Theorem dochsatshp 38589

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 2823	. . . 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 38248	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
7		dochsatshp.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
8	2, 3, 6, 7	lsatssv 36136	. . 3 ⊢ (𝜑 → 𝑄 ⊆ (Base‘𝑈))
9		eqid 2823	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
10		dochsatshp.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
11	4, 5, 2, 9, 10	dochlss 38492	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑄) ∈ (LSubSp‘𝑈))
12	1, 8, 11	syl2anc 586	. 2 ⊢ (𝜑 → ( ⊥ ‘𝑄) ∈ (LSubSp‘𝑈))
13		eqid 2823	. . . 4 ⊢ (0g‘𝑈) = (0g‘𝑈)
14	13, 3, 6, 7	lsatn0 36137	. . 3 ⊢ (𝜑 → 𝑄 ≠ {(0g‘𝑈)})
15	4, 5, 10, 2, 13	doch0 38496	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{(0g‘𝑈)}) = (Base‘𝑈))
16	1, 15	syl 17	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘{(0g‘𝑈)}) = (Base‘𝑈))
17	16	eqeq2d 2834	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘𝑄) = ( ⊥ ‘{(0g‘𝑈)}) ↔ ( ⊥ ‘𝑄) = (Base‘𝑈)))
18		eqid 2823	. . . . . 6 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
19	4, 5, 18, 3	dih1dimat 38468	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) → 𝑄 ∈ ran ((DIsoH‘𝐾)‘𝑊))
20	1, 7, 19	syl2anc 586	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ran ((DIsoH‘𝐾)‘𝑊))
21	4, 18, 5, 13	dih0rn 38422	. . . . . . 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 38511	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘𝑄) = ( ⊥ ‘{(0g‘𝑈)}) ↔ 𝑄 = {(0g‘𝑈)}))
24	17, 23	bitr3d 283	. . . 4 ⊢ (𝜑 → (( ⊥ ‘𝑄) = (Base‘𝑈) ↔ 𝑄 = {(0g‘𝑈)}))
25	24	necon3bid 3062	. . 3 ⊢ (𝜑 → (( ⊥ ‘𝑄) ≠ (Base‘𝑈) ↔ 𝑄 ≠ {(0g‘𝑈)}))
26	14, 25	mpbird 259	. 2 ⊢ (𝜑 → ( ⊥ ‘𝑄) ≠ (Base‘𝑈))
27		eqid 2823	. . . . . 6 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
28	2, 27, 13, 3	islsat 36129	. . . . 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 234	. . 3 ⊢ (𝜑 → ∃𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)})𝑄 = ((LSpan‘𝑈)‘{𝑣}))
31		eldifi 4105	. . . . . . 7 ⊢ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) → 𝑣 ∈ (Base‘𝑈))
32	31	adantr 483	. . . . . 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 19756	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ LMod ∧ ( ⊥ ‘𝑄) ∈ (LSubSp‘𝑈)) → ((LSpan‘𝑈)‘( ⊥ ‘𝑄)) = ( ⊥ ‘𝑄))
35	6, 12, 34	syl2anc 586	. . . . . . . . . . 11 ⊢ (𝜑 → ((LSpan‘𝑈)‘( ⊥ ‘𝑄)) = ( ⊥ ‘𝑄))
36	35	uneq1d 4140	. . . . . . . . . 10 ⊢ (𝜑 → (((LSpan‘𝑈)‘( ⊥ ‘𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣})) = (( ⊥ ‘𝑄) ∪ ((LSpan‘𝑈)‘{𝑣})))
37	36	fveq2d 6676	. . . . . . . . 9 ⊢ (𝜑 → ((LSpan‘𝑈)‘(((LSpan‘𝑈)‘( ⊥ ‘𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣}))) = ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ ((LSpan‘𝑈)‘{𝑣}))))
38	37	adantr 483	. . . . . . . 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 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → 𝑈 ∈ LMod)
40	2, 9	lssss 19710	. . . . . . . . . . 11 ⊢ (( ⊥ ‘𝑄) ∈ (LSubSp‘𝑈) → ( ⊥ ‘𝑄) ⊆ (Base‘𝑈))
41	12, 40	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ( ⊥ ‘𝑄) ⊆ (Base‘𝑈))
42	41	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ( ⊥ ‘𝑄) ⊆ (Base‘𝑈))
43	31	snssd 4744	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) → {𝑣} ⊆ (Base‘𝑈))
44	43	adantr 483	. . . . . . . . . 10 ⊢ ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → {𝑣} ⊆ (Base‘𝑈))
45	44	adantl 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → {𝑣} ⊆ (Base‘𝑈))
46	2, 27	lspun 19761	. . . . . . . . 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 1367	. . . . . . . 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 4135	. . . . . . . . . . 11 ⊢ (𝑄 = ((LSpan‘𝑈)‘{𝑣}) → (( ⊥ ‘𝑄) ∪ 𝑄) = (( ⊥ ‘𝑄) ∪ ((LSpan‘𝑈)‘{𝑣})))
49	48	fveq2d 6676	. . . . . . . . . 10 ⊢ (𝑄 = ((LSpan‘𝑈)‘{𝑣}) → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)) = ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ ((LSpan‘𝑈)‘{𝑣}))))
50	49	adantl 484	. . . . . . . . 9 ⊢ ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)) = ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ ((LSpan‘𝑈)‘{𝑣}))))
51	50	adantl 484	. . . . . . . 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 2868	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ {𝑣})) = ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)))
53		eqid 2823	. . . . . . . . . . 11 ⊢ ((joinH‘𝐾)‘𝑊) = ((joinH‘𝐾)‘𝑊)
54		eqid 2823	. . . . . . . . . . 11 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
55	4, 18, 5, 2, 10	dochcl 38491	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊))
56	1, 8, 55	syl2anc 586	. . . . . . . . . . 11 ⊢ (𝜑 → ( ⊥ ‘𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊))
57	4, 18, 53, 5, 54, 3, 1, 56, 7	dihjat2 38569	. . . . . . . . . 10 ⊢ (𝜑 → (( ⊥ ‘𝑄)((joinH‘𝐾)‘𝑊)𝑄) = (( ⊥ ‘𝑄)(LSSum‘𝑈)𝑄))
58	4, 5, 2, 53, 1, 41, 8	djhcom 38543	. . . . . . . . . 10 ⊢ (𝜑 → (( ⊥ ‘𝑄)((joinH‘𝐾)‘𝑊)𝑄) = (𝑄((joinH‘𝐾)‘𝑊)( ⊥ ‘𝑄)))
59	9, 3, 6, 7	lsatlssel 36135	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ (LSubSp‘𝑈))
60	9, 27, 54	lsmsp 19860	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ ( ⊥ ‘𝑄) ∈ (LSubSp‘𝑈) ∧ 𝑄 ∈ (LSubSp‘𝑈)) → (( ⊥ ‘𝑄)(LSSum‘𝑈)𝑄) = ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)))
61	6, 12, 59, 60	syl3anc 1367	. . . . . . . . . 10 ⊢ (𝜑 → (( ⊥ ‘𝑄)(LSSum‘𝑈)𝑄) = ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)))
62	57, 58, 61	3eqtr3rd 2867	. . . . . . . . 9 ⊢ (𝜑 → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)) = (𝑄((joinH‘𝐾)‘𝑊)( ⊥ ‘𝑄)))
63	4, 5, 2, 10, 53	djhexmid 38549	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ⊆ (Base‘𝑈)) → (𝑄((joinH‘𝐾)‘𝑊)( ⊥ ‘𝑄)) = (Base‘𝑈))
64	1, 8, 63	syl2anc 586	. . . . . . . . 9 ⊢ (𝜑 → (𝑄((joinH‘𝐾)‘𝑊)( ⊥ ‘𝑄)) = (Base‘𝑈))
65	62, 64	eqtrd 2858	. . . . . . . 8 ⊢ (𝜑 → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)) = (Base‘𝑈))
66	65	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)) = (Base‘𝑈))
67	52, 66	eqtrd 2858	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ {𝑣})) = (Base‘𝑈))
68	67	ex 415	. . . . 5 ⊢ (𝜑 → ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ {𝑣})) = (Base‘𝑈)))
69	33, 68	jcad 515	. . . 4 ⊢ (𝜑 → ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → (𝑣 ∈ (Base‘𝑈) ∧ ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ {𝑣})) = (Base‘𝑈))))
70	69	reximdv2 3273	. . 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 38247	. . 3 ⊢ (𝜑 → 𝑈 ∈ LVec)
73		dochsatshp.y	. . . 4 ⊢ 𝑌 = (LSHyp‘𝑈)
74	2, 27, 9, 73	islshp 36117	. . 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 1338	1 ⊢ (𝜑 → ( ⊥ ‘𝑄) ∈ 𝑌)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ∃wrex 3141   ∖ cdif 3935   ∪ cun 3936   ⊆ wss 3938  {csn 4569  ran crn 5558  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  0_gc0g 16715  LSSumclsm 18761  LModclmod 19636  LSubSpclss 19705  LSpanclspn 19745  LVecclvec 19876  LSAtomsclsa 36112  LSHypclsh 36113  HLchlt 36488  LHypclh 37122  DVecHcdvh 38216  DIsoHcdih 38366  ocHcoch 38485  joinHcdjh 38532

