Theorem dochsatshp 41408

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 2740	. . . 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 41067	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
7		dochsatshp.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
8	2, 3, 6, 7	lsatssv 38954	. . 3 ⊢ (𝜑 → 𝑄 ⊆ (Base‘𝑈))
9		eqid 2740	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
10		dochsatshp.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
11	4, 5, 2, 9, 10	dochlss 41311	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑄) ∈ (LSubSp‘𝑈))
12	1, 8, 11	syl2anc 583	. 2 ⊢ (𝜑 → ( ⊥ ‘𝑄) ∈ (LSubSp‘𝑈))
13		eqid 2740	. . . 4 ⊢ (0g‘𝑈) = (0g‘𝑈)
14	13, 3, 6, 7	lsatn0 38955	. . 3 ⊢ (𝜑 → 𝑄 ≠ {(0g‘𝑈)})
15	4, 5, 10, 2, 13	doch0 41315	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{(0g‘𝑈)}) = (Base‘𝑈))
16	1, 15	syl 17	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘{(0g‘𝑈)}) = (Base‘𝑈))
17	16	eqeq2d 2751	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘𝑄) = ( ⊥ ‘{(0g‘𝑈)}) ↔ ( ⊥ ‘𝑄) = (Base‘𝑈)))
18		eqid 2740	. . . . . 6 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
19	4, 5, 18, 3	dih1dimat 41287	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) → 𝑄 ∈ ran ((DIsoH‘𝐾)‘𝑊))
20	1, 7, 19	syl2anc 583	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ran ((DIsoH‘𝐾)‘𝑊))
21	4, 18, 5, 13	dih0rn 41241	. . . . . . 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 41330	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘𝑄) = ( ⊥ ‘{(0g‘𝑈)}) ↔ 𝑄 = {(0g‘𝑈)}))
24	17, 23	bitr3d 281	. . . 4 ⊢ (𝜑 → (( ⊥ ‘𝑄) = (Base‘𝑈) ↔ 𝑄 = {(0g‘𝑈)}))
25	24	necon3bid 2991	. . 3 ⊢ (𝜑 → (( ⊥ ‘𝑄) ≠ (Base‘𝑈) ↔ 𝑄 ≠ {(0g‘𝑈)}))
26	14, 25	mpbird 257	. 2 ⊢ (𝜑 → ( ⊥ ‘𝑄) ≠ (Base‘𝑈))
27		eqid 2740	. . . . . 6 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
28	2, 27, 13, 3	islsat 38947	. . . . 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 4154	. . . . . . 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 21003	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ LMod ∧ ( ⊥ ‘𝑄) ∈ (LSubSp‘𝑈)) → ((LSpan‘𝑈)‘( ⊥ ‘𝑄)) = ( ⊥ ‘𝑄))
35	6, 12, 34	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → ((LSpan‘𝑈)‘( ⊥ ‘𝑄)) = ( ⊥ ‘𝑄))
36	35	uneq1d 4190	. . . . . . . . . 10 ⊢ (𝜑 → (((LSpan‘𝑈)‘( ⊥ ‘𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣})) = (( ⊥ ‘𝑄) ∪ ((LSpan‘𝑈)‘{𝑣})))
37	36	fveq2d 6924	. . . . . . . . 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 20957	. . . . . . . . . . 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 4834	. . . . . . . . . . 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 21008	. . . . . . . . 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 1371	. . . . . . . 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 4185	. . . . . . . . . . 11 ⊢ (𝑄 = ((LSpan‘𝑈)‘{𝑣}) → (( ⊥ ‘𝑄) ∪ 𝑄) = (( ⊥ ‘𝑄) ∪ ((LSpan‘𝑈)‘{𝑣})))
49	48	fveq2d 6924	. . . . . . . . . 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 2790	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ {𝑣})) = ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)))
53		eqid 2740	. . . . . . . . . . 11 ⊢ ((joinH‘𝐾)‘𝑊) = ((joinH‘𝐾)‘𝑊)
54		eqid 2740	. . . . . . . . . . 11 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
55	4, 18, 5, 2, 10	dochcl 41310	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊))
56	1, 8, 55	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → ( ⊥ ‘𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊))
57	4, 18, 53, 5, 54, 3, 1, 56, 7	dihjat2 41388	. . . . . . . . . 10 ⊢ (𝜑 → (( ⊥ ‘𝑄)((joinH‘𝐾)‘𝑊)𝑄) = (( ⊥ ‘𝑄)(LSSum‘𝑈)𝑄))
58	4, 5, 2, 53, 1, 41, 8	djhcom 41362	. . . . . . . . . 10 ⊢ (𝜑 → (( ⊥ ‘𝑄)((joinH‘𝐾)‘𝑊)𝑄) = (𝑄((joinH‘𝐾)‘𝑊)( ⊥ ‘𝑄)))
59	9, 3, 6, 7	lsatlssel 38953	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ (LSubSp‘𝑈))
60	9, 27, 54	lsmsp 21108	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ ( ⊥ ‘𝑄) ∈ (LSubSp‘𝑈) ∧ 𝑄 ∈ (LSubSp‘𝑈)) → (( ⊥ ‘𝑄)(LSSum‘𝑈)𝑄) = ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)))
61	6, 12, 59, 60	syl3anc 1371	. . . . . . . . . 10 ⊢ (𝜑 → (( ⊥ ‘𝑄)(LSSum‘𝑈)𝑄) = ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)))
62	57, 58, 61	3eqtr3rd 2789	. . . . . . . . 9 ⊢ (𝜑 → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)) = (𝑄((joinH‘𝐾)‘𝑊)( ⊥ ‘𝑄)))
63	4, 5, 2, 10, 53	djhexmid 41368	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ⊆ (Base‘𝑈)) → (𝑄((joinH‘𝐾)‘𝑊)( ⊥ ‘𝑄)) = (Base‘𝑈))
64	1, 8, 63	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → (𝑄((joinH‘𝐾)‘𝑊)( ⊥ ‘𝑄)) = (Base‘𝑈))
65	62, 64	eqtrd 2780	. . . . . . . 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 2780	. . . . . 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 3170	. . 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 41066	. . 3 ⊢ (𝜑 → 𝑈 ∈ LVec)
73		dochsatshp.y	. . . 4 ⊢ 𝑌 = (LSHyp‘𝑈)
74	2, 27, 9, 73	islshp 38935	. . 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 1342	1 ⊢ (𝜑 → ( ⊥ ‘𝑄) ∈ 𝑌)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∃wrex 3076   ∖ cdif 3973   ∪ cun 3974   ⊆ wss 3976  {csn 4648  ran crn 5701  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  0_gc0g 17499  LSSumclsm 19676  LModclmod 20880  LSubSpclss 20952  LSpanclspn 20992  LVecclvec 21124  LSAtomsclsa 38930  LSHypclsh 38931  HLchlt 39306  LHypclh 39941  DVecHcdvh 41035  DIsoHcdih 41185  ocHcoch 41304  joinHcdjh 41351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-riotaBAD 38909

