lclkrlem2s - Mathbox for Norm Megill

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Theorem lclkrlem2s 40700

Description: Lemma for lclkr 40708. Thus, the sum has a closed kernel when 𝐵 is zero. (Contributed by NM, 18-Jan-2015.)

**Hypotheses**
Ref	Expression
lclkrlem2m.v	⊢ 𝑉 = (Base‘𝑈)
lclkrlem2m.t	⊢ · = ( ·_𝑠 ‘𝑈)
lclkrlem2m.s	⊢ 𝑆 = (Scalar‘𝑈)
lclkrlem2m.q	⊢ × = (._r‘𝑆)
lclkrlem2m.z	⊢ 0 = (0_g‘𝑆)
lclkrlem2m.i	⊢ 𝐼 = (inv_r‘𝑆)
lclkrlem2m.m	⊢ − = (-_g‘𝑈)
lclkrlem2m.f	⊢ 𝐹 = (LFnl‘𝑈)
lclkrlem2m.d	⊢ 𝐷 = (LDual‘𝑈)
lclkrlem2m.p	⊢ + = (+_g‘𝐷)
lclkrlem2m.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
lclkrlem2m.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
lclkrlem2m.e	⊢ (𝜑 → 𝐸 ∈ 𝐹)
lclkrlem2m.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
lclkrlem2n.n	⊢ 𝑁 = (LSpan‘𝑈)
lclkrlem2n.l	⊢ 𝐿 = (LKer‘𝑈)
lclkrlem2o.h	⊢ 𝐻 = (LHyp‘𝐾)
lclkrlem2o.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lclkrlem2o.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lclkrlem2o.a	⊢ ⊕ = (LSSum‘𝑈)
lclkrlem2o.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lclkrlem2q.le	⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
lclkrlem2q.lg	⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
lclkrlem2q.b	⊢ 𝐵 = (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌))
lclkrlem2q.n	⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )
lclkrlem2r.bn	⊢ (𝜑 → 𝐵 = (0_g‘𝑈))

