Theorem lcfrlem25 41585

Description: Lemma for lcfr 41603. Special case of lcfrlem35 41595 when ((𝐽‘𝑌)‘𝐼) is zero. (Contributed by NM, 11-Mar-2015.)

Hypotheses

Ref	Expression
lcfrlem17.h	⊢ 𝐻 = (LHyp‘𝐾)
lcfrlem17.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lcfrlem17.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lcfrlem17.v	⊢ 𝑉 = (Base‘𝑈)
lcfrlem17.p	⊢ + = (+g‘𝑈)
lcfrlem17.z	⊢ 0 = (0g‘𝑈)
lcfrlem17.n	⊢ 𝑁 = (LSpan‘𝑈)
lcfrlem17.a	⊢ 𝐴 = (LSAtoms‘𝑈)
lcfrlem17.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lcfrlem17.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
lcfrlem17.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
lcfrlem17.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
lcfrlem22.b	⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))
lcfrlem24.t	⊢ · = ( ·𝑠 ‘𝑈)
lcfrlem24.s	⊢ 𝑆 = (Scalar‘𝑈)
lcfrlem24.q	⊢ 𝑄 = (0g‘𝑆)
lcfrlem24.r	⊢ 𝑅 = (Base‘𝑆)
lcfrlem24.j	⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
lcfrlem24.ib	⊢ (𝜑 → 𝐼 ∈ 𝐵)
lcfrlem24.l	⊢ 𝐿 = (LKer‘𝑈)
lcfrlem25.d	⊢ 𝐷 = (LDual‘𝑈)
lcfrlem25.jz	⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) = 𝑄)
lcfrlem25.in	⊢ (𝜑 → 𝐼 ≠ 0 )

Assertion

Ref	Expression
lcfrlem25	⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) = (𝐿‘(𝐽‘𝑌)))

Distinct variable groups:   𝑣,𝑘,𝑤,𝑥, ⊥   + ,𝑘,𝑣,𝑤,𝑥   𝑅,𝑘,𝑣,𝑥   𝑆,𝑘   · ,𝑘,𝑣,𝑤,𝑥   𝑣,𝑉,𝑥   𝑘,𝑋,𝑣,𝑤,𝑥   𝑘,𝑌,𝑣,𝑤,𝑥   𝑥, 0

Allowed substitution hints:   𝜑(𝑥,𝑤,𝑣,𝑘)   𝐴(𝑥,𝑤,𝑣,𝑘)   𝐵(𝑥,𝑤,𝑣,𝑘)   𝐷(𝑥,𝑤,𝑣,𝑘)   𝑄(𝑥,𝑤,𝑣,𝑘)   𝑅(𝑤)   𝑆(𝑥,𝑤,𝑣)   𝑈(𝑥,𝑤,𝑣,𝑘)   𝐻(𝑥,𝑤,𝑣,𝑘)   𝐼(𝑥,𝑤,𝑣,𝑘)   𝐽(𝑥,𝑤,𝑣,𝑘)   𝐾(𝑥,𝑤,𝑣,𝑘)   𝐿(𝑥,𝑤,𝑣,𝑘)   𝑁(𝑥,𝑤,𝑣,𝑘)   𝑉(𝑤,𝑘)   𝑊(𝑥,𝑤,𝑣,𝑘)   0 (𝑤,𝑣,𝑘)

