Theorem lcfrlem25 42203

Description: Lemma for lcfr 42221. Special case of lcfrlem35 42213 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 2765	. . . 4 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	lcfrlem23 42201	. . 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 42202	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑋, 𝑌}) = ((𝐿‘(𝐽‘𝑋)) ∩ (𝐿‘(𝐽‘𝑌))))
24		inss2 4192	. . . . 5 ⊢ ((𝐿‘(𝐽‘𝑋)) ∩ (𝐿‘(𝐽‘𝑌))) ⊆ (𝐿‘(𝐽‘𝑌))
25	23, 24	eqsstrdi 3983	. . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝑋, 𝑌}) ⊆ (𝐿‘(𝐽‘𝑌)))
26	1, 3, 9	dvhlvec 41745	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec)
27	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	lcfrlem22 42200	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
28		lcfrlem25.in	. . . . . 6 ⊢ (𝜑 → 𝐼 ≠ 0 )
29	6, 7, 8, 26, 27, 21, 28	lsatel 39641	. . . . 5 ⊢ (𝜑 → 𝐵 = (𝑁‘{𝐼}))
30		eqid 2765	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
31	1, 3, 9	dvhlmod 41746	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
32		eqid 2765	. . . . . . . 8 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
33		lcfrlem25.d	. . . . . . . 8 ⊢ 𝐷 = (LDual‘𝑈)
34		eqid 2765	. . . . . . . 8 ⊢ (0g‘𝐷) = (0g‘𝐷)
35		eqid 2765	. . . . . . . 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 42188	. . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (LFnl‘𝑈))
37	32, 22, 30	lkrlss 39731	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑌) ∈ (LFnl‘𝑈)) → (𝐿‘(𝐽‘𝑌)) ∈ (LSubSp‘𝑈))
38	31, 36, 37	syl2anc 595	. . . . . 6 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑌)) ∈ (LSubSp‘𝑈))
39		lcfrlem25.jz	. . . . . . 7 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) = 𝑄)
40	4, 8, 31, 27	lsatssv 39634	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ 𝑉)
41	40, 21	sseldd 3940	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
42	4, 17, 18, 32, 22, 31, 36, 41	ellkr2 39727	. . . . . . 7 ⊢ (𝜑 → (𝐼 ∈ (𝐿‘(𝐽‘𝑌)) ↔ ((𝐽‘𝑌)‘𝐼) = 𝑄))
43	39, 42	mpbird 260	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (𝐿‘(𝐽‘𝑌)))
44	30, 7, 31, 38, 43	ellspsn5 21086	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝐼}) ⊆ (𝐿‘(𝐽‘𝑌)))
45	29, 44	eqsstrd 3973	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ (𝐿‘(𝐽‘𝑌)))
46	30	lsssssubg 21048	. . . . . . 7 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
47	31, 46	syl 18	. . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
48	10	eldifad 3919	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
49	11	eldifad 3919	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
50		prssi 4782	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ⊆ 𝑉)
51	48, 49, 50	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ 𝑉)
52	1, 3, 4, 30, 2	dochlss 41990	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋, 𝑌} ⊆ 𝑉) → ( ⊥ ‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
53	9, 51, 52	syl2anc 595	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
54	47, 53	sseldd 3940	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑋, 𝑌}) ∈ (SubGrp‘𝑈))
55	4, 30, 7, 31, 48, 49	lspprcl 21068	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
56	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	lcfrlem17 42195	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑉 ∖ { 0 }))
57	56	eldifad 3919	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉)
58	57	snssd 4748	. . . . . . . . 9 ⊢ (𝜑 → {(𝑋 + 𝑌)} ⊆ 𝑉)
59	1, 3, 4, 30, 2	dochlss 41990	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {(𝑋 + 𝑌)} ⊆ 𝑉) → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈))
60	9, 58, 59	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈))
61	30	lssincl 21055	. . . . . . . 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 1394	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ (LSubSp‘𝑈))
63	13, 62	eqeltrid 2869	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (LSubSp‘𝑈))
64	47, 63	sseldd 3940	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘𝑈))
65	47, 38	sseldd 3940	. . . . 5 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑌)) ∈ (SubGrp‘𝑈))
66	14	lsmlub 19725	. . . . 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 1394	. . . 4 ⊢ (𝜑 → ((( ⊥ ‘{𝑋, 𝑌}) ⊆ (𝐿‘(𝐽‘𝑌)) ∧ 𝐵 ⊆ (𝐿‘(𝐽‘𝑌))) ↔ (( ⊥ ‘{𝑋, 𝑌})(LSSum‘𝑈)𝐵) ⊆ (𝐿‘(𝐽‘𝑌))))
68	25, 45, 67	mpbi2and 724	. . 3 ⊢ (𝜑 → (( ⊥ ‘{𝑋, 𝑌})(LSSum‘𝑈)𝐵) ⊆ (𝐿‘(𝐽‘𝑌)))
69	15, 68	eqsstrrd 3974	. 2 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) ⊆ (𝐿‘(𝐽‘𝑌)))
70		eqid 2765	. . 3 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈)
71	1, 2, 3, 4, 6, 70, 9, 56	dochsnshp 42089	. . 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 42191	. . . . 5 ⊢ (𝜑 → (𝐽‘𝑌) ∈ ({𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∖ {(0g‘𝐷)}))
73		eldifsni 4753	. . . . 5 ⊢ ((𝐽‘𝑌) ∈ ({𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∖ {(0g‘𝐷)}) → (𝐽‘𝑌) ≠ (0g‘𝐷))
74	72, 73	syl 18	. . . 4 ⊢ (𝜑 → (𝐽‘𝑌) ≠ (0g‘𝐷))
75	70, 32, 22, 33, 34, 26, 36	lduallkr3 39798	. . . 4 ⊢ (𝜑 → ((𝐿‘(𝐽‘𝑌)) ∈ (LSHyp‘𝑈) ↔ (𝐽‘𝑌) ≠ (0g‘𝐷)))
76	74, 75	mpbird 260	. . 3 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑌)) ∈ (LSHyp‘𝑈))
77	70, 26, 71, 76	lshpcmp 39624	. 2 ⊢ (𝜑 → (( ⊥ ‘{(𝑋 + 𝑌)}) ⊆ (𝐿‘(𝐽‘𝑌)) ↔ ( ⊥ ‘{(𝑋 + 𝑌)}) = (𝐿‘(𝐽‘𝑌))))
78	69, 77	mpbid 235	1 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) = (𝐿‘(𝐽‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∃wrex 3089  {crab 3417   ∖ cdif 3904   ∩ cin 3906   ⊆ wss 3907  {csn 4585  {cpr 4587   ↦ cmpt 5186  ‘cfv 6525  ℩crio 7356  (class class class)co 7400  Basecbs 17259  +_gcplusg 17300  Scalarcsca 17303   ·_𝑠 cvsca 17304  0_gc0g 17482  SubGrpcsubg 19177  LSSumclsm 19695  LModclmod 20950  LSubSpclss 21021  LSpanclspn 21061  LSAtomsclsa 39610  LSHypclsh 39611  LFnlclfn 39693  LKerclk 39721  LDualcld 39759  HLchlt 39986  LHypclh 40620  DVecHcdvh 41714  ocHcoch 41983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-riotaBAD 39589

