Theorem dochsatshp 39465

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 2738	. . . 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 39124	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
7		dochsatshp.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
8	2, 3, 6, 7	lsatssv 37012	. . 3 ⊢ (𝜑 → 𝑄 ⊆ (Base‘𝑈))
9		eqid 2738	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
10		dochsatshp.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
11	4, 5, 2, 9, 10	dochlss 39368	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑄) ∈ (LSubSp‘𝑈))
12	1, 8, 11	syl2anc 584	. 2 ⊢ (𝜑 → ( ⊥ ‘𝑄) ∈ (LSubSp‘𝑈))
13		eqid 2738	. . . 4 ⊢ (0g‘𝑈) = (0g‘𝑈)
14	13, 3, 6, 7	lsatn0 37013	. . 3 ⊢ (𝜑 → 𝑄 ≠ {(0g‘𝑈)})
15	4, 5, 10, 2, 13	doch0 39372	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{(0g‘𝑈)}) = (Base‘𝑈))
16	1, 15	syl 17	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘{(0g‘𝑈)}) = (Base‘𝑈))
17	16	eqeq2d 2749	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘𝑄) = ( ⊥ ‘{(0g‘𝑈)}) ↔ ( ⊥ ‘𝑄) = (Base‘𝑈)))
18		eqid 2738	. . . . . 6 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
19	4, 5, 18, 3	dih1dimat 39344	. . . . . . 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 39298	. . . . . . 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 39387	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘𝑄) = ( ⊥ ‘{(0g‘𝑈)}) ↔ 𝑄 = {(0g‘𝑈)}))
24	17, 23	bitr3d 280	. . . 4 ⊢ (𝜑 → (( ⊥ ‘𝑄) = (Base‘𝑈) ↔ 𝑄 = {(0g‘𝑈)}))
25	24	necon3bid 2988	. . 3 ⊢ (𝜑 → (( ⊥ ‘𝑄) ≠ (Base‘𝑈) ↔ 𝑄 ≠ {(0g‘𝑈)}))
26	14, 25	mpbird 256	. 2 ⊢ (𝜑 → ( ⊥ ‘𝑄) ≠ (Base‘𝑈))
27		eqid 2738	. . . . . 6 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
28	2, 27, 13, 3	islsat 37005	. . . . 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 231	. . 3 ⊢ (𝜑 → ∃𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)})𝑄 = ((LSpan‘𝑈)‘{𝑣}))
31		eldifi 4061	. . . . . . 7 ⊢ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) → 𝑣 ∈ (Base‘𝑈))
32	31	adantr 481	. . . . . 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 20244	. . . . . . . . . . . 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 4096	. . . . . . . . . 10 ⊢ (𝜑 → (((LSpan‘𝑈)‘( ⊥ ‘𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣})) = (( ⊥ ‘𝑄) ∪ ((LSpan‘𝑈)‘{𝑣})))
37	36	fveq2d 6778	. . . . . . . . 9 ⊢ (𝜑 → ((LSpan‘𝑈)‘(((LSpan‘𝑈)‘( ⊥ ‘𝑄)) ∪ ((LSpan‘𝑈)‘{𝑣}))) = ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ ((LSpan‘𝑈)‘{𝑣}))))
38	37	adantr 481	. . . . . . . 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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → 𝑈 ∈ LMod)
40	2, 9	lssss 20198	. . . . . . . . . . 11 ⊢ (( ⊥ ‘𝑄) ∈ (LSubSp‘𝑈) → ( ⊥ ‘𝑄) ⊆ (Base‘𝑈))
41	12, 40	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ( ⊥ ‘𝑄) ⊆ (Base‘𝑈))
42	41	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ( ⊥ ‘𝑄) ⊆ (Base‘𝑈))
43	31	snssd 4742	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) → {𝑣} ⊆ (Base‘𝑈))
44	43	adantr 481	. . . . . . . . . 10 ⊢ ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → {𝑣} ⊆ (Base‘𝑈))
45	44	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → {𝑣} ⊆ (Base‘𝑈))
46	2, 27	lspun 20249	. . . . . . . . 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 1370	. . . . . . . 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 4091	. . . . . . . . . . 11 ⊢ (𝑄 = ((LSpan‘𝑈)‘{𝑣}) → (( ⊥ ‘𝑄) ∪ 𝑄) = (( ⊥ ‘𝑄) ∪ ((LSpan‘𝑈)‘{𝑣})))
49	48	fveq2d 6778	. . . . . . . . . 10 ⊢ (𝑄 = ((LSpan‘𝑈)‘{𝑣}) → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)) = ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ ((LSpan‘𝑈)‘{𝑣}))))
50	49	adantl 482	. . . . . . . . 9 ⊢ ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)) = ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ ((LSpan‘𝑈)‘{𝑣}))))
51	50	adantl 482	. . . . . . . 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 2788	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ {𝑣})) = ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)))
53		eqid 2738	. . . . . . . . . . 11 ⊢ ((joinH‘𝐾)‘𝑊) = ((joinH‘𝐾)‘𝑊)
54		eqid 2738	. . . . . . . . . . 11 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
55	4, 18, 5, 2, 10	dochcl 39367	. . . . . . . . . . . 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 39445	. . . . . . . . . 10 ⊢ (𝜑 → (( ⊥ ‘𝑄)((joinH‘𝐾)‘𝑊)𝑄) = (( ⊥ ‘𝑄)(LSSum‘𝑈)𝑄))
58	4, 5, 2, 53, 1, 41, 8	djhcom 39419	. . . . . . . . . 10 ⊢ (𝜑 → (( ⊥ ‘𝑄)((joinH‘𝐾)‘𝑊)𝑄) = (𝑄((joinH‘𝐾)‘𝑊)( ⊥ ‘𝑄)))
59	9, 3, 6, 7	lsatlssel 37011	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ (LSubSp‘𝑈))
60	9, 27, 54	lsmsp 20348	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ ( ⊥ ‘𝑄) ∈ (LSubSp‘𝑈) ∧ 𝑄 ∈ (LSubSp‘𝑈)) → (( ⊥ ‘𝑄)(LSSum‘𝑈)𝑄) = ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)))
61	6, 12, 59, 60	syl3anc 1370	. . . . . . . . . 10 ⊢ (𝜑 → (( ⊥ ‘𝑄)(LSSum‘𝑈)𝑄) = ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)))
62	57, 58, 61	3eqtr3rd 2787	. . . . . . . . 9 ⊢ (𝜑 → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)) = (𝑄((joinH‘𝐾)‘𝑊)( ⊥ ‘𝑄)))
63	4, 5, 2, 10, 53	djhexmid 39425	. . . . . . . . . 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 2778	. . . . . . . 8 ⊢ (𝜑 → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)) = (Base‘𝑈))
66	65	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ 𝑄)) = (Base‘𝑈))
67	52, 66	eqtrd 2778	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣}))) → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ {𝑣})) = (Base‘𝑈))
68	67	ex 413	. . . . 5 ⊢ (𝜑 → ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ {𝑣})) = (Base‘𝑈)))
69	33, 68	jcad 513	. . . 4 ⊢ (𝜑 → ((𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ∧ 𝑄 = ((LSpan‘𝑈)‘{𝑣})) → (𝑣 ∈ (Base‘𝑈) ∧ ((LSpan‘𝑈)‘(( ⊥ ‘𝑄) ∪ {𝑣})) = (Base‘𝑈))))
70	69	reximdv2 3199	. . 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 39123	. . 3 ⊢ (𝜑 → 𝑈 ∈ LVec)
73		dochsatshp.y	. . . 4 ⊢ 𝑌 = (LSHyp‘𝑈)
74	2, 27, 9, 73	islshp 36993	. . 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 1341	1 ⊢ (𝜑 → ( ⊥ ‘𝑄) ∈ 𝑌)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3065   ∖ cdif 3884   ∪ cun 3885   ⊆ wss 3887  {csn 4561  ran crn 5590  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  0_gc0g 17150  LSSumclsm 19239  LModclmod 20123  LSubSpclss 20193  LSpanclspn 20233  LVecclvec 20364  LSAtomsclsa 36988  LSHypclsh 36989  HLchlt 37364  LHypclh 37998  DVecHcdvh 39092  DIsoHcdih 39242  ocHcoch 39361  joinHcdjh 39408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-riotaBAD 36967

