Theorem lclkrslem2 41521

Description: The set of functionals having closed kernels and majorizing the orthocomplement of a given subspace 𝑄 is closed under scalar product. (Contributed by NM, 28-Jan-2015.)

Hypotheses

Ref	Expression
lclkrslem1.h	⊢ 𝐻 = (LHyp‘𝐾)
lclkrslem1.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lclkrslem1.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lclkrslem1.s	⊢ 𝑆 = (LSubSp‘𝑈)
lclkrslem1.f	⊢ 𝐹 = (LFnl‘𝑈)
lclkrslem1.l	⊢ 𝐿 = (LKer‘𝑈)
lclkrslem1.d	⊢ 𝐷 = (LDual‘𝑈)
lclkrslem1.r	⊢ 𝑅 = (Scalar‘𝑈)
lclkrslem1.b	⊢ 𝐵 = (Base‘𝑅)
lclkrslem1.t	⊢ · = ( ·𝑠 ‘𝐷)
lclkrslem1.c	⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑄)}
lclkrslem1.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lclkrslem1.q	⊢ (𝜑 → 𝑄 ∈ 𝑆)
lclkrslem1.g	⊢ (𝜑 → 𝐺 ∈ 𝐶)
lclkrslem2.p	⊢ + = (+g‘𝐷)
lclkrslem2.e	⊢ (𝜑 → 𝐸 ∈ 𝐶)

Assertion

Ref	Expression
lclkrslem2	⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐶)

Distinct variable groups:   ⊥ ,𝑓   𝑓,𝐹   𝑓,𝐺   𝑓,𝐿   𝑄,𝑓   · ,𝑓   𝑓,𝐸   + ,𝑓

Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑓)   𝐶(𝑓)   𝐷(𝑓)   𝑅(𝑓)   𝑆(𝑓)   𝑈(𝑓)   𝐻(𝑓)   𝐾(𝑓)   𝑊(𝑓)

