Theorem dochsatshp 41447

Description: The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 27-Jul-2014.) (Revised by Mario Carneiro, 1-Oct-2014.)

Hypotheses

Ref	Expression
dochsatshp.h	⊢ 𝐻 = (LHyp‘𝐾)
dochsatshp.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dochsatshp.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
dochsatshp.a	⊢ 𝐴 = (LSAtoms‘𝑈)
dochsatshp.y	⊢ 𝑌 = (LSHyp‘𝑈)
dochsatshp.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dochsatshp.q	⊢ (𝜑 → 𝑄 ∈ 𝐴)

Assertion

Ref	Expression
dochsatshp	⊢ (𝜑 → ( ⊥ ‘𝑄) ∈ 𝑌)

Proof of Theorem dochsatshp

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dochsatshp.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		eqid 2729	. . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈)
3		dochsatshp.a	. . . 4 ⊢ 𝐴 = (LSAtoms‘𝑈)
4		dochsatshp.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
5		dochsatshp.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
6	4, 5, 1	dvhlmod 41106	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
7		dochsatshp.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
8	2, 3, 6, 7	lsatssv 38994	. . 3 ⊢ (𝜑 → 𝑄 ⊆ (Base‘𝑈))
9		eqid 2729	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
10		dochsatshp.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
11	4, 5, 2, 9, 10	dochlss 41350	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑄) ∈ (LSubSp‘𝑈))
12	1, 8, 11	syl2anc 584	. 2 ⊢ (𝜑 → ( ⊥ ‘𝑄) ∈ (LSubSp‘𝑈))
13		eqid 2729	. . . 4 ⊢ (0g‘𝑈) = (0g‘𝑈)
14	13, 3, 6, 7	lsatn0 38995	. . 3 ⊢ (𝜑 → 𝑄 ≠ {(0g‘𝑈)})
15	4, 5, 10, 2, 13	doch0 41354	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{(0g‘𝑈)}) = (Base‘𝑈))
16	1, 15	syl 17	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘{(0g‘𝑈)}) = (Base‘𝑈))
17	16	eqeq2d 2740	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘𝑄) = ( ⊥ ‘{(0g‘𝑈)}) ↔ ( ⊥ ‘𝑄) = (Base‘𝑈)))
18		eqid 2729	. . . . . 6 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
19	4, 5, 18, 3	dih1dimat 41326	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) → 𝑄 ∈ ran ((DIsoH‘𝐾)‘𝑊))
20	1, 7, 19	syl2anc 584	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ran ((DIsoH‘𝐾)‘𝑊))
21	4, 18, 5, 13	dih0rn 41280	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → {(0g‘𝑈)} ∈ ran ((DIsoH‘𝐾)‘𝑊))
22	1, 21	syl 17	. . . . . 6 ⊢ (𝜑 → {(0g‘𝑈)} ∈ ran ((DIsoH‘𝐾)‘𝑊))
23	4, 18, 10, 1, 20, 22	doch11 41369	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘𝑄) = ( ⊥ ‘{(0g‘𝑈)}) ↔ 𝑄 = {(0g‘𝑈)}))
24	17, 23	bitr3d 281	. . . 4 ⊢ (𝜑 → (( ⊥ ‘𝑄) = (Base‘𝑈) ↔ 𝑄 = {(0g‘𝑈)}))
25	24	necon3bid 2969	. . 3 ⊢ (𝜑 → (( ⊥ ‘𝑄) ≠ (Base‘𝑈) ↔ 𝑄 ≠ {(0g‘𝑈)}))
26	14, 25	mpbird 257	. 2 ⊢ (𝜑 → ( ⊥ ‘𝑄) ≠ (Base‘𝑈))
27		eqid 2729	. . . . . 6 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
28	2, 27, 13, 3	islsat 38987	. . . . 5 ⊢ (𝑈 ∈ LMod → (𝑄 ∈ 𝐴 ↔ ∃𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)})𝑄 = ((LSpan‘𝑈)‘{𝑣})))
29	6, 28	syl 17	. . . 4 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ↔ ∃𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)})𝑄 = ((LSpan‘𝑈)‘{𝑣})))
30	7, 29	mpbid 232	. . 3 ⊢ (𝜑 → ∃𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)})𝑄 = ((LSpan‘𝑈)‘{𝑣}))
31		eldifi 4078	. . . . . . 7 ⊢ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) → 𝑣 ∈ (Base‘𝑈))
