lclkrslem2 - Mathbox for Norm Megill

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Theorem lclkrslem2 40922

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 2726	. . 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 40920	. . . . 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 40920	. . . . 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 40916	. 2 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
20		eqid 2726	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
21	1, 3, 9	dvhlmod 40494	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
22	11	lcfls1lem 40918	. . . . . . . 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 1139	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝐹)
25	11	lcfls1lem 40918	. . . . . . . 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 1139	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
28	4, 6, 7, 21, 24, 27	ldualvaddcl 38513	. . . . 5 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹)
29	20, 4, 5, 21, 28	lkrssv 38479	. . . 4 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ⊆ (Base‘𝑈))
30	4, 5, 6, 7, 21, 24, 27	lkrin 38547	. . . 4 ⊢ (𝜑 → ((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊆ (𝐿‘(𝐸 + 𝐺)))
31	1, 3, 20, 2	dochss 40749	. . . 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 1368	. . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) ⊆ ( ⊥ ‘((𝐿‘𝐸) ∩ (𝐿‘𝐺))))
33		eqid 2726	. . . . . 6 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
34		eqid 2726	. . . . . 6 ⊢ ((joinH‘𝐾)‘𝑊) = ((joinH‘𝐾)‘𝑊)
35	23	simp2d 1140	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐸))) = (𝐿‘𝐸))
36	1, 33, 2, 3, 4, 5, 9, 24	lcfl5a 40881	. . . . . . 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 1140	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
39	1, 33, 2, 3, 4, 5, 9, 27	lcfl5a 40881	. . . . . . 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 40794	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘((𝐿‘𝐸) ∩ (𝐿‘𝐺))) = (( ⊥ ‘(𝐿‘𝐸))((joinH‘𝐾)‘𝑊)( ⊥ ‘(𝐿‘𝐺))))
42		eqid 2726	. . . . . . 7 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
43	20, 4, 5, 21, 24	lkrssv 38479	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘𝐸) ⊆ (Base‘𝑈))
44	1, 33, 3, 20, 2	dochcl 40737	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐸) ⊆ (Base‘𝑈)) → ( ⊥ ‘(𝐿‘𝐸)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
45	9, 43, 44	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐸)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
46	1, 33, 2, 3, 42, 4, 5, 9, 45, 27	dochkrsm 40842	. . . . . 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 40738	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐸) ⊆ (Base‘𝑈)) → ( ⊥ ‘(𝐿‘𝐸)) ∈ 𝑆)
49	9, 43, 48	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐸)) ∈ 𝑆)
50	20, 4, 5, 21, 27	lkrssv 38479	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (Base‘𝑈))
51	1, 3, 20, 47, 2	dochlss 40738	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐺) ⊆ (Base‘𝑈)) → ( ⊥ ‘(𝐿‘𝐺)) ∈ 𝑆)
52	9, 50, 51	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ∈ 𝑆)
53	1, 3, 20, 47, 42, 33, 34, 9, 49, 52	djhlsmcl 40798	. . . . . 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 2769	. . . 4 ⊢ (𝜑 → ( ⊥ ‘((𝐿‘𝐸) ∩ (𝐿‘𝐺))) = (( ⊥ ‘(𝐿‘𝐸))(LSSum‘𝑈)( ⊥ ‘(𝐿‘𝐺))))
56	23	simp3d 1141	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐸)) ⊆ 𝑄)
57	26	simp3d 1141	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄)
58	47	lsssssubg 20805	. . . . . . . 8 ⊢ (𝑈 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑈))
59	21, 58	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑈))
60	59, 49	sseldd 3978	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐸)) ∈ (SubGrp‘𝑈))
61	59, 52	sseldd 3978	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ∈ (SubGrp‘𝑈))
62		lclkrslem1.q	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ 𝑆)
63	59, 62	sseldd 3978	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑈))
64	42	lsmlub 19584	. . . . . 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 1368	. . . . 5 ⊢ (𝜑 → ((( ⊥ ‘(𝐿‘𝐸)) ⊆ 𝑄 ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄) ↔ (( ⊥ ‘(𝐿‘𝐸))(LSSum‘𝑈)( ⊥ ‘(𝐿‘𝐺))) ⊆ 𝑄))
66	56, 57, 65	mpbi2and 709	. . . 4 ⊢ (𝜑 → (( ⊥ ‘(𝐿‘𝐸))(LSSum‘𝑈)( ⊥ ‘(𝐿‘𝐺))) ⊆ 𝑄)
67	55, 66	eqsstrd 4015	. . 3 ⊢ (𝜑 → ( ⊥ ‘((𝐿‘𝐸) ∩ (𝐿‘𝐺))) ⊆ 𝑄)
68	32, 67	sstrd 3987	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) ⊆ 𝑄)
69	11, 8	lcfls1c 40920	. 2 ⊢ ((𝐸 + 𝐺) ∈ 𝐶 ↔ ((𝐸 + 𝐺) ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∧ ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) ⊆ 𝑄))
70	19, 68, 69	sylanbrc 582	1 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 {crab 3426 ∩ cin 3942 ⊆ wss 3943 ran crn 5670 ‘cfv 6537 (class class class)co 7405 Basecbs 17153 +_gcplusg 17206 Scalarcsca 17209 ·_𝑠 cvsca 17210 SubGrpcsubg 19047 LSSumclsm 19554 LModclmod 20706 LSubSpclss 20778 LFnlclfn 38440 LKerclk 38468 LDualcld 38506 HLchlt 38733 LHypclh 39368 DVecHcdvh 40462 DIsoHcdih 40612 ocHcoch 40731 joinHcdjh 40778

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-riotaBAD 38336

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-tpos 8212 df-undef 8259 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-0g 17396 df-mre 17539 df-mrc 17540 df-acs 17542 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-p1 18391 df-lat 18397 df-clat 18464 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19050 df-cntz 19233 df-oppg 19262 df-lsm 19556 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-dvr 20303 df-drng 20589 df-lmod 20708 df-lss 20779 df-lsp 20819 df-lvec 20951 df-lsatoms 38359 df-lshyp 38360 df-lcv 38402 df-lfl 38441 df-lkr 38469 df-ldual 38507 df-oposet 38559 df-ol 38561 df-oml 38562 df-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 df-hlat 38734 df-llines 38882 df-lplanes 38883 df-lvols 38884 df-lines 38885 df-psubsp 38887 df-pmap 38888 df-padd 39180 df-lhyp 39372 df-laut 39373 df-ldil 39488 df-ltrn 39489 df-trl 39543 df-tgrp 40127 df-tendo 40139 df-edring 40141 df-dveca 40387 df-disoa 40413 df-dvech 40463 df-dib 40523 df-dic 40557 df-dih 40613 df-doch 40732 df-djh 40779

This theorem is referenced by: lclkrs 40923

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