Proof of Theorem lcfrlem25

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lcfrlem17.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		lcfrlem17.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
3		lcfrlem17.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		lcfrlem17.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
5		lcfrlem17.p	. . . 4 ⊢ + = (+g‘𝑈)
6		lcfrlem17.z	. . . 4 ⊢ 0 = (0g‘𝑈)
7		lcfrlem17.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑈)
8		lcfrlem17.a	. . . 4 ⊢ 𝐴 = (LSAtoms‘𝑈)
9		lcfrlem17.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10		lcfrlem17.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
11		lcfrlem17.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
12		lcfrlem17.ne	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
13		lcfrlem22.b	. . . 4 ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))
14		eqid 2730	. . . 4 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	lcfrlem23 41583	. . 3 ⊢ (𝜑 → (( ⊥ ‘{𝑋, 𝑌})(LSSum‘𝑈)𝐵) = ( ⊥ ‘{(𝑋 + 𝑌)}))
16		lcfrlem24.t	. . . . . 6 ⊢ · = ( ·𝑠 ‘𝑈)
17		lcfrlem24.s	. . . . . 6 ⊢ 𝑆 = (Scalar‘𝑈)
18		lcfrlem24.q	. . . . . 6 ⊢ 𝑄 = (0g‘𝑆)
19		lcfrlem24.r	. . . . . 6 ⊢ 𝑅 = (Base‘𝑆)
20		lcfrlem24.j	. . . . . 6 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
21		lcfrlem24.ib	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
22		lcfrlem24.l	. . . . . 6 ⊢ 𝐿 = (LKer‘𝑈)
23	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22	lcfrlem24 41584	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑋, 𝑌}) = ((𝐿‘(𝐽‘𝑋)) ∩ (𝐿‘(𝐽‘𝑌))))
24		inss2 4186	. . . . 5 ⊢ ((𝐿‘(𝐽‘𝑋)) ∩ (𝐿‘(𝐽‘𝑌))) ⊆ (𝐿‘(𝐽‘𝑌))
25	23, 24	eqsstrdi 3977	. . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝑋, 𝑌}) ⊆ (𝐿‘(𝐽‘𝑌)))
26	1, 3, 9	dvhlvec 41127	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec)
27	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	lcfrlem22 41582	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
28		lcfrlem25.in	. . . . . 6 ⊢ (𝜑 → 𝐼 ≠ 0 )
29	6, 7, 8, 26, 27, 21, 28	lsatel 39023	. . . . 5 ⊢ (𝜑 → 𝐵 = (𝑁‘{𝐼}))
30		eqid 2730	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
31	1, 3, 9	dvhlmod 41128	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
32		eqid 2730	. . . . . . . 8 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
33		lcfrlem25.d	. . . . . . . 8 ⊢ 𝐷 = (LDual‘𝑈)
34		eqid 2730	. . . . . . . 8 ⊢ (0g‘𝐷) = (0g‘𝐷)
35		eqid 2730	. . . . . . . 8 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
36	1, 2, 3, 4, 5, 16, 17, 19, 6, 32, 22, 33, 34, 35, 20, 9, 11	lcfrlem10 41570	. . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (LFnl‘𝑈))
37	32, 22, 30	lkrlss 39113	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑌) ∈ (LFnl‘𝑈)) → (𝐿‘(𝐽‘𝑌)) ∈ (LSubSp‘𝑈))
38	31, 36, 37	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑌)) ∈ (LSubSp‘𝑈))
39		lcfrlem25.jz	. . . . . . 7 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) = 𝑄)
40	4, 8, 31, 27	lsatssv 39016	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ 𝑉)
41	40, 21	sseldd 3933	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
42	4, 17, 18, 32, 22, 31, 36, 41	ellkr2 39109	. . . . . . 7 ⊢ (𝜑 → (𝐼 ∈ (𝐿‘(𝐽‘𝑌)) ↔ ((𝐽‘𝑌)‘𝐼) = 𝑄))
43	39, 42	mpbird 257	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (𝐿‘(𝐽‘𝑌)))
44	30, 7, 31, 38, 43	ellspsn5 20922	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝐼}) ⊆ (𝐿‘(𝐽‘𝑌)))
45	29, 44	eqsstrd 3967	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ (𝐿‘(𝐽‘𝑌)))
46	30	lsssssubg 20884	. . . . . . 7 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
47	31, 46	syl 17	. . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
48	10	eldifad 3912	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