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-tpos 8042  df-undef 8089  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-sca 16978  df-vsca 16979  df-0g 17152  df-proset 18013  df-poset 18031  df-plt 18048  df-lub 18064  df-glb 18065  df-join 18066  df-meet 18067  df-p0 18143  df-p1 18144  df-lat 18150  df-clat 18217  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-submnd 18431  df-grp 18580  df-minusg 18581  df-sbg 18582  df-subg 18752  df-cntz 18923  df-lsm 19241  df-cmn 19388  df-abl 19389  df-mgp 19721  df-ur 19738  df-ring 19785  df-oppr 19862  df-dvdsr 19883  df-unit 19884  df-invr 19914  df-dvr 19925  df-drng 19993  df-lmod 20125  df-lss 20194  df-lsp 20234  df-lvec 20365  df-lsatoms 36990  df-lshyp 36991  df-oposet 37190  df-ol 37192  df-oml 37193  df-covers 37280  df-ats 37281  df-atl 37312  df-cvlat 37336  df-hlat 37365  df-llines 37512  df-lplanes 37513  df-lvols 37514  df-lines 37515  df-psubsp 37517  df-pmap 37518  df-padd 37810  df-lhyp 38002  df-laut 38003  df-ldil 38118  df-ltrn 38119  df-trl 38173  df-tgrp 38757  df-tendo 38769  df-edring 38771  df-dveca 39017  df-disoa 39043  df-dvech 39093  df-dib 39153  df-dic 39187  df-dih 39243  df-doch 39362  df-djh 39409

This theorem is referenced by:  dochsatshpb  39466  dochsnshp  39467  dochpolN  39504  lclkrlem2c  39523  lclkrlem2e  39525  mapdordlem2  39651