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 41941

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 2765	. . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈)
6	1, 2, 3, 4, 5	dvhvbase 41921	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝑈) = (𝑇 × 𝐸))
7	6	eqcomd 2771	. 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 41929	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸))) → (𝑓 + 𝑔) ∈ (𝑇 × 𝐸))
13	12	3impb 1132	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (𝑓 + 𝑔) ∈ (𝑇 × 𝐸))
14	1, 2, 3, 4, 10, 11, 8	dvhvaddass 41931	. 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 40984	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝑇)
17		eqid 2765	. . . . . . . 8 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
18	1, 17, 4, 10	dvhsca 41916	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
19	1, 17	erngdv 41827	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
20	18, 19	eqeltrd 2865	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ DivRing)
21		drnggrp 20889	. . . . . 6 ⊢ (𝐷 ∈ DivRing → 𝐷 ∈ Grp)
22	20, 21	syl 18	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp)
23		eqid 2765	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
24		dvhgrp.o	. . . . . 6 ⊢ 0 = (0_g‘𝐷)
25	23, 24	grpidcl 19078	. . . . 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 41917	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸)
28	26, 27	eleqtrd 2867	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 ∈ 𝐸)
29		opelxpi 5700	. . 3 ⊢ ((( I ↾ 𝐵) ∈ 𝑇 ∧ 0 ∈ 𝐸) → ⟨( I ↾ 𝐵), 0 ⟩ ∈ (𝑇 × 𝐸))
30	16, 28, 29	syl2anc 596	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ⟨( I ↾ 𝐵), 0 ⟩ ∈ (𝑇 × 𝐸))
31		simpl 488	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
32	16	adantr 486	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ( I ↾ 𝐵) ∈ 𝑇)
33	28	adantr 486	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 0 ∈ 𝐸)
34		xp1st 8024	. . . . . 6 ⊢ (𝑓 ∈ (𝑇 × 𝐸) → (1^st ‘𝑓) ∈ 𝑇)
35	34	adantl 487	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1^st ‘𝑓) ∈ 𝑇)
36		xp2nd 8025	. . . . . 6 ⊢ (𝑓 ∈ (𝑇 × 𝐸) → (2^nd ‘𝑓) ∈ 𝐸)
37	36	adantl 487	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (2^nd ‘𝑓) ∈ 𝐸)
38	1, 2, 3, 4, 10, 8, 11	dvhopvadd 41927	. . . . 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 1406	. . . 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 40958	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (1^st ‘𝑓) ∈ 𝑇) → (1^st ‘𝑓):𝐵–1-1-onto→𝐵)
41	35, 40	syldan 603	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (1^st ‘𝑓):𝐵–1-1-onto→𝐵)
42		f1of 6824	. . . . . 6 ⊢ ((1^st ‘𝑓):𝐵–1-1-onto→𝐵 → (1^st ‘𝑓):𝐵⟶𝐵)
43		fcoi2 6757	. . . . . 6 ⊢ ((1^st ‘𝑓):𝐵⟶𝐵 → (( I ↾ 𝐵) ∘ (1^st ‘𝑓)) = (1^st ‘𝑓))
44	41, 42, 43	3syl 19	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (( I ↾ 𝐵) ∘ (1^st ‘𝑓)) = (1^st ‘𝑓))
45	22	adantr 486	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 𝐷 ∈ Grp)
46	27	adantr 486	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (Base‘𝐷) = 𝐸)
47	37, 46	eleqtrrd 2868	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (2^nd ‘𝑓) ∈ (Base‘𝐷))
48	23, 11, 24	grplid 19080	. . . . . 6 ⊢ ((𝐷 ∈ Grp ∧ (2^nd ‘𝑓) ∈ (Base‘𝐷)) → ( 0 ⨣ (2^nd ‘𝑓)) = (2^nd ‘𝑓))
49	45, 47, 48	syl2anc 596	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ( 0 ⨣ (2^nd ‘𝑓)) = (2^nd ‘𝑓))
50	44, 49	opeq12d 4848	. . . 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 2800	. . 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 8031	. . . . 5 ⊢ (𝑓 ∈ (𝑇 × 𝐸) → 𝑓 = ⟨(1^st ‘𝑓), (2^nd ‘𝑓)⟩)
53	52	adantl 487	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → 𝑓 = ⟨(1^st ‘𝑓), (2^nd ‘𝑓)⟩)
54	53	oveq2d 7435	. . 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 2810	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (⟨( I ↾ 𝐵), 0 ⟩ + 𝑓) = 𝑓)
56	1, 2	ltrncnv 40980	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (1^st ‘𝑓) ∈ 𝑇) → ^◡(1^st ‘𝑓) ∈ 𝑇)
57	35, 56	syldan 603	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ^◡(1^st ‘𝑓) ∈ 𝑇)
58		dvhgrp.i	. . . . . 6 ⊢ 𝐼 = (inv_g‘𝐷)
59	23, 58	grpinvcl 19100	. . . . 5 ⊢ ((𝐷 ∈ Grp ∧ (2^nd ‘𝑓) ∈ (Base‘𝐷)) → (𝐼‘(2^nd ‘𝑓)) ∈ (Base‘𝐷))
60	45, 47, 59	syl2anc 596	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐼‘(2^nd ‘𝑓)) ∈ (Base‘𝐷))
61	60, 46	eleqtrd 2867	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (𝐼‘(2^nd ‘𝑓)) ∈ 𝐸)
62		opelxpi 5700	. . 3 ⊢ ((^◡(1^st ‘𝑓) ∈ 𝑇 ∧ (𝐼‘(2^nd ‘𝑓)) ∈ 𝐸) → ⟨^◡(1^st ‘𝑓), (𝐼‘(2^nd ‘𝑓))⟩ ∈ (𝑇 × 𝐸))
63	57, 61, 62	syl2anc 596	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ⟨^◡(1^st ‘𝑓), (𝐼‘(2^nd ‘𝑓))⟩ ∈ (𝑇 × 𝐸))
64	53	oveq2d 7435	. . 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 41927	. . . . 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 1406	. . . 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 6854	. . . . . 6 ⊢ ((1^st ‘𝑓):𝐵–1-1-onto→𝐵 → (^◡(1^st ‘𝑓) ∘ (1^st ‘𝑓)) = ( I ↾ 𝐵))
68	41, 67	syl 18	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → (^◡(1^st ‘𝑓) ∘ (1^st ‘𝑓)) = ( I ↾ 𝐵))
69	23, 11, 24, 58	grplinv 19102	. . . . . 6 ⊢ ((𝐷 ∈ Grp ∧ (2^nd ‘𝑓) ∈ (Base‘𝐷)) → ((𝐼‘(2^nd ‘𝑓)) ⨣ (2^nd ‘𝑓)) = 0 )
70	45, 47, 69	syl2anc 596	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸)) → ((𝐼‘(2^nd ‘𝑓)) ⨣ (2^nd ‘𝑓)) = 0 )
71	68, 70	opeq12d 4848	. . . 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 2800	. . 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 2800	. 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 19071	1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ Grp)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ⟨cop 4597 I cid 5557 × cxp 5661 ^◡ccnv 5662 ↾ cres 5665 ∘ ccom 5667 ⟶wf 6536 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7419 1^st c1st 7990 2^nd c2nd 7991 Basecbs 17293 +_gcplusg 17334 Scalarcsca 17337 0_gc0g 17516 Grpcgrp 19046 inv_gcminusg 19047 DivRingcdr 20879 HLchlt 40184 LHypclh 40818 LTrncltrn 40935 TEndoctendo 41586 EDRingcedring 41587 DVecHcdvh 41912