Proof of Theorem lclkrslem2

Step	Hyp	Ref	Expression
1		lclkrslem1.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		lclkrslem1.o	. . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
3		lclkrslem1.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		lclkrslem1.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑈)
5		lclkrslem1.l	. . 3 ⊢ 𝐿 = (LKer‘𝑈)
6		lclkrslem1.d	. . 3 ⊢ 𝐷 = (LDual‘𝑈)
7		lclkrslem2.p	. . 3 ⊢ + = (+g‘𝐷)
8		eqid 2729	. . 3 ⊢ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
9		lclkrslem1.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10		lclkrslem2.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐶)
11		lclkrslem1.c	. . . . . 6 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑄)}
12	11, 8	lcfls1c 41519	. . . . 5 ⊢ (𝐸 ∈ 𝐶 ↔ (𝐸 ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∧ ( ⊥ ‘(𝐿‘𝐸)) ⊆ 𝑄))
13	12	simplbi 497	. . . 4 ⊢ (𝐸 ∈ 𝐶 → 𝐸 ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
14	10, 13	syl 17	. . 3 ⊢ (𝜑 → 𝐸 ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
15		lclkrslem1.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐶)
16	11, 8	lcfls1c 41519	. . . . 5 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))
17	16	simplbi 497	. . . 4 ⊢ (𝐺 ∈ 𝐶 → 𝐺 ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
18	15, 17	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 18	lclkrlem2 41515	. 2 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
20		eqid 2729	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
21	1, 3, 9	dvhlmod 41093	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
22	11	lcfls1lem 41517	. . . . . . . 8 ⊢ (𝐸 ∈ 𝐶 ↔ (𝐸 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐸))) = (𝐿‘𝐸) ∧ ( ⊥ ‘(𝐿‘𝐸)) ⊆ 𝑄))
23	10, 22	sylib 218	. . . . . . 7 ⊢ (𝜑 → (𝐸 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐸))) = (𝐿‘𝐸) ∧ ( ⊥ ‘(𝐿‘𝐸)) ⊆ 𝑄))
24	23	simp1d 1142	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝐹)
25	11	lcfls1lem 41517	. . . . . . . 8 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))
26	15, 25	sylib 218	. . . . . . 7 ⊢ (𝜑 → (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))
27	26	simp1d 1142	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
28	4, 6, 7, 21, 24, 27	ldualvaddcl 39113	. . . . 5 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹)
29	20, 4, 5, 21, 28	lkrssv 39079	. . . 4 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ⊆ (Base‘𝑈))
30	4, 5, 6, 7, 21, 24, 27	lkrin 39147	. . . 4 ⊢ (𝜑 → ((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊆ (𝐿‘(𝐸 + 𝐺)))
31	1, 3, 20, 2	dochss 41348	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘(𝐸 + 𝐺)) ⊆ (Base‘𝑈) ∧ ((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊆ (𝐿‘(𝐸 + 𝐺))) → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) ⊆ ( ⊥ ‘((𝐿‘𝐸) ∩ (𝐿‘𝐺))))
32	9, 29, 30, 31	syl3anc 1373	. . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) ⊆ ( ⊥ ‘((𝐿‘𝐸) ∩ (𝐿‘𝐺))))
33		eqid 2729	. . . . . 6 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
34		eqid 2729	. . . . . 6 ⊢ ((joinH‘𝐾)‘𝑊) = ((joinH‘𝐾)‘𝑊)
35	23	simp2d 1143	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐸))) = (𝐿‘𝐸))
36	1, 33, 2, 3, 4, 5, 9, 24	lcfl5a 41480	. . . . . . 7 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐸))) = (𝐿‘𝐸) ↔ (𝐿‘𝐸) ∈ ran ((DIsoH‘𝐾)‘𝑊)))
37	35, 36	mpbid 232	. . . . . 6 ⊢ (𝜑 → (𝐿‘𝐸) ∈ ran ((DIsoH‘𝐾)‘𝑊))
38	26	simp2d 1143	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
39	1, 33, 2, 3, 4, 5, 9, 27	lcfl5a 41480	. . . . . . 7 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ↔ (𝐿‘𝐺) ∈ ran ((DIsoH‘𝐾)‘𝑊)))
40	38, 39	mpbid 232	. . . . . 6 ⊢ (𝜑 → (𝐿‘𝐺) ∈ ran ((DIsoH‘𝐾)‘𝑊))
41	1, 33, 3, 20, 2, 34, 9, 37, 40	dochdmm1 41393	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘((𝐿‘𝐸) ∩ (𝐿‘𝐺))) = (( ⊥ ‘(𝐿‘𝐸))((joinH‘𝐾)‘𝑊)( ⊥ ‘(𝐿‘𝐺))))
42		eqid 2729	. . . . . . 7 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
43	20, 4, 5, 21, 24	lkrssv 39079	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘𝐸) ⊆ (Base‘𝑈))
44	1, 33, 3, 20, 2	dochcl 41336	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐸) ⊆ (Base‘𝑈)) → ( ⊥ ‘(𝐿‘𝐸)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
45	9, 43, 44	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐸)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
46	1, 33, 2, 3, 42, 4, 5, 9, 45, 27	dochkrsm 41441	. . . . . 6 ⊢ (𝜑 → (( ⊥ ‘(𝐿‘𝐸))(LSSum‘𝑈)( ⊥ ‘(𝐿‘𝐺))) ∈ ran ((DIsoH‘𝐾)‘𝑊))
47		lclkrslem1.s	. . . . . . 7 ⊢ 𝑆 = (LSubSp‘𝑈)
48	1, 3, 20, 47, 2	dochlss 41337	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐸) ⊆ (Base‘𝑈)) → ( ⊥ ‘(𝐿‘𝐸)) ∈ 𝑆)
49	9, 43, 48	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐸)) ∈ 𝑆)
50	20, 4, 5, 21, 27	lkrssv 39079	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (Base‘𝑈))
51	1, 3, 20, 47, 2	dochlss 41337	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐺) ⊆ (Base‘𝑈)) → ( ⊥ ‘(𝐿‘𝐺)) ∈ 𝑆)
52	9, 50, 51	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ∈ 𝑆)
53	1, 3, 20, 47, 42, 33, 34, 9, 49, 52	djhlsmcl 41397	. . . . . 6 ⊢ (𝜑 → ((( ⊥ ‘(𝐿‘𝐸))(LSSum‘𝑈)( ⊥ ‘(𝐿‘𝐺))) ∈ ran ((DIsoH‘𝐾)‘𝑊) ↔ (( ⊥ ‘(𝐿‘𝐸))(LSSum‘𝑈)( ⊥ ‘(𝐿‘𝐺))) = (( ⊥ ‘(𝐿‘𝐸))((joinH‘𝐾)‘𝑊)( ⊥ ‘(𝐿‘𝐺)))))
54	46, 53	mpbid 232	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘(𝐿‘𝐸))(LSSum‘𝑈)( ⊥ ‘(𝐿‘𝐺))) = (( ⊥ ‘(𝐿‘𝐸))((joinH‘𝐾)‘𝑊)( ⊥ ‘(𝐿‘𝐺))))
55	41, 54	eqtr4d 2767	. . . 4 ⊢ (𝜑 → ( ⊥ ‘((𝐿‘𝐸) ∩ (𝐿‘𝐺))) = (( ⊥ ‘(𝐿‘𝐸))(LSSum‘𝑈)( ⊥ ‘(𝐿‘𝐺))))
56	23	simp3d 1144	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐸)) ⊆ 𝑄)
57	26	simp3d 1144	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄)
58	47	lsssssubg 20861	. . . . . . . 8 ⊢ (𝑈 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑈))
59	21, 58	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑈))
60	59, 49	sseldd 3936	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐸)) ∈ (SubGrp‘𝑈))
61	59, 52	sseldd 3936	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ∈ (SubGrp‘𝑈))
62		lclkrslem1.q	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ 𝑆)
63	59, 62	sseldd 3936	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑈))
64	42	lsmlub 19543	. . . . . 6 ⊢ ((( ⊥ ‘(𝐿‘𝐸)) ∈ (SubGrp‘𝑈) ∧ ( ⊥ ‘(𝐿‘𝐺)) ∈ (SubGrp‘𝑈) ∧ 𝑄 ∈ (SubGrp‘𝑈)) → ((( ⊥ ‘(𝐿‘𝐸)) ⊆ 𝑄 ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄) ↔ (( ⊥ ‘(𝐿‘𝐸))(LSSum‘𝑈)( ⊥ ‘(𝐿‘𝐺))) ⊆ 𝑄))
65	60, 61, 63, 64	syl3anc 1373	. . . . 5 ⊢ (𝜑 → ((( ⊥ ‘(𝐿‘𝐸)) ⊆ 𝑄 ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄) ↔ (( ⊥ ‘(𝐿‘𝐸))(LSSum‘𝑈)( ⊥ ‘(𝐿‘𝐺))) ⊆ 𝑄))
66	56, 57, 65	mpbi2and 712	. . . 4 ⊢ (𝜑 → (( ⊥ ‘(𝐿‘𝐸))(LSSum‘𝑈)( ⊥ ‘(𝐿‘𝐺))) ⊆ 𝑄)
67	55, 66	eqsstrd 3970	. . 3 ⊢ (𝜑 → ( ⊥ ‘((𝐿‘𝐸) ∩ (𝐿‘𝐺))) ⊆ 𝑄)
68	32, 67	sstrd 3946	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) ⊆ 𝑄)
69	11, 8	lcfls1c 41519	. 2 ⊢ ((𝐸 + 𝐺) ∈ 𝐶 ↔ ((𝐸 + 𝐺) ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∧ ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) ⊆ 𝑄))
70	19, 68, 69	sylanbrc 583	1 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {crab 3394   ∩ cin 3902   ⊆ wss 3903  ran crn 5620  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  +_gcplusg 17161  Scalarcsca 17164   ·_𝑠 cvsca 17165  SubGrpcsubg 18999  LSSumclsm 19513  LModclmod 20763  LSubSpclss 20834  LFnlclfn 39040  LKerclk 39068  LDualcld 39106  HLchlt 39333  LHypclh 39967  DVecHcdvh 41061  DIsoHcdih 41211  ocHcoch 41330  joinHcdjh 41377