32	31	adantr 480	. . . . . 6 ⊢ ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → 𝑣 ∈ (Base‘𝑈))
33	32	a1i 11	. . . . 5 ⊢ (𝜑 → ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → 𝑣 ∈ (Base‘𝑈)))
34	9, 27	lspid 20869	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ LMod ∧ ( ⊥ ‘𝑄) ∈ (LSubSp‘𝑈)) → ((LSpan‘𝑈)‘( ⊥ ‘𝑄)) = ( ⊥ ‘𝑄))
35	6, 12, 34	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → ((LSpan‘𝑈)‘( ⊥ ‘𝑄)) = ( ⊥ ‘𝑄))
36	35	uneq1d 4114	. . . . . . . . . 10 ⊢ (𝜑 → (((LSpan‘𝑈)‘( ⊥ ‘𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣})) = (( ⊥ ‘𝑄) ∪ ((LSpan‘𝑈)‘{𝑣})))
37	36	fveq2d 6820	. . . . . . . . 9 ⊢ (𝜑 → ((LSpan‘𝑈)‘(((LSpan‘𝑈)‘( ⊥ ‘𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣}))) = ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ ((LSpan‘𝑈)‘{𝑣}))))
38	37	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(((LSpan‘𝑈)‘( ⊥ ‘𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣}))) = ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ ((LSpan‘𝑈)‘{𝑣}))))
39	6	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → 𝑈 ∈ LMod)
40	2, 9	lssss 20823	. . . . . . . . . . 11 ⊢ (( ⊥ ‘𝑄) ∈ (LSubSp‘𝑈) → ( ⊥ ‘𝑄) ⊆ (Base‘𝑈))
41	12, 40	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ( ⊥ ‘𝑄) ⊆ (Base‘𝑈))
42	41	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ( ⊥ ‘𝑄) ⊆ (Base‘𝑈))
43	31	snssd 4758	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) → {𝑣} ⊆ (Base‘𝑈))
44	43	adantr 480	. . . . . . . . . 10 ⊢ ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → {𝑣} ⊆ (Base‘𝑈))
45	44	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → {𝑣} ⊆ (Base‘𝑈))
46	2, 27	lspun 20874	. . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ ( ⊥ ‘𝑄) ⊆ (Base‘𝑈) ∧ {𝑣} ⊆ (Base‘𝑈)) → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ {𝑣})) = ((LSpan‘𝑈)‘(((LSpan‘𝑈)‘( ⊥ ‘𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣}))))
47	39, 42, 45, 46	syl3anc 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ {𝑣})) = ((LSpan‘𝑈)‘(((LSpan‘𝑈)‘( ⊥ ‘𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣}))))
48		uneq2 4109	. . . . . . . . . . 11 ⊢ (𝑄 = ((LSpan‘𝑈)‘{𝑣}) → (( ⊥ ‘𝑄) ∪ 𝑄) = (( ⊥ ‘𝑄) ∪ ((LSpan‘𝑈)‘{𝑣})))
49	48	fveq2d 6820	. . . . . . . . . 10 ⊢ (𝑄 = ((LSpan‘𝑈)‘{𝑣}) → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)) = ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ ((LSpan‘𝑈)‘{𝑣}))))
50	49	adantl 481	. . . . . . . . 9 ⊢ ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)) = ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ ((LSpan‘𝑈)‘{𝑣}))))
51	50	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)) = ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ ((LSpan‘𝑈)‘{𝑣}))))
52	38, 47, 51	3eqtr4d 2774	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ {𝑣})) = ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)))
53		eqid 2729	. . . . . . . . . . 11 ⊢ ((joinH‘𝐾)‘𝑊) = ((joinH‘𝐾)‘𝑊)
54		eqid 2729	. . . . . . . . . . 11 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
55	4, 18, 5, 2, 10	dochcl 41349	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊))
56	1, 8, 55	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → ( ⊥ ‘𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊))
57	4, 18, 53, 5, 54, 3, 1, 56, 7	dihjat2 41427	. . . . . . . . . 10 ⊢ (𝜑 → (( ⊥ ‘𝑄)((joinH‘𝐾)‘𝑊)𝑄) = (( ⊥ ‘𝑄)(LSSum‘𝑈)𝑄))