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194 ax-riotaBAD 39787

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-tpos 8228 df-undef 8275 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-nn 12251 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-n0 12522 df-z 12609 df-uz 12881 df-fz 13554 df-struct 17231 df-sets 17248 df-slot 17266 df-ndx 17278 df-base 17294 df-ress 17315 df-plusg 17347 df-mulr 17348 df-sca 17350 df-vsca 17351 df-0g 17518 df-proset 18374 df-poset 18393 df-plt 18408 df-lub 18424 df-glb 18425 df-join 18426 df-meet 18427 df-p0 18503 df-p1 18504 df-lat 18512 df-clat 18579 df-mgm 18722 df-sgrp 18811 df-mnd 18827 df-grp 19049 df-minusg 19050 df-cmn 19898 df-abl 19899 df-mgp 20263 df-rng 20277 df-ur 20310 df-ring 20363 df-oppr 20467 df-dvdsr 20487 df-unit 20488 df-invr 20518 df-dvr 20531 df-drng 20881 df-oposet 40010 df-ol 40012 df-oml 40013 df-covers 40100 df-ats 40101 df-atl 40132 df-cvlat 40156 df-hlat 40185 df-llines 40332 df-lplanes 40333 df-lvols 40334 df-lines 40335 df-psubsp 40337 df-pmap 40338 df-padd 40630 df-lhyp 40822 df-laut 40823 df-ldil 40938 df-ltrn 40939 df-trl 40993 df-tendo 41589 df-edring 41591 df-dvech 41913

This theorem is used by: dvhlveclem 41942

W3C validator