Theorem lcfrlem25 41605

Description: Lemma for lcfr 41623. Special case of lcfrlem35 41615 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 2731	. . . 4 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	lcfrlem23 41603	. . 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 41604	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑋, 𝑌}) = ((𝐿‘(𝐽‘𝑋)) ∩ (𝐿‘(𝐽‘𝑌))))
24		inss2 4188	. . . . 5 ⊢ ((𝐿‘(𝐽‘𝑋)) ∩ (𝐿‘(𝐽‘𝑌))) ⊆ (𝐿‘(𝐽‘𝑌))
25	23, 24	eqsstrdi 3979	. . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝑋, 𝑌}) ⊆ (𝐿‘(𝐽‘𝑌)))
26	1, 3, 9	dvhlvec 41147	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec)
27	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	lcfrlem22 41602	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
28		lcfrlem25.in	. . . . . 6 ⊢ (𝜑 → 𝐼 ≠ 0 )
29	6, 7, 8, 26, 27, 21, 28	lsatel 39043	. . . . 5 ⊢ (𝜑 → 𝐵 = (𝑁‘{𝐼}))
30		eqid 2731	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
31	1, 3, 9	dvhlmod 41148	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
32		eqid 2731	. . . . . . . 8 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
33		lcfrlem25.d	. . . . . . . 8 ⊢ 𝐷 = (LDual‘𝑈)
34		eqid 2731	. . . . . . . 8 ⊢ (0g‘𝐷) = (0g‘𝐷)
35		eqid 2731	. . . . . . . 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 41590	. . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (LFnl‘𝑈))
37	32, 22, 30	lkrlss 39133	. . . . . . 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 39036	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ 𝑉)
41	40, 21	sseldd 3935	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
42	4, 17, 18, 32, 22, 31, 36, 41	ellkr2 39129	. . . . . . 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 20927	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝐼}) ⊆ (𝐿‘(𝐽‘𝑌)))
45	29, 44	eqsstrd 3969	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ (𝐿‘(𝐽‘𝑌)))
46	30	lsssssubg 20889	. . . . . . 7 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
47	31, 46	syl 17	. . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
48	10	eldifad 3914	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
49	11	eldifad 3914	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
50		prssi 4773	. . . . . . . 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 41392	. . . . . . 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 3935	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑋, 𝑌}) ∈ (SubGrp‘𝑈))
55	4, 30, 7, 31, 48, 49	lspprcl 20909	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
56	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	lcfrlem17 41597	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑉 ∖ { 0 }))
57	56	eldifad 3914	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉)
58	57	snssd 4761	. . . . . . . . 9 ⊢ (𝜑 → {(𝑋 + 𝑌)} ⊆ 𝑉)
59	1, 3, 4, 30, 2	dochlss 41392	. . . . . . . . 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 20896	. . . . . . . 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 2835	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (LSubSp‘𝑈))
64	47, 63	sseldd 3935	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘𝑈))
65	47, 38	sseldd 3935	. . . . 5 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑌)) ∈ (SubGrp‘𝑈))
66	14	lsmlub 19574	. . . . 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 3970	. 2 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) ⊆ (𝐿‘(𝐽‘𝑌)))
70		eqid 2731	. . 3 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈)
71	1, 2, 3, 4, 6, 70, 9, 56	dochsnshp 41491	. . 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 41593	. . . . 5 ⊢ (𝜑 → (𝐽‘𝑌) ∈ ({𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∖ {(0g‘𝐷)}))
73		eldifsni 4742	. . . . 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 39200	. . . 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 39026	. 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 2111   ≠ wne 2928  ∃wrex 3056  {crab 3395   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902  {csn 4576  {cpr 4578   ↦ cmpt 5172  ‘cfv 6481  ℩crio 7302  (class class class)co 7346  Basecbs 17117  +_gcplusg 17158  Scalarcsca 17161   ·_𝑠 cvsca 17162  0_gc0g 17340  SubGrpcsubg 19030  LSSumclsm 19544  LModclmod 20791  LSubSpclss 20862  LSpanclspn 20902  LSAtomsclsa 39012  LSHypclsh 39013  LFnlclfn 39095  LKerclk 39123  LDualcld 39161  HLchlt 39388  LHypclh 40022  DVecHcdvh 41116  ocHcoch 41385

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080  ax-riotaBAD 38991

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 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-iin 4944  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-of 7610  df-om 7797  df-1st 7921  df-2nd 7922  df-tpos 8156  df-undef 8203  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-nn 12123  df-2 12185  df-3 12186  df-4 12187  df-5 12188  df-6 12189  df-n0 12379  df-z 12466  df-uz 12730  df-fz 13405  df-struct 17055  df-sets 17072  df-slot 17090  df-ndx 17102  df-base 17118  df-ress 17139  df-plusg 17171  df-mulr 17172  df-sca 17174  df-vsca 17175  df-0g 17342  df-mre 17485  df-mrc 17486  df-acs 17488  df-proset 18197  df-poset 18216  df-plt 18231  df-lub 18247  df-glb 18248  df-join 18249  df-meet 18250  df-p0 18326  df-p1 18327  df-lat 18335  df-clat 18402  df-mgm 18545  df-sgrp 18624  df-mnd 18640  df-submnd 18689  df-grp 18846  df-minusg 18847  df-sbg 18848  df-subg 19033  df-cntz 19227  df-oppg 19256  df-lsm 19546  df-cmn 19692  df-abl 19693  df-mgp 20057  df-rng 20069  df-ur 20098  df-ring 20151  df-oppr 20253  df-dvdsr 20273  df-unit 20274  df-invr 20304  df-dvr 20317  df-drng 20644  df-lmod 20793  df-lss 20863  df-lsp 20903  df-lvec 21035  df-lsatoms 39014  df-lshyp 39015  df-lcv 39057  df-lfl 39096  df-lkr 39124  df-ldual 39162  df-oposet 39214  df-ol 39216  df-oml 39217  df-covers 39304  df-ats 39305  df-atl 39336  df-cvlat 39360  df-hlat 39389  df-llines 39536  df-lplanes 39537  df-lvols 39538  df-lines 39539  df-psubsp 39541  df-pmap 39542  df-padd 39834  df-lhyp 40026  df-laut 40027  df-ldil 40142  df-ltrn 40143  df-trl 40197  df-tgrp 40781  df-tendo 40793  df-edring 40795  df-dveca 41041  df-disoa 41067  df-dvech 41117  df-dib 41177  df-dic 41211  df-dih 41267  df-doch 41386  df-djh 41433

This theorem is referenced by:  lcfrlem26  41606