dvhgrp - Mathbox for Norm Megill

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Theorem dvhgrp 41154

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 2731	. . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈)
6	1, 2, 3, 4, 5	dvhvbase 41134	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝑈) = (𝑇 × 𝐸))
7	6	eqcomd 2737	. 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 41142	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸))) → (𝑓 + 𝑔) ∈ (𝑇 × 𝐸))
13	12	3impb 1114	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (𝑓 + 𝑔) ∈ (𝑇 × 𝐸))
14	1, 2, 3, 4, 10, 11, 8	dvhvaddass 41144	. 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 40197	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝑇)
17		eqid 2731	. . . . . . . 8 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
18	1, 17, 4, 10	dvhsca 41129	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
19	1, 17	erngdv 41040	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
20	18, 19	eqeltrd 2831	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ DivRing)
21		drnggrp 20654	. . . . . 6 ⊢ (𝐷 ∈ DivRing → 𝐷 ∈ Grp)
22	20, 21	syl 17	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp)
23		eqid 2731	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
24		dvhgrp.o	. . . . . 6 ⊢ 0 = (0_g‘𝐷)
25	23, 24	grpidcl 18878	. . . . 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 41130	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸)
28	26, 27	eleqtrd 2833	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 ∈ 𝐸)
29		opelxpi 5651	. . 3 ⊢ ((( I ↾ 𝐵) ∈ 𝑇 ∧ 0 ∈ 𝐸) → ⟨( I ↾ 𝐵), 0 ⟩ ∈ (𝑇 × 𝐸))
30	16, 28, 29	syl2anc 584	. 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 7953	. . . . . 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 7954	. . . . . 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 41140	. . . . 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 1381	. . . 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 40171	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (1^st ‘𝑓) ∈ 𝑇) → (1^st ‘𝑓):𝐵–1-1-onto→𝐵)
41	35, 40	syldan 591	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1^st ‘𝑓):𝐵–1-1-onto→𝐵)
42		f1of 6763	. . . . . 6 ⊢ ((1^st ‘𝑓):𝐵–1-1-onto→𝐵 → (1^st ‘𝑓):𝐵⟶𝐵)
43		fcoi2 6698	. . . . . 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 2834	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (2^nd ‘𝑓) ∈ (Base‘𝐷))
48	23, 11, 24	grplid 18880	. . . . . 6 ⊢ ((𝐷 ∈ Grp ∧ (2^nd ‘𝑓) ∈ (Base‘𝐷)) → ( 0 ⨣ (2^nd ‘𝑓)) = (2^nd ‘𝑓))
49	45, 47, 48	syl2anc 584	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ( 0 ⨣ (2^nd ‘𝑓)) = (2^nd ‘𝑓))
50	44, 49	opeq12d 4830	. . . 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 2766	. . 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 7960	. . . . 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 7362	. . 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 2776	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0 ⟩ + 𝑓) = 𝑓)
56	1, 2	ltrncnv 40193	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (1^st ‘𝑓) ∈ 𝑇) → ^◡(1^st ‘𝑓) ∈ 𝑇)
57	35, 56	syldan 591	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ^◡(1^st ‘𝑓) ∈ 𝑇)
58		dvhgrp.i	. . . . . 6 ⊢ 𝐼 = (inv_g‘𝐷)
59	23, 58	grpinvcl 18900	. . . . 5 ⊢ ((𝐷 ∈ Grp ∧ (2^nd ‘𝑓) ∈ (Base‘𝐷)) → (𝐼‘(2^nd ‘𝑓)) ∈ (Base‘𝐷))
60	45, 47, 59	syl2anc 584	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐼‘(2^nd ‘𝑓)) ∈ (Base‘𝐷))
61	60, 46	eleqtrd 2833	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐼‘(2^nd ‘𝑓)) ∈ 𝐸)
62		opelxpi 5651	. . 3 ⊢ ((^◡(1^st ‘𝑓) ∈ 𝑇 ∧ (𝐼‘(2^nd ‘𝑓)) ∈ 𝐸) → ⟨^◡(1^st ‘𝑓), (𝐼‘(2^nd ‘𝑓))⟩ ∈ (𝑇 × 𝐸))
63	57, 61, 62	syl2anc 584	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨^◡(1^st ‘𝑓), (𝐼‘(2^nd ‘𝑓))⟩ ∈ (𝑇 × 𝐸))
64	53	oveq2d 7362	. . 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 41140	. . . . 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 1381	. . . 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 6792	. . . . . 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 18902	. . . . . 6 ⊢ ((𝐷 ∈ Grp ∧ (2^nd ‘𝑓) ∈ (Base‘𝐷)) → ((𝐼‘(2^nd ‘𝑓)) ⨣ (2^nd ‘𝑓)) = 0 )
70	45, 47, 69	syl2anc 584	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ((𝐼‘(2^nd ‘𝑓)) ⨣ (2^nd ‘𝑓)) = 0 )
71	68, 70	opeq12d 4830	. . . 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 2766	. . 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 2766	. 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 18871	1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ Grp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ⟨cop 4579 I cid 5508 × cxp 5612 ^◡ccnv 5613 ↾ cres 5616 ∘ ccom 5618 ⟶wf 6477 –1-1-onto→wf1o 6480 ‘cfv 6481 (class class class)co 7346 1^st c1st 7919 2^nd c2nd 7920 Basecbs 17120 +_gcplusg 17161 Scalarcsca 17164 0_gc0g 17343 Grpcgrp 18846 inv_gcminusg 18847 DivRingcdr 20644 HLchlt 39397 LHypclh 40031 LTrncltrn 40148 TEndoctendo 40799 EDRingcedring 40800 DVecHcdvh 41125

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-riotaBAD 39000

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-tpos 8156 df-undef 8203 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-n0 12382 df-z 12469 df-uz 12733 df-fz 13408 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-0g 17345 df-proset 18200 df-poset 18219 df-plt 18234 df-lub 18250 df-glb 18251 df-join 18252 df-meet 18253 df-p0 18329 df-p1 18330 df-lat 18338 df-clat 18405 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-grp 18849 df-minusg 18850 df-cmn 19694 df-abl 19695 df-mgp 20059 df-rng 20071 df-ur 20100 df-ring 20153 df-oppr 20255 df-dvdsr 20275 df-unit 20276 df-invr 20306 df-dvr 20319 df-drng 20646 df-oposet 39223 df-ol 39225 df-oml 39226 df-covers 39313 df-ats 39314 df-atl 39345 df-cvlat 39369 df-hlat 39398 df-llines 39545 df-lplanes 39546 df-lvols 39547 df-lines 39548 df-psubsp 39550 df-pmap 39551 df-padd 39843 df-lhyp 40035 df-laut 40036 df-ldil 40151 df-ltrn 40152 df-trl 40206 df-tendo 40802 df-edring 40804 df-dvech 41126

This theorem is referenced by: dvhlveclem 41155

W3C validator