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-iin 4955  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-of 7664  df-om 7851  df-1st 7974  df-2nd 7975  df-tpos 8210  df-undef 8257  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-er 8682  df-map 8814  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-2 12294  df-3 12295  df-4 12296  df-5 12297  df-6 12298  df-n0 12496  df-z 12583  df-uz 12854  df-fz 13527  df-struct 17197  df-sets 17214  df-slot 17232  df-ndx 17244  df-base 17260  df-ress 17281  df-plusg 17313  df-mulr 17314  df-sca 17316  df-vsca 17317  df-0g 17484  df-mre 17628  df-mrc 17629  df-acs 17631  df-proset 18340  df-poset 18359  df-plt 18374  df-lub 18390  df-glb 18391  df-join 18392  df-meet 18393  df-p0 18469  df-p1 18470  df-lat 18478  df-clat 18545  df-mgm 18688  df-sgrp 18767  df-mnd 18783  df-submnd 18832  df-grp 18993  df-minusg 18994  df-sbg 18995  df-subg 19180  df-cntz 19378  df-oppg 19407  df-lsm 19697  df-cmn 19843  df-abl 19844  df-mgp 20208  df-rng 20222  df-ur 20255  df-ring 20308  df-oppr 20410  df-dvdsr 20430  df-unit 20431  df-invr 20461  df-dvr 20474  df-drng 20806  df-lmod 20952  df-lss 21022  df-lsp 21062  df-lvec 21193  df-lsatoms 39612  df-lshyp 39613  df-lcv 39655  df-lfl 39694  df-lkr 39722  df-ldual 39760  df-oposet 39812  df-ol 39814  df-oml 39815  df-covers 39902  df-ats 39903  df-atl 39934  df-cvlat 39958  df-hlat 39987  df-llines 40134  df-lplanes 40135  df-lvols 40136  df-lines 40137  df-psubsp 40139  df-pmap 40140  df-padd 40432  df-lhyp 40624  df-laut 40625  df-ldil 40740  df-ltrn 40741  df-trl 40795  df-tgrp 41379  df-tendo 41391  df-edring 41393  df-dveca 41639  df-disoa 41665  df-dvech 41715  df-dib 41775  df-dic 41809  df-dih 41865  df-doch 41984  df-djh 42031

This theorem is referenced by:  lcfrlem26  42204