Theorem lclkrslem2 42476

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 2760	. . 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 42474	. . . . 5 ⊢ (𝐸 ∈ 𝐶 ↔ (𝐸 ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∧ ( ⊥ ‘(𝐿‘𝐸)) ⊆ 𝑄))
13	12	simplbi 502	. . . 4 ⊢ (𝐸 ∈ 𝐶 → 𝐸 ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
14	10, 13	syl 18	. . 3 ⊢ (𝜑 → 𝐸 ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
15		lclkrslem1.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐶)
16	11, 8	lcfls1c 42474	. . . . 5 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))
17	16	simplbi 502	. . . 4 ⊢ (𝐺 ∈ 𝐶 → 𝐺 ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
18	15, 17	syl 18	. . 3 ⊢ (𝜑 → 𝐺 ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 18	lclkrlem2 42470	. 2 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
20		eqid 2760	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
21	1, 3, 9	dvhlmod 42048	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
22	11	lcfls1lem 42472	. . . . . . . 8 ⊢ (𝐸 ∈ 𝐶 ↔ (𝐸 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐸))) = (𝐿‘𝐸) ∧ ( ⊥ ‘(𝐿‘𝐸)) ⊆ 𝑄))
23	10, 22	sylib 221	. . . . . . 7 ⊢ (𝜑 → (𝐸 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐸))) = (𝐿‘𝐸) ∧ ( ⊥ ‘(𝐿‘𝐸)) ⊆ 𝑄))
24	23	simp1d 1160	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝐹)
25	11	lcfls1lem 42472	. . . . . . . 8 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))
26	15, 25	sylib 221	. . . . . . 7 ⊢ (𝜑 → (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))
27	26	simp1d 1160	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
28	4, 6, 7, 21, 24, 27	ldualvaddcl 40068	. . . . 5 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹)
29	20, 4, 5, 21, 28	lkrssv 40034	. . . 4 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ⊆ (Base‘𝑈))
30	4, 5, 6, 7, 21, 24, 27	lkrin 40102	. . . 4 ⊢ (𝜑 → ((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊆ (𝐿‘(𝐸 + 𝐺)))
31	1, 3, 20, 2	dochss 42303	. . . 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 1398	. . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) ⊆ ( ⊥ ‘((𝐿‘𝐸) ∩ (𝐿‘𝐺))))
33		eqid 2760	. . . . . 6 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
34		eqid 2760	. . . . . 6 ⊢ ((joinH‘𝐾)‘𝑊) = ((joinH‘𝐾)‘𝑊)
35	23	simp2d 1161	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐸))) = (𝐿‘𝐸))
36	1, 33, 2, 3, 4, 5, 9, 24	lcfl5a 42435	. . . . . . 7 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐸))) = (𝐿‘𝐸) ↔ (𝐿‘𝐸) ∈ ran ((DIsoH‘𝐾)‘𝑊)))
37	35, 36	mpbid 235	. . . . . 6 ⊢ (𝜑 → (𝐿‘𝐸) ∈ ran ((DIsoH‘𝐾)‘𝑊))
38	26	simp2d 1161	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
39	1, 33, 2, 3, 4, 5, 9, 27	lcfl5a 42435	. . . . . . 7 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ↔ (𝐿‘𝐺) ∈ ran ((DIsoH‘𝐾)‘𝑊)))
40	38, 39	mpbid 235	. . . . . 6 ⊢ (𝜑 → (𝐿‘𝐺) ∈ ran ((DIsoH‘𝐾)‘𝑊))
41	1, 33, 3, 20, 2, 34, 9, 37, 40	dochdmm1 42348	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘((𝐿‘𝐸) ∩ (𝐿‘𝐺))) = (( ⊥ ‘(𝐿‘𝐸))((joinH‘𝐾)‘𝑊)( ⊥ ‘(𝐿‘𝐺))))
42		eqid 2760	. . . . . . 7 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
43	20, 4, 5, 21, 24	lkrssv 40034	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘𝐸) ⊆ (Base‘𝑈))
44	1, 33, 3, 20, 2	dochcl 42291	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐸) ⊆ (Base‘𝑈)) → ( ⊥ ‘(𝐿‘𝐸)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
45	9, 43, 44	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐸)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
46	1, 33, 2, 3, 42, 4, 5, 9, 45, 27	dochkrsm 42396	. . . . . 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 42292	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐸) ⊆ (Base‘𝑈)) → ( ⊥ ‘(𝐿‘𝐸)) ∈ 𝑆)
49	9, 43, 48	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐸)) ∈ 𝑆)
50	20, 4, 5, 21, 27	lkrssv 40034	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (Base‘𝑈))
51	1, 3, 20, 47, 2	dochlss 42292	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐺) ⊆ (Base‘𝑈)) → ( ⊥ ‘(𝐿‘𝐺)) ∈ 𝑆)
52	9, 50, 51	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ∈ 𝑆)
53	1, 3, 20, 47, 42, 33, 34, 9, 49, 52	djhlsmcl 42352	. . . . . 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 235	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘(𝐿‘𝐸))(LSSum‘𝑈)( ⊥ ‘(𝐿‘𝐺))) = (( ⊥ ‘(𝐿‘𝐸))((joinH‘𝐾)‘𝑊)( ⊥ ‘(𝐿‘𝐺))))
55	41, 54	eqtr4d 2798	. . . 4 ⊢ (𝜑 → ( ⊥ ‘((𝐿‘𝐸) ∩ (𝐿‘𝐺))) = (( ⊥ ‘(𝐿‘𝐸))(LSSum‘𝑈)( ⊥ ‘(𝐿‘𝐺))))
56	23	simp3d 1162	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐸)) ⊆ 𝑄)
57	26	simp3d 1162	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄)
58	47	lsssssubg 21172	. . . . . . . 8 ⊢ (𝑈 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑈))
59	21, 58	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑈))
60	59, 49	sseldd 3932	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐸)) ∈ (SubGrp‘𝑈))
61	59, 52	sseldd 3932	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ∈ (SubGrp‘𝑈))
62		lclkrslem1.q	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ 𝑆)
63	59, 62	sseldd 3932	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑈))
64	42	lsmlub 19817	. . . . . 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 1398	. . . . 5 ⊢ (𝜑 → ((( ⊥ ‘(𝐿‘𝐸)) ⊆ 𝑄 ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄) ↔ (( ⊥ ‘(𝐿‘𝐸))(LSSum‘𝑈)( ⊥ ‘(𝐿‘𝐺))) ⊆ 𝑄))
66	56, 57, 65	mpbi2and 725	. . . 4 ⊢ (𝜑 → (( ⊥ ‘(𝐿‘𝐸))(LSSum‘𝑈)( ⊥ ‘(𝐿‘𝐺))) ⊆ 𝑄)
67	55, 66	eqsstrd 3965	. . 3 ⊢ (𝜑 → ( ⊥ ‘((𝐿‘𝐸) ∩ (𝐿‘𝐺))) ⊆ 𝑄)
68	32, 67	sstrd 3941	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) ⊆ 𝑄)
69	11, 8	lcfls1c 42474	. 2 ⊢ ((𝐸 + 𝐺) ∈ 𝐶 ↔ ((𝐸 + 𝐺) ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∧ ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) ⊆ 𝑄))
70	19, 68, 69	sylanbrc 595	1 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐶)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {crab 3412   ∩ cin 3898   ⊆ wss 3899  ran crn 5656  ‘cfv 6535  (class class class)co 7416  Basecbs 17326  +_gcplusg 17367  Scalarcsca 17370   ·_𝑠 cvsca 17371  SubGrpcsubg 19269  LSSumclsm 19787  LModclmod 21074  LSubSpclss 21145  LFnlclfn 39995  LKerclk 40023  LDualcld 40061  HLchlt 40288  LHypclh 40922  DVecHcdvh 42016  DIsoHcdih 42166  ocHcoch 42285  joinHcdjh 42332