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-tpos 8267  df-undef 8314  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-0g 17501  df-proset 18365  df-poset 18383  df-plt 18400  df-lub 18416  df-glb 18417  df-join 18418  df-meet 18419  df-p0 18495  df-p1 18496  df-lat 18502  df-clat 18569  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-submnd 18819  df-grp 18976  df-minusg 18977  df-sbg 18978  df-subg 19163  df-cntz 19357  df-lsm 19678  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-ring 20262  df-oppr 20360  df-dvdsr 20383  df-unit 20384  df-invr 20414  df-dvr 20427  df-drng 20753  df-lmod 20882  df-lss 20953  df-lsp 20993  df-lvec 21125  df-lsatoms 38932  df-lshyp 38933  df-oposet 39132  df-ol 39134  df-oml 39135  df-covers 39222  df-ats 39223  df-atl 39254  df-cvlat 39278  df-hlat 39307  df-llines 39455  df-lplanes 39456  df-lvols 39457  df-lines 39458  df-psubsp 39460  df-pmap 39461  df-padd 39753  df-lhyp 39945  df-laut 39946  df-ldil 40061  df-ltrn 40062  df-trl 40116  df-tgrp 40700  df-tendo 40712  df-edring 40714  df-dveca 40960  df-disoa 40986  df-dvech 41036  df-dib 41096  df-dic 41130  df-dih 41186  df-doch 41305  df-djh 41352

This theorem is referenced by:  dochsatshpb  41409  dochsnshp  41410  dochpolN  41447  lclkrlem2c  41466  lclkrlem2e  41468  mapdordlem2  41594