21		drnggrp 20789	. . . . . 6 ⊢ (𝐷 ∈ DivRing → 𝐷 ∈ Grp)
22	20, 21	syl 17	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp)
23		eqid 2762	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
24		dvhgrp.o	. . . . . 6 ⊢ 0 = (0_g‘𝐷)
25	23, 24	grpidcl 19007	. . . . 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 41707	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸)
28	26, 27	eleqtrd 2864	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 ∈ 𝐸)
29		opelxpi 5684	. . 3 ⊢ ((( I ↾ 𝐵) ∈ 𝑇 ∧ 0 ∈ 𝐸) → ⟨( I ↾ 𝐵), 0 ⟩ ∈ (𝑇 × 𝐸))
30	16, 28, 29	syl2anc 593	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ⟨( I ↾ 𝐵), 0 ⟩ ∈ (𝑇 × 𝐸))
31		simpl 486	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
32	16	adantr 484	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ( I ↾ 𝐵) ∈ 𝑇)
33	28	adantr 484	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 0 ∈ 𝐸)
34		xp1st 8002	. . . . . 6 ⊢ (𝑓 ∈ (𝑇 × 𝐸) → (1^st ‘𝑓) ∈ 𝑇)
35	34	adantl 485	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1^st ‘𝑓) ∈ 𝑇)
36		xp2nd 8003	. . . . . 6 ⊢ (𝑓 ∈ (𝑇 × 𝐸) → (2^nd ‘𝑓) ∈ 𝐸)
37	36	adantl 485	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (2^nd ‘𝑓) ∈ 𝐸)
38	1, 2, 3, 4, 10, 8, 11	dvhopvadd 41717	. . . . 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 1398	. . . 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 40748	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (1^st ‘𝑓) ∈ 𝑇) → (1^st ‘𝑓):𝐵–1-1-onto→𝐵)
41	35, 40	syldan 600	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1^st ‘𝑓):𝐵–1-1-onto→𝐵)
42		f1of 6806	. . . . . 6 ⊢ ((1^st ‘𝑓):𝐵–1-1-onto→𝐵 → (1^st ‘𝑓):𝐵⟶𝐵)
43		fcoi2 6739	. . . . . 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 484	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 𝐷 ∈ Grp)
46	27	adantr 484	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (Base‘𝐷) = 𝐸)
47	37, 46	eleqtrrd 2865	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (2^nd ‘𝑓) ∈ (Base‘𝐷))
48	23, 11, 24	grplid 19009	. . . . . 6 ⊢ ((𝐷 ∈ Grp ∧ (2^nd ‘𝑓) ∈ (Base‘𝐷)) → ( 0 ⨣ (2^nd ‘𝑓)) = (2^nd ‘𝑓))
49	45, 47, 48	syl2anc 593	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ( 0 ⨣ (2^nd ‘𝑓)) = (2^nd ‘𝑓))
50	44, 49	opeq12d 4839	. . . 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 2797	. . 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 8009	. . . . 5 ⊢ (𝑓 ∈ (𝑇 × 𝐸) → 𝑓 = ⟨(1^st ‘𝑓), (2^nd ‘𝑓)⟩)
53	52	adantl 485	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 𝑓 = ⟨(1^st ‘𝑓), (2^nd ‘𝑓)⟩)
54	53	oveq2d 7412	. . 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 2807	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0 ⟩ + 𝑓) = 𝑓)
56	1, 2	ltrncnv 40770	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (1^st ‘𝑓) ∈ 𝑇) → ^◡(1^st ‘𝑓) ∈ 𝑇)
57	35, 56	syldan 600	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ^◡(1^st ‘𝑓) ∈ 𝑇)
58		dvhgrp.i	. . . . . 6 ⊢ 𝐼 = (inv_g‘𝐷)
59	23, 58	grpinvcl 19029	. . . . 5 ⊢ ((𝐷 ∈ Grp ∧ (2^nd ‘𝑓) ∈ (Base‘𝐷)) → (𝐼‘(2^nd ‘𝑓)) ∈ (Base‘𝐷))
60	45, 47, 59	syl2anc 593	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐼‘(2^nd ‘𝑓)) ∈ (Base‘𝐷))
61	60, 46	eleqtrd 2864	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐼‘(2^nd ‘𝑓)) ∈ 𝐸)
62		opelxpi 5684	. . 3 ⊢ ((^◡(1^st ‘𝑓) ∈ 𝑇 ∧ (𝐼‘(2^nd ‘𝑓)) ∈ 𝐸) → ⟨^◡(1^st ‘𝑓), (𝐼‘(2^nd ‘𝑓))⟩ ∈ (𝑇 × 𝐸))
63	57, 61, 62	syl2anc 593	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨^◡(1^st ‘𝑓), (𝐼‘(2^nd ‘𝑓))⟩ ∈ (𝑇 × 𝐸))
64	53	oveq2d 7412	. . 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 41717	. . . . 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 1398	. . . 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 6836	. . . . . 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 19031	. . . . . 6 ⊢ ((𝐷 ∈ Grp ∧ (2^nd ‘𝑓) ∈ (Base‘𝐷)) → ((𝐼‘(2^nd ‘𝑓)) ⨣ (2^nd ‘𝑓)) = 0 )
70	45, 47, 69	syl2anc 593	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ((𝐼‘(2^nd ‘𝑓)) ⨣ (2^nd ‘𝑓)) = 0 )
71	68, 70	opeq12d 4839	. . . 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 2797	. . 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 2797	. 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 19000	1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ Grp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1560 ∈ wcel 2142 ⟨cop 4588 I cid 5541 × cxp 5645 ^◡ccnv 5646 ↾ cres 5649 ∘ ccom 5651 ⟶wf 6517 –1-1-onto→wf1o 6520 ‘cfv 6521 (class class class)co 7396 1^st c1st 7968 2^nd c2nd 7969 Basecbs 17245 +_gcplusg 17286 Scalarcsca 17289 0_gc0g 17468 Grpcgrp 18975 inv_gcminusg 18976 DivRingcdr 20779 HLchlt 39974 LHypclh 40608 LTrncltrn 40725 TEndoctendo 41376 EDRingcedring 41377 DVecHcdvh 41702

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-riotaBAD 39577

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4951 df-iin 4952 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-tpos 8206 df-undef 8253 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-er 8678 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-n0 12482 df-z 12569 df-uz 12840 df-fz 13513 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-sca 17302 df-vsca 17303 df-0g 17470 df-proset 18326 df-poset 18345 df-plt 18360 df-lub 18376 df-glb 18377 df-join 18378 df-meet 18379 df-p0 18455 df-p1 18456 df-lat 18464 df-clat 18531 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-grp 18978 df-minusg 18979 df-cmn 19822 df-abl 19823 df-mgp 20187 df-rng 20199 df-ur 20232 df-ring 20285 df-oppr 20386 df-dvdsr 20406 df-unit 20407 df-invr 20437 df-dvr 20450 df-drng 20781 df-oposet 39800 df-ol 39802 df-oml 39803 df-covers 39890 df-ats 39891 df-atl 39922 df-cvlat 39946 df-hlat 39975 df-llines 40122 df-lplanes 40123 df-lvols 40124 df-lines 40125 df-psubsp 40127 df-pmap 40128 df-padd 40420 df-lhyp 40612 df-laut 40613 df-ldil 40728 df-ltrn 40729 df-trl 40783 df-tendo 41379 df-edring 41381 df-dvech 41703

This theorem is referenced by: dvhlveclem 41732

W3C validator