lclkrslem2 - Mathbox for Norm Megill

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Theorem lclkrslem2 40397

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 2732	. . 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 40395	. . . . 5 ⊢ (𝐸 ∈ 𝐶 ↔ (𝐸 ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∧ ( ⊥ ‘(𝐿‘𝐸)) ⊆ 𝑄))
13	12	simplbi 498	. . . 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 40395	. . . . 5 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))
17	16	simplbi 498	. . . 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 40391	. 2 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
20		eqid 2732	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
21	1, 3, 9	dvhlmod 39969	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
22	11	lcfls1lem 40393	. . . . . . . 8 ⊢ (𝐸 ∈ 𝐶 ↔ (𝐸 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐸))) = (𝐿‘𝐸) ∧ ( ⊥ ‘(𝐿‘𝐸)) ⊆ 𝑄))
23	10, 22	sylib 217	. . . . . . 7 ⊢ (𝜑 → (𝐸 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐸))) = (𝐿‘𝐸) ∧ ( ⊥ ‘(𝐿‘𝐸)) ⊆ 𝑄))
24	23	simp1d 1142	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝐹)
25	11	lcfls1lem 40393	. . . . . . . 8 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))
26	15, 25	sylib 217	. . . . . . 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 37988	. . . . 5 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹)
29	20, 4, 5, 21, 28	lkrssv 37954	. . . 4 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ⊆ (Base‘𝑈))
30	4, 5, 6, 7, 21, 24, 27	lkrin 38022	. . . 4 ⊢ (𝜑 → ((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊆ (𝐿‘(𝐸 + 𝐺)))
31	1, 3, 20, 2	dochss 40224	. . . 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 1371	. . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) ⊆ ( ⊥ ‘((𝐿‘𝐸) ∩ (𝐿‘𝐺))))
33		eqid 2732	. . . . . 6 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
34		eqid 2732	. . . . . 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 40356	. . . . . . 7 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐸))) = (𝐿‘𝐸) ↔ (𝐿‘𝐸) ∈ ran ((DIsoH‘𝐾)‘𝑊)))
37	35, 36	mpbid 231	. . . . . 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 40356	. . . . . . 7 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ↔ (𝐿‘𝐺) ∈ ran ((DIsoH‘𝐾)‘𝑊)))
40	38, 39	mpbid 231	. . . . . 6 ⊢ (𝜑 → (𝐿‘𝐺) ∈ ran ((DIsoH‘𝐾)‘𝑊))
41	1, 33, 3, 20, 2, 34, 9, 37, 40	dochdmm1 40269	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘((𝐿‘𝐸) ∩ (𝐿‘𝐺))) = (( ⊥ ‘(𝐿‘𝐸))((joinH‘𝐾)‘𝑊)( ⊥ ‘(𝐿‘𝐺))))
42		eqid 2732	. . . . . . 7 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
43	20, 4, 5, 21, 24	lkrssv 37954	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘𝐸) ⊆ (Base‘𝑈))
44	1, 33, 3, 20, 2	dochcl 40212	. . . . . . . 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 40317	. . . . . 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 40213	. . . . . . . 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 37954	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (Base‘𝑈))
51	1, 3, 20, 47, 2	dochlss 40213	. . . . . . . 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 40273	. . . . . 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 231	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘(𝐿‘𝐸))(LSSum‘𝑈)( ⊥ ‘(𝐿‘𝐺))) = (( ⊥ ‘(𝐿‘𝐸))((joinH‘𝐾)‘𝑊)( ⊥ ‘(𝐿‘𝐺))))
55	41, 54	eqtr4d 2775	. . . 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 20561	. . . . . . . 8 ⊢ (𝑈 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑈))
59	21, 58	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑈))
60	59, 49	sseldd 3982	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐸)) ∈ (SubGrp‘𝑈))
61	59, 52	sseldd 3982	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ∈ (SubGrp‘𝑈))
62		lclkrslem1.q	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ 𝑆)
63	59, 62	sseldd 3982	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑈))
64	42	lsmlub 19526	. . . . . 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 1371	. . . . 5 ⊢ (𝜑 → ((( ⊥ ‘(𝐿‘𝐸)) ⊆ 𝑄 ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄) ↔ (( ⊥ ‘(𝐿‘𝐸))(LSSum‘𝑈)( ⊥ ‘(𝐿‘𝐺))) ⊆ 𝑄))
66	56, 57, 65	mpbi2and 710	. . . 4 ⊢ (𝜑 → (( ⊥ ‘(𝐿‘𝐸))(LSSum‘𝑈)( ⊥ ‘(𝐿‘𝐺))) ⊆ 𝑄)
67	55, 66	eqsstrd 4019	. . 3 ⊢ (𝜑 → ( ⊥ ‘((𝐿‘𝐸) ∩ (𝐿‘𝐺))) ⊆ 𝑄)
68	32, 67	sstrd 3991	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) ⊆ 𝑄)
69	11, 8	lcfls1c 40395	. 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)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {crab 3432 ∩ cin 3946 ⊆ wss 3947 ran crn 5676 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 +_gcplusg 17193 Scalarcsca 17196 ·_𝑠 cvsca 17197 SubGrpcsubg 18994 LSSumclsm 19496 LModclmod 20463 LSubSpclss 20534 LFnlclfn 37915 LKerclk 37943 LDualcld 37981 HLchlt 38208 LHypclh 38843 DVecHcdvh 39937 DIsoHcdih 40087 ocHcoch 40206 joinHcdjh 40253

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-riotaBAD 37811

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8207 df-undef 8254 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-0g 17383 df-mre 17526 df-mrc 17527 df-acs 17529 df-proset 18244 df-poset 18262 df-plt 18279 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-p1 18375 df-lat 18381 df-clat 18448 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-cntz 19175 df-oppg 19204 df-lsm 19498 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-dvr 20207 df-drng 20309 df-lmod 20465 df-lss 20535 df-lsp 20575 df-lvec 20706 df-lsatoms 37834 df-lshyp 37835 df-lcv 37877 df-lfl 37916 df-lkr 37944 df-ldual 37982 df-oposet 38034 df-ol 38036 df-oml 38037 df-covers 38124 df-ats 38125 df-atl 38156 df-cvlat 38180 df-hlat 38209 df-llines 38357 df-lplanes 38358 df-lvols 38359 df-lines 38360 df-psubsp 38362 df-pmap 38363 df-padd 38655 df-lhyp 38847 df-laut 38848 df-ldil 38963 df-ltrn 38964 df-trl 39018 df-tgrp 39602 df-tendo 39614 df-edring 39616 df-dveca 39862 df-disoa 39888 df-dvech 39938 df-dib 39998 df-dic 40032 df-dih 40088 df-doch 40207 df-djh 40254

This theorem is referenced by: lclkrs 40398

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