49	11	eldifad 3912	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
50		prssi 4771	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ⊆ 𝑉)
51	48, 49, 50	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ 𝑉)
52	1, 3, 4, 30, 2	dochlss 41372	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋, 𝑌} ⊆ 𝑉) → ( ⊥ ‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
53	9, 51, 52	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
54	47, 53	sseldd 3933	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑋, 𝑌}) ∈ (SubGrp‘𝑈))
55	4, 30, 7, 31, 48, 49	lspprcl 20904	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
56	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	lcfrlem17 41577	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑉 ∖ { 0 }))
57	56	eldifad 3912	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉)
58	57	snssd 4759	. . . . . . . . 9 ⊢ (𝜑 → {(𝑋 + 𝑌)} ⊆ 𝑉)
59	1, 3, 4, 30, 2	dochlss 41372	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {(𝑋 + 𝑌)} ⊆ 𝑉) → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈))
60	9, 58, 59	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈))
61	30	lssincl 20891	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈) ∧ ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈)) → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ (LSubSp‘𝑈))
62	31, 55, 60, 61	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ (LSubSp‘𝑈))
63	13, 62	eqeltrid 2833	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (LSubSp‘𝑈))
64	47, 63	sseldd 3933	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘𝑈))
65	47, 38	sseldd 3933	. . . . 5 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑌)) ∈ (SubGrp‘𝑈))
66	14	lsmlub 19569	. . . . 5 ⊢ ((( ⊥ ‘{𝑋, 𝑌}) ∈ (SubGrp‘𝑈) ∧ 𝐵 ∈ (SubGrp‘𝑈) ∧ (𝐿‘(𝐽‘𝑌)) ∈ (SubGrp‘𝑈)) → ((( ⊥ ‘{𝑋, 𝑌}) ⊆ (𝐿‘(𝐽‘𝑌)) ∧ 𝐵 ⊆ (𝐿‘(𝐽‘𝑌))) ↔ (( ⊥ ‘{𝑋, 𝑌})(LSSum‘𝑈)𝐵) ⊆ (𝐿‘(𝐽‘𝑌))))
67	54, 64, 65, 66	syl3anc 1373	. . . 4 ⊢ (𝜑 → ((( ⊥ ‘{𝑋, 𝑌}) ⊆ (𝐿‘(𝐽‘𝑌)) ∧ 𝐵 ⊆ (𝐿‘(𝐽‘𝑌))) ↔ (( ⊥ ‘{𝑋, 𝑌})(LSSum‘𝑈)𝐵) ⊆ (𝐿‘(𝐽‘𝑌))))
68	25, 45, 67	mpbi2and 712	. . 3 ⊢ (𝜑 → (( ⊥ ‘{𝑋, 𝑌})(LSSum‘𝑈)𝐵) ⊆ (𝐿‘(𝐽‘𝑌)))
69	15, 68	eqsstrrd 3968	. 2 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) ⊆ (𝐿‘(𝐽‘𝑌)))
70		eqid 2730	. . 3 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈)
71	1, 2, 3, 4, 6, 70, 9, 56	dochsnshp 41471	. . 3 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ (LSHyp‘𝑈))
72	1, 2, 3, 4, 5, 16, 17, 19, 6, 32, 22, 33, 34, 35, 20, 9, 11	lcfrlem13 41573	. . . . 5 ⊢ (𝜑 → (𝐽‘𝑌) ∈ ({𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∖ {(0g‘𝐷)}))
73		eldifsni 4740	. . . . 5 ⊢ ((𝐽‘𝑌) ∈ ({𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∖ {(0g‘𝐷)}) → (𝐽‘𝑌) ≠ (0g‘𝐷))
74	72, 73	syl 17	. . . 4 ⊢ (𝜑 → (𝐽‘𝑌) ≠ (0g‘𝐷))
75	70, 32, 22, 33, 34, 26, 36	lduallkr3 39180	. . . 4 ⊢ (𝜑 → ((𝐿‘(𝐽‘𝑌)) ∈ (LSHyp‘𝑈) ↔ (𝐽‘𝑌) ≠ (0g‘𝐷)))
76	74, 75	mpbird 257	. . 3 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑌)) ∈ (LSHyp‘𝑈))
77	70, 26, 71, 76	lshpcmp 39006	. 2 ⊢ (𝜑 → (( ⊥ ‘{(𝑋 + 𝑌)}) ⊆ (𝐿‘(𝐽‘𝑌)) ↔ ( ⊥ ‘{(𝑋 + 𝑌)}) = (𝐿‘(𝐽‘𝑌))))
78	69, 77	mpbid 232	1 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) = (𝐿‘(𝐽‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2110   ≠ wne 2926  ∃wrex 3054  {crab 3393   ∖ cdif 3897   ∩ cin 3899   ⊆ wss 3900  {csn 4574  {cpr 4576   ↦ cmpt 5170  ‘cfv 6477  ℩crio 7297  (class class class)co 7341  Basecbs 17112  +_gcplusg 17153  Scalarcsca 17156   ·_𝑠 cvsca 17157  0_gc0g 17335  SubGrpcsubg 19025  LSSumclsm 19539  LModclmod 20786  LSubSpclss 20857  LSpanclspn 20897  LSAtomsclsa 38992  LSHypclsh 38993  LFnlclfn 39075  LKerclk 39103  LDualcld 39141  HLchlt 39368  LHypclh 40002  DVecHcdvh 41096  ocHcoch 41365

