lclkrlem2s - Mathbox for Norm Megill

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Theorem lclkrlem2s 39185

Description: Lemma for lclkr 39193. 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 4698	. . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
4		lclkrlem2o.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
5		eqid 2739	. . . . . . . 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 39013	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑌} ⊆ 𝑉) → ( ⊥ ‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
10	1, 3, 9	syl2anc 587	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
11	4, 5, 8	dochoc 39027	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑌}))) = ( ⊥ ‘{𝑌}))
12	1, 10, 11	syl2anc 587	. . . . 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 39184	. . . . . . . . 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 2739	. . . . . . . . 9 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈)
37	4, 6, 1	dvhlvec 38769	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ LVec)
38	37	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → 𝑈 ∈ LVec)
39		simplr 769	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → (𝐿‘𝐺) ∈ (LSHyp‘𝑈))
40		simpr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈))
41	36, 38, 39, 40	lshpcmp 36648	. . . . . . . 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 235	. . . . . . 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 2776	. . . . . 6 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → (𝐿‘(𝐸 + 𝐺)) = ( ⊥ ‘{𝑌}))
45	44	fveq2d 6681	. . . . 5 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) = ( ⊥ ‘( ⊥ ‘{𝑌})))
46	45	fveq2d 6681	. . . 4 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑌}))))
47	13, 46, 44	3eqtr4d 2784	. . 3 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
48	4, 6, 8, 7, 1	dochoc1 39021	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
49	48	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) = 𝑉) → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
50		simpr 488	. . . . . 6 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) = 𝑉) → (𝐿‘(𝐸 + 𝐺)) = 𝑉)
51	50	fveq2d 6681	. . . . 5 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) = 𝑉) → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) = ( ⊥ ‘𝑉))
52	51	fveq2d 6681	. . . 4 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = ( ⊥ ‘( ⊥ ‘𝑉)))
53	49, 52, 50	3eqtr4d 2784	. . 3 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
54	4, 6, 1	dvhlmod 38770	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
55	20, 21, 22, 54, 24, 25	ldualvaddcl 36790	. . . . 5 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹)
56	7, 36, 20, 27, 37, 55	lkrshpor 36767	. . . 4 ⊢ (𝜑 → ((𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈) ∨ (𝐿‘(𝐸 + 𝐺)) = 𝑉))
57	56	adantr 484	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) → ((𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈) ∨ (𝐿‘(𝐸 + 𝐺)) = 𝑉))
58	47, 53, 57	mpjaodan 958	. 2 ⊢ ((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
59	48	adantr 484	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
60	7, 20, 27, 54, 55	lkrssv 36756	. . . . . . 7 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ⊆ 𝑉)
61	60	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → (𝐿‘(𝐸 + 𝐺)) ⊆ 𝑉)
62		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → (𝐿‘𝐺) = 𝑉)
63	34	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → (𝐿‘𝐺) ⊆ (𝐿‘(𝐸 + 𝐺)))
64	62, 63	eqsstrrd 3917	. . . . . 6 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → 𝑉 ⊆ (𝐿‘(𝐸 + 𝐺)))
65	61, 64	eqssd 3895	. . . . 5 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → (𝐿‘(𝐸 + 𝐺)) = 𝑉)
66	65	fveq2d 6681	. . . 4 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) = ( ⊥ ‘𝑉))
67	66	fveq2d 6681	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = ( ⊥ ‘( ⊥ ‘𝑉)))
68	59, 67, 65	3eqtr4d 2784	. 2 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
69	7, 36, 20, 27, 37, 25	lkrshpor 36767	. 2 ⊢ (𝜑 → ((𝐿‘𝐺) ∈ (LSHyp‘𝑈) ∨ (𝐿‘𝐺) = 𝑉))
70	58, 68, 69	mpjaodan 958	1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∨ wo 846 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 ⊆ wss 3844 {csn 4517 ran crn 5527 ‘cfv 6340 (class class class)co 7173 Basecbs 16589 +_gcplusg 16671 ._rcmulr 16672 Scalarcsca 16674 ·_𝑠 cvsca 16675 0_gc0g 16819 -_gcsg 18224 LSSumclsm 18880 inv_rcinvr 19546 LSpanclspn 19865 LVecclvec 19996 LSHypclsh 36635 LFnlclfn 36717 LKerclk 36745 LDualcld 36783 HLchlt 37010 LHypclh 37644 DVecHcdvh 38738 DIsoHcdih 38888 ocHcoch 39007

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 ax-riotaBAD 36613

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-iin 4885 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-of 7428 df-om 7603 df-1st 7717 df-2nd 7718 df-tpos 7924 df-undef 7971 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-er 8323 df-map 8442 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-nn 11720 df-2 11782 df-3 11783 df-4 11784 df-5 11785 df-6 11786 df-n0 11980 df-z 12066 df-uz 12328 df-fz 12985 df-struct 16591 df-ndx 16592 df-slot 16593 df-base 16595 df-sets 16596 df-ress 16597 df-plusg 16684 df-mulr 16685 df-sca 16687 df-vsca 16688 df-0g 16821 df-proset 17657 df-poset 17675 df-plt 17687 df-lub 17703 df-glb 17704 df-join 17705 df-meet 17706 df-p0 17768 df-p1 17769 df-lat 17775 df-clat 17837 df-mgm 17971 df-sgrp 18020 df-mnd 18031 df-submnd 18076 df-grp 18225 df-minusg 18226 df-sbg 18227 df-subg 18397 df-cntz 18568 df-lsm 18882 df-cmn 19029 df-abl 19030 df-mgp 19362 df-ur 19374 df-ring 19421 df-oppr 19498 df-dvdsr 19516 df-unit 19517 df-invr 19547 df-dvr 19558 df-drng 19626 df-lmod 19758 df-lss 19826 df-lsp 19866 df-lvec 19997 df-lsatoms 36636 df-lshyp 36637 df-lfl 36718 df-lkr 36746 df-ldual 36784 df-oposet 36836 df-ol 36838 df-oml 36839 df-covers 36926 df-ats 36927 df-atl 36958 df-cvlat 36982 df-hlat 37011 df-llines 37158 df-lplanes 37159 df-lvols 37160 df-lines 37161 df-psubsp 37163 df-pmap 37164 df-padd 37456 df-lhyp 37648 df-laut 37649 df-ldil 37764 df-ltrn 37765 df-trl 37819 df-tendo 38415 df-edring 38417 df-disoa 38689 df-dvech 38739 df-dib 38799 df-dic 38833 df-dih 38889 df-doch 39008

This theorem is referenced by: lclkrlem2t 39186

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