**Assertion**
Ref	Expression
lclkrlem2s	⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

**Proof of Theorem lclkrlem2s**
Step	Hyp	Ref	Expression
1		lclkrlem2o.k	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		lclkrlem2m.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
3	2	snssd 4813	. . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
4		lclkrlem2o.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
5		eqid 2731	. . . . . . . 8 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
6		lclkrlem2o.u	. . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		lclkrlem2m.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑈)
8		lclkrlem2o.o	. . . . . . . 8 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
9	4, 5, 6, 7, 8	dochcl 40528	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑌} ⊆ 𝑉) → ( ⊥ ‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
10	1, 3, 9	syl2anc 583	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
11	4, 5, 8	dochoc 40542	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑌}))) = ( ⊥ ‘{𝑌}))
12	1, 10, 11	syl2anc 583	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑌}))) = ( ⊥ ‘{𝑌}))
13	12	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑌}))) = ( ⊥ ‘{𝑌}))
14		lclkrlem2m.t	. . . . . . . . . 10 ⊢ · = ( ·_𝑠 ‘𝑈)
15		lclkrlem2m.s	. . . . . . . . . 10 ⊢ 𝑆 = (Scalar‘𝑈)
16		lclkrlem2m.q	. . . . . . . . . 10 ⊢ × = (._r‘𝑆)
17		lclkrlem2m.z	. . . . . . . . . 10 ⊢ 0 = (0_g‘𝑆)
18		lclkrlem2m.i	. . . . . . . . . 10 ⊢ 𝐼 = (inv_r‘𝑆)
19		lclkrlem2m.m	. . . . . . . . . 10 ⊢ − = (-_g‘𝑈)
20		lclkrlem2m.f	. . . . . . . . . 10 ⊢ 𝐹 = (LFnl‘𝑈)
21		lclkrlem2m.d	. . . . . . . . . 10 ⊢ 𝐷 = (LDual‘𝑈)
22		lclkrlem2m.p	. . . . . . . . . 10 ⊢ + = (+_g‘𝐷)
23		lclkrlem2m.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
24		lclkrlem2m.e	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ 𝐹)
25		lclkrlem2m.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
26		lclkrlem2n.n	. . . . . . . . . 10 ⊢ 𝑁 = (LSpan‘𝑈)
27		lclkrlem2n.l	. . . . . . . . . 10 ⊢ 𝐿 = (LKer‘𝑈)
28		lclkrlem2o.a	. . . . . . . . . 10 ⊢ ⊕ = (LSSum‘𝑈)
29		lclkrlem2q.le	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
30		lclkrlem2q.lg	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
31		lclkrlem2q.b	. . . . . . . . . 10 ⊢ 𝐵 = (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌))
32		lclkrlem2q.n	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )
33		lclkrlem2r.bn	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = (0_g‘𝑈))
34	7, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 2, 24, 25, 26, 27, 4, 8, 6, 28, 1, 29, 30, 31, 32, 33	lclkrlem2r 40699	. . . . . . . . 9 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (𝐿‘(𝐸 + 𝐺)))
35	34	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → (𝐿‘𝐺) ⊆ (𝐿‘(𝐸 + 𝐺)))
36		eqid 2731	. . . . . . . . 9 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈)
37	4, 6, 1	dvhlvec 40284	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ LVec)
38	37	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → 𝑈 ∈ LVec)
39		simplr 766	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → (𝐿‘𝐺) ∈ (LSHyp‘𝑈))
40		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈))
41	36, 38, 39, 40	lshpcmp 38162	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → ((𝐿‘𝐺) ⊆ (𝐿‘(𝐸 + 𝐺)) ↔ (𝐿‘𝐺) = (𝐿‘(𝐸 + 𝐺))))
42	35, 41	mpbid 231	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → (𝐿‘𝐺) = (𝐿‘(𝐸 + 𝐺)))
43	30	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
44	42, 43	eqtr3d 2773	. . . . . 6 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → (𝐿‘(𝐸 + 𝐺)) = ( ⊥ ‘{𝑌}))
45	44	fveq2d 6896	. . . . 5 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) = ( ⊥ ‘( ⊥ ‘{𝑌})))
46	45	fveq2d 6896	. . . 4 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑌}))))
47	13, 46, 44	3eqtr4d 2781	. . 3 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
48	4, 6, 8, 7, 1	dochoc1 40536	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
49	48	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) = 𝑉) → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
50		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) = 𝑉) → (𝐿‘(𝐸 + 𝐺)) = 𝑉)
51	50	fveq2d 6896	. . . . 5 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) = 𝑉) → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) = ( ⊥ ‘𝑉))
52	51	fveq2d 6896	. . . 4 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = ( ⊥ ‘( ⊥ ‘𝑉)))
53	49, 52, 50	3eqtr4d 2781	. . 3 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
54	4, 6, 1	dvhlmod 40285	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
55	20, 21, 22, 54, 24, 25	ldualvaddcl 38304	. . . . 5 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹)
56	7, 36, 20, 27, 37, 55	lkrshpor 38281	. . . 4 ⊢ (𝜑 → ((𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈) ∨ (𝐿‘(𝐸 + 𝐺)) = 𝑉))
57	56	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) → ((𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈) ∨ (𝐿‘(𝐸 + 𝐺)) = 𝑉))
58	47, 53, 57	mpjaodan 956	. 2 ⊢ ((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
59	48	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
60	7, 20, 27, 54, 55	lkrssv 38270	. . . . . . 7 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ⊆ 𝑉)
61	60	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → (𝐿‘(𝐸 + 𝐺)) ⊆ 𝑉)
62		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → (𝐿‘𝐺) = 𝑉)
63	34	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → (𝐿‘𝐺) ⊆ (𝐿‘(𝐸 + 𝐺)))
64	62, 63	eqsstrrd 4022	. . . . . 6 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → 𝑉 ⊆ (𝐿‘(𝐸 + 𝐺)))
65	61, 64	eqssd 4000	. . . . 5 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → (𝐿‘(𝐸 + 𝐺)) = 𝑉)
66	65	fveq2d 6896	. . . 4 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) = ( ⊥ ‘𝑉))
67	66	fveq2d 6896	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = ( ⊥ ‘( ⊥ ‘𝑉)))
68	59, 67, 65	3eqtr4d 2781	. 2 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
69	7, 36, 20, 27, 37, 25	lkrshpor 38281	. 2 ⊢ (𝜑 → ((𝐿‘𝐺) ∈ (LSHyp‘𝑈) ∨ (𝐿‘𝐺) = 𝑉))
70	58, 68, 69	mpjaodan 956	1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 844 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ⊆ wss 3949 {csn 4629 ran crn 5678 ‘cfv 6544 (class class class)co 7412 Basecbs 17149 +_gcplusg 17202 ._rcmulr 17203 Scalarcsca 17205 ·_𝑠 cvsca 17206 0_gc0g 17390 -_gcsg 18858 LSSumclsm 19544 inv_rcinvr 20279 LSpanclspn 20727 LVecclvec 20858 LSHypclsh 38149 LFnlclfn 38231 LKerclk 38259 LDualcld 38297 HLchlt 38524 LHypclh 39159 DVecHcdvh 40253 DIsoHcdih 40403 ocHcoch 40522

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-riotaBAD 38127

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7673 df-om 7859 df-1st 7978 df-2nd 7979 df-tpos 8214 df-undef 8261 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-n0 12478 df-z 12564 df-uz 12828 df-fz 13490 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-0g 17392 df-proset 18253 df-poset 18271 df-plt 18288 df-lub 18304 df-glb 18305 df-join 18306 df-meet 18307 df-p0 18383 df-p1 18384 df-lat 18390 df-clat 18457 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-grp 18859 df-minusg 18860 df-sbg 18861 df-subg 19040 df-cntz 19223 df-lsm 19546 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-dvr 20293 df-drng 20503 df-lmod 20617 df-lss 20688 df-lsp 20728 df-lvec 20859 df-lsatoms 38150 df-lshyp 38151 df-lfl 38232 df-lkr 38260 df-ldual 38298 df-oposet 38350 df-ol 38352 df-oml 38353 df-covers 38440 df-ats 38441 df-atl 38472 df-cvlat 38496 df-hlat 38525 df-llines 38673 df-lplanes 38674 df-lvols 38675 df-lines 38676 df-psubsp 38678 df-pmap 38679 df-padd 38971 df-lhyp 39163 df-laut 39164 df-ldil 39279 df-ltrn 39280 df-trl 39334 df-tendo 39930 df-edring 39932 df-disoa 40204 df-dvech 40254 df-dib 40314 df-dic 40348 df-dih 40404 df-doch 40523

This theorem is referenced by: lclkrlem2t 40701

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