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-riotaBAD 36091

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-tpos 7894  df-undef 7941  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-map 8410  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-n0 11901  df-z 11985  df-uz 12247  df-fz 12896  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-mulr 16581  df-sca 16583  df-vsca 16584  df-0g 16717  df-proset 17540  df-poset 17558  df-plt 17570  df-lub 17586  df-glb 17587  df-join 17588  df-meet 17589  df-p0 17651  df-p1 17652  df-lat 17658  df-clat 17720  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-submnd 17959  df-grp 18108  df-minusg 18109  df-sbg 18110  df-subg 18278  df-cntz 18449  df-lsm 18763  df-cmn 18910  df-abl 18911  df-mgp 19242  df-ur 19254  df-ring 19301  df-oppr 19375  df-dvdsr 19393  df-unit 19394  df-invr 19424  df-dvr 19435  df-drng 19506  df-lmod 19638  df-lss 19706  df-lsp 19746  df-lvec 19877  df-lsatoms 36114  df-lshyp 36115  df-oposet 36314  df-ol 36316  df-oml 36317  df-covers 36404  df-ats 36405  df-atl 36436  df-cvlat 36460  df-hlat 36489  df-llines 36636  df-lplanes 36637  df-lvols 36638  df-lines 36639  df-psubsp 36641  df-pmap 36642  df-padd 36934  df-lhyp 37126  df-laut 37127  df-ldil 37242  df-ltrn 37243  df-trl 37297  df-tgrp 37881  df-tendo 37893  df-edring 37895  df-dveca 38141  df-disoa 38167  df-dvech 38217  df-dib 38277  df-dic 38311  df-dih 38367  df-doch 38486  df-djh 38533

This theorem is referenced by:  dochsatshpb  38590  dochsnshp  38591  dochpolN  38628  lclkrlem2c  38647  lclkrlem2e  38649  mapdordlem2  38775