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 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-cnex 11054  ax-resscn 11055  ax-1cn 11056  ax-icn 11057  ax-addcl 11058  ax-addrcl 11059  ax-mulcl 11060  ax-mulrcl 11061  ax-mulcom 11062  ax-addass 11063  ax-mulass 11064  ax-distr 11065  ax-i2m1 11066  ax-1ne0 11067  ax-1rid 11068  ax-rnegex 11069  ax-rrecex 11070  ax-cnre 11071  ax-pre-lttri 11072  ax-pre-lttrn 11073  ax-pre-ltadd 11074  ax-pre-mulgt0 11075  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 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-tp 4579  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7605  df-om 7792  df-1st 7916  df-2nd 7917  df-tpos 8151  df-undef 8198  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-er 8617  df-map 8747  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144  df-sub 11338  df-neg 11339  df-nn 12118  df-2 12180  df-3 12181  df-4 12182  df-5 12183  df-6 12184  df-n0 12374  df-z 12461  df-uz 12725  df-fz 13400  df-struct 17050  df-sets 17067  df-slot 17085  df-ndx 17097  df-base 17113  df-ress 17134  df-plusg 17166  df-mulr 17167  df-sca 17169  df-vsca 17170  df-0g 17337  df-mre 17480  df-mrc 17481  df-acs 17483  df-proset 18192  df-poset 18211  df-plt 18226  df-lub 18242  df-glb 18243  df-join 18244  df-meet 18245  df-p0 18321  df-p1 18322  df-lat 18330  df-clat 18397  df-mgm 18540  df-sgrp 18619  df-mnd 18635  df-submnd 18684  df-grp 18841  df-minusg 18842  df-sbg 18843  df-subg 19028  df-cntz 19222  df-oppg 19251  df-lsm 19541  df-cmn 19687  df-abl 19688  df-mgp 20052  df-rng 20064  df-ur 20093  df-ring 20146  df-oppr 20248  df-dvdsr 20268  df-unit 20269  df-invr 20299  df-dvr 20312  df-drng 20639  df-lmod 20788  df-lss 20858  df-lsp 20898  df-lvec 21030  df-lsatoms 38994  df-lshyp 38995  df-lcv 39037  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 39516  df-lplanes 39517  df-lvols 39518  df-lines 39519  df-psubsp 39521  df-pmap 39522  df-padd 39814  df-lhyp 40006  df-laut 40007  df-ldil 40122  df-ltrn 40123  df-trl 40177  df-tgrp 40761  df-tendo 40773  df-edring 40775  df-dveca 41021  df-disoa 41047  df-dvech 41097  df-dib 41157  df-dic 41191  df-dih 41247  df-doch 41366  df-djh 41413

This theorem is referenced by:  lcfrlem26  41586