58	4, 5, 2, 53, 1, 41, 8	djhcom 41401	. . . . . . . . . 10 ⊢ (𝜑 → (( ⊥ ‘𝑄)((joinH‘𝐾)‘𝑊)𝑄) = (𝑄((joinH‘𝐾)‘𝑊)( ⊥ ‘𝑄)))
59	9, 3, 6, 7	lsatlssel 38993	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ (LSubSp‘𝑈))
60	9, 27, 54	lsmsp 20974	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ ( ⊥ ‘𝑄) ∈ (LSubSp‘𝑈) ∧ 𝑄 ∈ (LSubSp‘𝑈)) → (( ⊥ ‘𝑄)(LSSum‘𝑈)𝑄) = ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)))
61	6, 12, 59, 60	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → (( ⊥ ‘𝑄)(LSSum‘𝑈)𝑄) = ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)))
62	57, 58, 61	3eqtr3rd 2773	. . . . . . . . 9 ⊢ (𝜑 → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)) = (𝑄((joinH‘𝐾)‘𝑊)( ⊥ ‘𝑄)))
63	4, 5, 2, 10, 53	djhexmid 41407	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ⊆ (Base‘𝑈)) → (𝑄((joinH‘𝐾)‘𝑊)( ⊥ ‘𝑄)) = (Base‘𝑈))
64	1, 8, 63	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑄((joinH‘𝐾)‘𝑊)( ⊥ ‘𝑄)) = (Base‘𝑈))
65	62, 64	eqtrd 2764	. . . . . . . 8 ⊢ (𝜑 → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)) = (Base‘𝑈))
66	65	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)) = (Base‘𝑈))
67	52, 66	eqtrd 2764	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ {𝑣})) = (Base‘𝑈))
68	67	ex 412	. . . . 5 ⊢ (𝜑 → ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ {𝑣})) = (Base‘𝑈)))
69	33, 68	jcad 512	. . . 4 ⊢ (𝜑 → ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → (𝑣 ∈ (Base‘𝑈) ∧ ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ {𝑣})) = (Base‘𝑈))))
70	69	reximdv2 3139	. . 3 ⊢ (𝜑 → (∃𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)})𝑄 = ((LSpan‘𝑈)‘{𝑣}) → ∃𝑣 ∈ (Base‘𝑈)((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ {𝑣})) = (Base‘𝑈)))
71	30, 70	mpd 15	. 2 ⊢ (𝜑 → ∃𝑣 ∈ (Base‘𝑈)((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ {𝑣})) = (Base‘𝑈))
72	4, 5, 1	dvhlvec 41105	. . 3 ⊢ (𝜑 → 𝑈 ∈ LVec)
73		dochsatshp.y	. . . 4 ⊢ 𝑌 = (LSHyp‘𝑈)
74	2, 27, 9, 73	islshp 38975	. . 3 ⊢ (𝑈 ∈ LVec → (( ⊥ ‘𝑄) ∈ 𝑌 ↔ (( ⊥ ‘𝑄) ∈ (LSubSp‘𝑈) ∧ ( ⊥ ‘𝑄) ≠ (Base‘𝑈) ∧ ∃𝑣 ∈ (Base‘𝑈)((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ {𝑣})) = (Base‘𝑈))))
75	72, 74	syl 17	. 2 ⊢ (𝜑 → (( ⊥ ‘𝑄) ∈ 𝑌 ↔ (( ⊥ ‘𝑄) ∈ (LSubSp‘𝑈) ∧ ( ⊥ ‘𝑄) ≠ (Base‘𝑈) ∧ ∃𝑣 ∈ (Base‘𝑈)((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ {𝑣})) = (Base‘𝑈))))
76	12, 26, 71, 75	mpbir3and 1343	1 ⊢ (𝜑 → ( ⊥ ‘𝑄) ∈ 𝑌)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899  {csn 4573  ran crn 5614  ‘cfv 6476  (class class class)co 7340  Basecbs 17107  0_gc0g 17330  LSSumclsm 19500  LModclmod 20747  LSubSpclss 20818  LSpanclspn 20858  LVecclvec 20990  LSAtomsclsa 38970  LSHypclsh 38971  HLchlt 39346  LHypclh 39980  DVecHcdvh 41074  DIsoHcdih 41224  ocHcoch 41343  joinHcdjh 41390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5214  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5367  ax-un 7662  ax-cnex 11053  ax-resscn 11054  ax-1cn 11055  ax-icn 11056  ax-addcl 11057  ax-addrcl 11058  ax-mulcl 11059  ax-mulrcl 11060  ax-mulcom 11061  ax-addass 11062  ax-mulass 11063  ax-distr 11064  ax-i2m1 11065  ax-1ne0 11066  ax-1rid 11067  ax-rnegex 11068  ax-rrecex 11069  ax-cnre 11070  ax-pre-lttri 11071  ax-pre-lttrn 11072  ax-pre-ltadd 11073  ax-pre-mulgt0 11074  ax-riotaBAD 38949

