lclkrlem2s - Mathbox for Norm Megill

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Theorem lclkrlem2s 41544

Description: Lemma for lclkr 41552. 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 4785	. . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
4		lclkrlem2o.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
5		eqid 2735	. . . . . . . 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 41372	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑌} ⊆ 𝑉) → ( ⊥ ‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
10	1, 3, 9	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
11	4, 5, 8	dochoc 41386	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑌}))) = ( ⊥ ‘{𝑌}))
12	1, 10, 11	syl2anc 584	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑌}))) = ( ⊥ ‘{𝑌}))
13	12	ad2antrr 726	. . . 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 41543	. . . . . . . . 9 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (𝐿‘(𝐸 + 𝐺)))
35	34	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → (𝐿‘𝐺) ⊆ (𝐿‘(𝐸 + 𝐺)))
36		eqid 2735	. . . . . . . . 9 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈)
37	4, 6, 1	dvhlvec 41128	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ LVec)
38	37	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → 𝑈 ∈ LVec)
39		simplr 768	. . . . . . . . 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 39006	. . . . . . . 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 232	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → (𝐿‘𝐺) = (𝐿‘(𝐸 + 𝐺)))
43	30	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
44	42, 43	eqtr3d 2772	. . . . . 6 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → (𝐿‘(𝐸 + 𝐺)) = ( ⊥ ‘{𝑌}))
45	44	fveq2d 6880	. . . . 5 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) = ( ⊥ ‘( ⊥ ‘{𝑌})))
46	45	fveq2d 6880	. . . 4 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑌}))))
47	13, 46, 44	3eqtr4d 2780	. . 3 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
48	4, 6, 8, 7, 1	dochoc1 41380	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
49	48	ad2antrr 726	. . . 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 6880	. . . . 5 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) = 𝑉) → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) = ( ⊥ ‘𝑉))
52	51	fveq2d 6880	. . . 4 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = ( ⊥ ‘( ⊥ ‘𝑉)))
53	49, 52, 50	3eqtr4d 2780	. . 3 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
54	4, 6, 1	dvhlmod 41129	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
55	20, 21, 22, 54, 24, 25	ldualvaddcl 39148	. . . . 5 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹)
56	7, 36, 20, 27, 37, 55	lkrshpor 39125	. . . 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 960	. 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 39114	. . . . . . 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 3994	. . . . . 6 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → 𝑉 ⊆ (𝐿‘(𝐸 + 𝐺)))
65	61, 64	eqssd 3976	. . . . 5 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → (𝐿‘(𝐸 + 𝐺)) = 𝑉)
66	65	fveq2d 6880	. . . 4 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) = ( ⊥ ‘𝑉))
67	66	fveq2d 6880	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = ( ⊥ ‘( ⊥ ‘𝑉)))
68	59, 67, 65	3eqtr4d 2780	. 2 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
69	7, 36, 20, 27, 37, 25	lkrshpor 39125	. 2 ⊢ (𝜑 → ((𝐿‘𝐺) ∈ (LSHyp‘𝑈) ∨ (𝐿‘𝐺) = 𝑉))
70	58, 68, 69	mpjaodan 960	1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ⊆ wss 3926 {csn 4601 ran crn 5655 ‘cfv 6531 (class class class)co 7405 Basecbs 17228 +_gcplusg 17271 ._rcmulr 17272 Scalarcsca 17274 ·_𝑠 cvsca 17275 0_gc0g 17453 -_gcsg 18918 LSSumclsm 19615 inv_rcinvr 20347 LSpanclspn 20928 LVecclvec 21060 LSHypclsh 38993 LFnlclfn 39075 LKerclk 39103 LDualcld 39141 HLchlt 39368 LHypclh 40003 DVecHcdvh 41097 DIsoHcdih 41247 ocHcoch 41366

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-riotaBAD 38971

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-tpos 8225 df-undef 8272 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-map 8842 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-n0 12502 df-z 12589 df-uz 12853 df-fz 13525 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-sca 17287 df-vsca 17288 df-0g 17455 df-proset 18306 df-poset 18325 df-plt 18340 df-lub 18356 df-glb 18357 df-join 18358 df-meet 18359 df-p0 18435 df-p1 18436 df-lat 18442 df-clat 18509 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-submnd 18762 df-grp 18919 df-minusg 18920 df-sbg 18921 df-subg 19106 df-cntz 19300 df-lsm 19617 df-cmn 19763 df-abl 19764 df-mgp 20101 df-rng 20113 df-ur 20142 df-ring 20195 df-oppr 20297 df-dvdsr 20317 df-unit 20318 df-invr 20348 df-dvr 20361 df-drng 20691 df-lmod 20819 df-lss 20889 df-lsp 20929 df-lvec 21061 df-lsatoms 38994 df-lshyp 38995 df-lfl 39076 df-lkr 39104 df-ldual 39142 df-oposet 39194 df-ol 39196 df-oml 39197 df-covers 39284 df-ats 39285 df-atl 39316 df-cvlat 39340 df-hlat 39369 df-llines 39517 df-lplanes 39518 df-lvols 39519 df-lines 39520 df-psubsp 39522 df-pmap 39523 df-padd 39815 df-lhyp 40007 df-laut 40008 df-ldil 40123 df-ltrn 40124 df-trl 40178 df-tendo 40774 df-edring 40776 df-disoa 41048 df-dvech 41098 df-dib 41158 df-dic 41192 df-dih 41248 df-doch 41367

This theorem is referenced by: lclkrlem2t 41545

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