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 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-riotaBAD 38936

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 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-om 7800  df-1st 7924  df-2nd 7925  df-tpos 8159  df-undef 8206  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-n0 12385  df-z 12472  df-uz 12736  df-fz 13411  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-mre 17488  df-mrc 17489  df-acs 17491  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 18514  df-sgrp 18593  df-mnd 18609  df-submnd 18658  df-grp 18815  df-minusg 18816  df-sbg 18817  df-subg 19002  df-cntz 19196  df-oppg 19225  df-lsm 19515  df-cmn 19661  df-abl 19662  df-mgp 20026  df-rng 20038  df-ur 20067  df-ring 20120  df-oppr 20222  df-dvdsr 20242  df-unit 20243  df-invr 20273  df-dvr 20286  df-drng 20616  df-lmod 20765  df-lss 20835  df-lsp 20875  df-lvec 21007  df-lsatoms 38959  df-lshyp 38960  df-lcv 39002  df-lfl 39041  df-lkr 39069  df-ldual 39107  df-oposet 39159  df-ol 39161  df-oml 39162  df-covers 39249  df-ats 39250  df-atl 39281  df-cvlat 39305  df-hlat 39334  df-llines 39481  df-lplanes 39482  df-lvols 39483  df-lines 39484  df-psubsp 39486  df-pmap 39487  df-padd 39779  df-lhyp 39971  df-laut 39972  df-ldil 40087  df-ltrn 40088  df-trl 40142  df-tgrp 40726  df-tendo 40738  df-edring 40740  df-dveca 40986  df-disoa 41012  df-dvech 41062  df-dib 41122  df-dic 41156  df-dih 41212  df-doch 41331  df-djh 41378

This theorem is referenced by:  lclkrs  41522