Theorem dochsatshp 41496

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 2731	. . . 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 41155	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
7		dochsatshp.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
8	2, 3, 6, 7	lsatssv 39043	. . 3 ⊢ (𝜑 → 𝑄 ⊆ (Base‘𝑈))
9		eqid 2731	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
10		dochsatshp.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
11	4, 5, 2, 9, 10	dochlss 41399	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑄) ∈ (LSubSp‘𝑈))
12	1, 8, 11	syl2anc 584	. 2 ⊢ (𝜑 → ( ⊥ ‘𝑄) ∈ (LSubSp‘𝑈))
13		eqid 2731	. . . 4 ⊢ (0g‘𝑈) = (0g‘𝑈)
14	13, 3, 6, 7	lsatn0 39044	. . 3 ⊢ (𝜑 → 𝑄 ≠ {(0g‘𝑈)})
15	4, 5, 10, 2, 13	doch0 41403	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{(0g‘𝑈)}) = (Base‘𝑈))
16	1, 15	syl 17	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘{(0g‘𝑈)}) = (Base‘𝑈))
17	16	eqeq2d 2742	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘𝑄) = ( ⊥ ‘{(0g‘𝑈)}) ↔ ( ⊥ ‘𝑄) = (Base‘𝑈)))
18		eqid 2731	. . . . . 6 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
19	4, 5, 18, 3	dih1dimat 41375	. . . . . . 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 41329	. . . . . . 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 41418	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘𝑄) = ( ⊥ ‘{(0g‘𝑈)}) ↔ 𝑄 = {(0g‘𝑈)}))
24	17, 23	bitr3d 281	. . . 4 ⊢ (𝜑 → (( ⊥ ‘𝑄) = (Base‘𝑈) ↔ 𝑄 = {(0g‘𝑈)}))
25	24	necon3bid 2972	. . 3 ⊢ (𝜑 → (( ⊥ ‘𝑄) ≠ (Base‘𝑈) ↔ 𝑄 ≠ {(0g‘𝑈)}))
26	14, 25	mpbird 257	. 2 ⊢ (𝜑 → ( ⊥ ‘𝑄) ≠ (Base‘𝑈))
27		eqid 2731	. . . . . 6 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
28	2, 27, 13, 3	islsat 39036	. . . . 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 4081	. . . . . . 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 20916	. . . . . . . . . . . 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 4117	. . . . . . . . . 10 ⊢ (𝜑 → (((LSpan‘𝑈)‘( ⊥ ‘𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣})) = (( ⊥ ‘𝑄) ∪ ((LSpan‘𝑈)‘{𝑣})))
37	36	fveq2d 6826	. . . . . . . . 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 20870	. . . . . . . . . . 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 4761	. . . . . . . . . . 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 20921	. . . . . . . . 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 4112	. . . . . . . . . . 11 ⊢ (𝑄 = ((LSpan‘𝑈)‘{𝑣}) → (( ⊥ ‘𝑄) ∪ 𝑄) = (( ⊥ ‘𝑄) ∪ ((LSpan‘𝑈)‘{𝑣})))
49	48	fveq2d 6826	. . . . . . . . . 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 2776	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ {𝑣})) = ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)))
53		eqid 2731	. . . . . . . . . . 11 ⊢ ((joinH‘𝐾)‘𝑊) = ((joinH‘𝐾)‘𝑊)
54		eqid 2731	. . . . . . . . . . 11 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
55	4, 18, 5, 2, 10	dochcl 41398	. . . . . . . . . . . 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 41476	. . . . . . . . . 10 ⊢ (𝜑 → (( ⊥ ‘𝑄)((joinH‘𝐾)‘𝑊)𝑄) = (( ⊥ ‘𝑄)(LSSum‘𝑈)𝑄))
58	4, 5, 2, 53, 1, 41, 8	djhcom 41450	. . . . . . . . . 10 ⊢ (𝜑 → (( ⊥ ‘𝑄)((joinH‘𝐾)‘𝑊)𝑄) = (𝑄((joinH‘𝐾)‘𝑊)( ⊥ ‘𝑄)))
59	9, 3, 6, 7	lsatlssel 39042	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ (LSubSp‘𝑈))
60	9, 27, 54	lsmsp 21021	. . . . . . . . . . 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 2775	. . . . . . . . 9 ⊢ (𝜑 → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)) = (𝑄((joinH‘𝐾)‘𝑊)( ⊥ ‘𝑄)))
63	4, 5, 2, 10, 53	djhexmid 41456	. . . . . . . . . 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 2766	. . . . . . . 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 2766	. . . . . 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 3142	. . 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 41154	. . 3 ⊢ (𝜑 → 𝑈 ∈ LVec)
73		dochsatshp.y	. . . 4 ⊢ 𝑌 = (LSHyp‘𝑈)
74	2, 27, 9, 73	islshp 39024	. . 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 1541   ∈ wcel 2111   ≠ wne 2928  ∃wrex 3056   ∖ cdif 3899   ∪ cun 3900   ⊆ wss 3902  {csn 4576  ran crn 5617  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  0_gc0g 17343  LSSumclsm 19547  LModclmod 20794  LSubSpclss 20865  LSpanclspn 20905  LVecclvec 21037  LSAtomsclsa 39019  LSHypclsh 39020  HLchlt 39395  LHypclh 40029  DVecHcdvh 41123  DIsoHcdih 41273  ocHcoch 41392  joinHcdjh 41439

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 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  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 38998

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 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-iin 4944  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  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-submnd 18692  df-grp 18849  df-minusg 18850  df-sbg 18851  df-subg 19036  df-cntz 19230  df-lsm 19549  df-cmn 19695  df-abl 19696  df-mgp 20060  df-rng 20072  df-ur 20101  df-ring 20154  df-oppr 20256  df-dvdsr 20276  df-unit 20277  df-invr 20307  df-dvr 20320  df-drng 20647  df-lmod 20796  df-lss 20866  df-lsp 20906  df-lvec 21038  df-lsatoms 39021  df-lshyp 39022  df-oposet 39221  df-ol 39223  df-oml 39224  df-covers 39311  df-ats 39312  df-atl 39343  df-cvlat 39367  df-hlat 39396  df-llines 39543  df-lplanes 39544  df-lvols 39545  df-lines 39546  df-psubsp 39548  df-pmap 39549  df-padd 39841  df-lhyp 40033  df-laut 40034  df-ldil 40149  df-ltrn 40150  df-trl 40204  df-tgrp 40788  df-tendo 40800  df-edring 40802  df-dveca 41048  df-disoa 41074  df-dvech 41124  df-dib 41184  df-dic 41218  df-dih 41274  df-doch 41393  df-djh 41440

This theorem is referenced by:  dochsatshpb  41497  dochsnshp  41498  dochpolN  41535  lclkrlem2c  41554  lclkrlem2e  41556  mapdordlem2  41682