lclkrlem2s - Mathbox for Norm Megill

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Theorem lclkrlem2s 40488

Description: Lemma for lclkr 40496. 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 4812	. . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
4		lclkrlem2o.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
5		eqid 2732	. . . . . . . 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 40316	. . . . . . 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 40330	. . . . . 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 724	. . . 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 40487	. . . . . . . . 9 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (𝐿‘(𝐸 + 𝐺)))
35	34	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → (𝐿‘𝐺) ⊆ (𝐿‘(𝐸 + 𝐺)))
36		eqid 2732	. . . . . . . . 9 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈)
37	4, 6, 1	dvhlvec 40072	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ LVec)
38	37	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → 𝑈 ∈ LVec)
39		simplr 767	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → (𝐿‘𝐺) ∈ (LSHyp‘𝑈))
40		simpr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈))
41	36, 38, 39, 40	lshpcmp 37950	. . . . . . . 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 724	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
44	42, 43	eqtr3d 2774	. . . . . 6 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → (𝐿‘(𝐸 + 𝐺)) = ( ⊥ ‘{𝑌}))
45	44	fveq2d 6895	. . . . 5 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) = ( ⊥ ‘( ⊥ ‘{𝑌})))
46	45	fveq2d 6895	. . . 4 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑌}))))
47	13, 46, 44	3eqtr4d 2782	. . 3 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
48	4, 6, 8, 7, 1	dochoc1 40324	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
49	48	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) = 𝑉) → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
50		simpr 485	. . . . . 6 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) = 𝑉) → (𝐿‘(𝐸 + 𝐺)) = 𝑉)
51	50	fveq2d 6895	. . . . 5 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) = 𝑉) → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) = ( ⊥ ‘𝑉))
52	51	fveq2d 6895	. . . 4 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = ( ⊥ ‘( ⊥ ‘𝑉)))
53	49, 52, 50	3eqtr4d 2782	. . 3 ⊢ (((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) ∧ (𝐿‘(𝐸 + 𝐺)) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
54	4, 6, 1	dvhlmod 40073	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
55	20, 21, 22, 54, 24, 25	ldualvaddcl 38092	. . . . 5 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹)
56	7, 36, 20, 27, 37, 55	lkrshpor 38069	. . . 4 ⊢ (𝜑 → ((𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈) ∨ (𝐿‘(𝐸 + 𝐺)) = 𝑉))
57	56	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) → ((𝐿‘(𝐸 + 𝐺)) ∈ (LSHyp‘𝑈) ∨ (𝐿‘(𝐸 + 𝐺)) = 𝑉))
58	47, 53, 57	mpjaodan 957	. 2 ⊢ ((𝜑 ∧ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
59	48	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
60	7, 20, 27, 54, 55	lkrssv 38058	. . . . . . 7 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ⊆ 𝑉)
61	60	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → (𝐿‘(𝐸 + 𝐺)) ⊆ 𝑉)
62		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → (𝐿‘𝐺) = 𝑉)
63	34	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → (𝐿‘𝐺) ⊆ (𝐿‘(𝐸 + 𝐺)))
64	62, 63	eqsstrrd 4021	. . . . . 6 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → 𝑉 ⊆ (𝐿‘(𝐸 + 𝐺)))
65	61, 64	eqssd 3999	. . . . 5 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → (𝐿‘(𝐸 + 𝐺)) = 𝑉)
66	65	fveq2d 6895	. . . 4 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘(𝐿‘(𝐸 + 𝐺))) = ( ⊥ ‘𝑉))
67	66	fveq2d 6895	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = ( ⊥ ‘( ⊥ ‘𝑉)))
68	59, 67, 65	3eqtr4d 2782	. 2 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
69	7, 36, 20, 27, 37, 25	lkrshpor 38069	. 2 ⊢ (𝜑 → ((𝐿‘𝐺) ∈ (LSHyp‘𝑈) ∨ (𝐿‘𝐺) = 𝑉))
70	58, 68, 69	mpjaodan 957	1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ⊆ wss 3948 {csn 4628 ran crn 5677 ‘cfv 6543 (class class class)co 7411 Basecbs 17146 +_gcplusg 17199 ._rcmulr 17200 Scalarcsca 17202 ·_𝑠 cvsca 17203 0_gc0g 17387 -_gcsg 18823 LSSumclsm 19504 inv_rcinvr 20205 LSpanclspn 20587 LVecclvec 20718 LSHypclsh 37937 LFnlclfn 38019 LKerclk 38047 LDualcld 38085 HLchlt 38312 LHypclh 38947 DVecHcdvh 40041 DIsoHcdih 40191 ocHcoch 40310

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 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 37915

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-undef 8260 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-n0 12475 df-z 12561 df-uz 12825 df-fz 13487 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-sca 17215 df-vsca 17216 df-0g 17389 df-proset 18250 df-poset 18268 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18387 df-clat 18454 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-submnd 18674 df-grp 18824 df-minusg 18825 df-sbg 18826 df-subg 19005 df-cntz 19183 df-lsm 19506 df-cmn 19652 df-abl 19653 df-mgp 19990 df-ur 20007 df-ring 20060 df-oppr 20154 df-dvdsr 20175 df-unit 20176 df-invr 20206 df-dvr 20219 df-drng 20363 df-lmod 20477 df-lss 20548 df-lsp 20588 df-lvec 20719 df-lsatoms 37938 df-lshyp 37939 df-lfl 38020 df-lkr 38048 df-ldual 38086 df-oposet 38138 df-ol 38140 df-oml 38141 df-covers 38228 df-ats 38229 df-atl 38260 df-cvlat 38284 df-hlat 38313 df-llines 38461 df-lplanes 38462 df-lvols 38463 df-lines 38464 df-psubsp 38466 df-pmap 38467 df-padd 38759 df-lhyp 38951 df-laut 38952 df-ldil 39067 df-ltrn 39068 df-trl 39122 df-tendo 39718 df-edring 39720 df-disoa 39992 df-dvech 40042 df-dib 40102 df-dic 40136 df-dih 40192 df-doch 40311

This theorem is referenced by: lclkrlem2t 40489

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