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226  ax-riotaBAD 39891

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6301  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7684  df-om 7869  df-1st 7992  df-2nd 7993  df-tpos 8229  df-undef 8276  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-1o 8462  df-2o 8463  df-er 8703  df-map 8835  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11492  df-neg 11493  df-nn 12283  df-2 12352  df-3 12353  df-4 12354  df-5 12355  df-6 12356  df-n0 12554  df-z 12641  df-uz 12913  df-fz 13587  df-struct 17264  df-sets 17281  df-slot 17299  df-ndx 17311  df-base 17327  df-ress 17348  df-plusg 17380  df-mulr 17381  df-sca 17383  df-vsca 17384  df-0g 17551  df-mre 17695  df-mrc 17696  df-acs 17698  df-proset 18407  df-poset 18426  df-plt 18441  df-lub 18457  df-glb 18458  df-join 18459  df-meet 18460  df-p0 18536  df-p1 18537  df-lat 18545  df-clat 18612  df-mgm 18755  df-sgrp 18847  df-mnd 18863  df-submnd 18918  df-grp 19086  df-minusg 19087  df-sbg 19088  df-subg 19272  df-cntz 19470  df-oppg 19499  df-lsm 19789  df-cmn 19935  df-abl 19936  df-mgp 20300  df-rng 20314  df-ur 20347  df-ring 20400  df-oppr 20506  df-dvdsr 20526  df-unit 20527  df-invr 20557  df-dvr 20570  df-drng 20921  df-lmod 21076  df-lss 21146  df-lsp 21186  df-lvec 21317  df-lsatoms 39914  df-lshyp 39915  df-lcv 39957  df-lfl 39996  df-lkr 40024  df-ldual 40062  df-oposet 40114  df-ol 40116  df-oml 40117  df-covers 40204  df-ats 40205  df-atl 40236  df-cvlat 40260  df-hlat 40289  df-llines 40436  df-lplanes 40437  df-lvols 40438  df-lines 40439  df-psubsp 40441  df-pmap 40442  df-padd 40734  df-lhyp 40926  df-laut 40927  df-ldil 41042  df-ltrn 41043  df-trl 41097  df-tgrp 41681  df-tendo 41693  df-edring 41695  df-dveca 41941  df-disoa 41967  df-dvech 42017  df-dib 42077  df-dic 42111  df-dih 42167  df-doch 42286  df-djh 42333

This theorem is used by:  lclkrs  42477