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3393  df-v 3435  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  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-int 4895  df-iun 4940  df-iin 4941  df-br 5089  df-opab 5151  df-mpt 5170  df-tr 5196  df-id 5508  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5566  df-we 5568  df-xp 5619  df-rel 5620  df-cnv 5621  df-co 5622  df-dm 5623  df-rn 5624  df-res 5625  df-ima 5626  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7297  df-ov 7343  df-oprab 7344  df-mpo 7345  df-om 7791  df-1st 7915  df-2nd 7916  df-tpos 8150  df-undef 8197  df-frecs 8205  df-wrecs 8236  df-recs 8285  df-rdg 8323  df-1o 8379  df-er 8616  df-map 8746  df-en 8864  df-dom 8865  df-sdom 8866  df-fin 8867  df-pnf 11139  df-mnf 11140  df-xr 11141  df-ltxr 11142  df-le 11143  df-sub 11337  df-neg 11338  df-nn 12117  df-2 12179  df-3 12180  df-4 12181  df-5 12182  df-6 12183  df-n0 12373  df-z 12460  df-uz 12724  df-fz 13399  df-struct 17045  df-sets 17062  df-slot 17080  df-ndx 17092  df-base 17108  df-ress 17129  df-plusg 17161  df-mulr 17162  df-sca 17164  df-vsca 17165  df-0g 17332  df-proset 18187  df-poset 18206  df-plt 18221  df-lub 18237  df-glb 18238  df-join 18239  df-meet 18240  df-p0 18316  df-p1 18317  df-lat 18325  df-clat 18392  df-mgm 18501  df-sgrp 18580  df-mnd 18596  df-submnd 18645  df-grp 18802  df-minusg 18803  df-sbg 18804  df-subg 18989  df-cntz 19183  df-lsm 19502  df-cmn 19648  df-abl 19649  df-mgp 20013  df-rng 20025  df-ur 20054  df-ring 20107  df-oppr 20209  df-dvdsr 20229  df-unit 20230  df-invr 20260  df-dvr 20273  df-drng 20600  df-lmod 20749  df-lss 20819  df-lsp 20859  df-lvec 20991  df-lsatoms 38972  df-lshyp 38973  df-oposet 39172  df-ol 39174  df-oml 39175  df-covers 39262  df-ats 39263  df-atl 39294  df-cvlat 39318  df-hlat 39347  df-llines 39494  df-lplanes 39495  df-lvols 39496  df-lines 39497  df-psubsp 39499  df-pmap 39500  df-padd 39792  df-lhyp 39984  df-laut 39985  df-ldil 40100  df-ltrn 40101  df-trl 40155  df-tgrp 40739  df-tendo 40751  df-edring 40753  df-dveca 40999  df-disoa 41025  df-dvech 41075  df-dib 41135  df-dic 41169  df-dih 41225  df-doch 41344  df-djh 41391

This theorem is referenced by:  dochsatshpb  41448  dochsnshp  41449  dochpolN  41486  lclkrlem2c  41505  lclkrlem2e  41507  mapdordlem2  41633