Theorem lcfrlem25 41524

Description: Lemma for lcfr 41542. Special case of lcfrlem35 41534 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 2740	. . . 4 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	lcfrlem23 41522	. . 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 41523	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑋, 𝑌}) = ((𝐿‘(𝐽‘𝑋)) ∩ (𝐿‘(𝐽‘𝑌))))
24		inss2 4259	. . . . 5 ⊢ ((𝐿‘(𝐽‘𝑋)) ∩ (𝐿‘(𝐽‘𝑌))) ⊆ (𝐿‘(𝐽‘𝑌))
25	23, 24	eqsstrdi 4063	. . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝑋, 𝑌}) ⊆ (𝐿‘(𝐽‘𝑌)))
26	1, 3, 9	dvhlvec 41066	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec)
27	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	lcfrlem22 41521	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
28		lcfrlem25.in	. . . . . 6 ⊢ (𝜑 → 𝐼 ≠ 0 )
29	6, 7, 8, 26, 27, 21, 28	lsatel 38961	. . . . 5 ⊢ (𝜑 → 𝐵 = (𝑁‘{𝐼}))
30		eqid 2740	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
31	1, 3, 9	dvhlmod 41067	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
32		eqid 2740	. . . . . . . 8 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
33		lcfrlem25.d	. . . . . . . 8 ⊢ 𝐷 = (LDual‘𝑈)
34		eqid 2740	. . . . . . . 8 ⊢ (0g‘𝐷) = (0g‘𝐷)
35		eqid 2740	. . . . . . . 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 41509	. . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (LFnl‘𝑈))
37	32, 22, 30	lkrlss 39051	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑌) ∈ (LFnl‘𝑈)) → (𝐿‘(𝐽‘𝑌)) ∈ (LSubSp‘𝑈))
38	31, 36, 37	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑌)) ∈ (LSubSp‘𝑈))
39		lcfrlem25.jz	. . . . . . 7 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) = 𝑄)
40	4, 8, 31, 27	lsatssv 38954	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ 𝑉)
41	40, 21	sseldd 4009	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
42	4, 17, 18, 32, 22, 31, 36, 41	ellkr2 39047	. . . . . . 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 21017	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝐼}) ⊆ (𝐿‘(𝐽‘𝑌)))
45	29, 44	eqsstrd 4047	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ (𝐿‘(𝐽‘𝑌)))
46	30	lsssssubg 20979	. . . . . . 7 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
47	31, 46	syl 17	. . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
48	10	eldifad 3988	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
49	11	eldifad 3988	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
50		prssi 4846	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ⊆ 𝑉)
51	48, 49, 50	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ 𝑉)
52	1, 3, 4, 30, 2	dochlss 41311	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋, 𝑌} ⊆ 𝑉) → ( ⊥ ‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
53	9, 51, 52	syl2anc 583	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
54	47, 53	sseldd 4009	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑋, 𝑌}) ∈ (SubGrp‘𝑈))
55	4, 30, 7, 31, 48, 49	lspprcl 20999	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
56	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	lcfrlem17 41516	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑉 ∖ { 0 }))
57	56	eldifad 3988	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉)
58	57	snssd 4834	. . . . . . . . 9 ⊢ (𝜑 → {(𝑋 + 𝑌)} ⊆ 𝑉)
59	1, 3, 4, 30, 2	dochlss 41311	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {(𝑋 + 𝑌)} ⊆ 𝑉) → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈))
60	9, 58, 59	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈))
61	30	lssincl 20986	. . . . . . . 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 1371	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ (LSubSp‘𝑈))
63	13, 62	eqeltrid 2848	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (LSubSp‘𝑈))
64	47, 63	sseldd 4009	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘𝑈))
65	47, 38	sseldd 4009	. . . . 5 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑌)) ∈ (SubGrp‘𝑈))
66	14	lsmlub 19706	. . . . 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 1371	. . . 4 ⊢ (𝜑 → ((( ⊥ ‘{𝑋, 𝑌}) ⊆ (𝐿‘(𝐽‘𝑌)) ∧ 𝐵 ⊆ (𝐿‘(𝐽‘𝑌))) ↔ (( ⊥ ‘{𝑋, 𝑌})(LSSum‘𝑈)𝐵) ⊆ (𝐿‘(𝐽‘𝑌))))
68	25, 45, 67	mpbi2and 711	. . 3 ⊢ (𝜑 → (( ⊥ ‘{𝑋, 𝑌})(LSSum‘𝑈)𝐵) ⊆ (𝐿‘(𝐽‘𝑌)))
69	15, 68	eqsstrrd 4048	. 2 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) ⊆ (𝐿‘(𝐽‘𝑌)))
70		eqid 2740	. . 3 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈)
71	1, 2, 3, 4, 6, 70, 9, 56	dochsnshp 41410	. . 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 41512	. . . . 5 ⊢ (𝜑 → (𝐽‘𝑌) ∈ ({𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∖ {(0g‘𝐷)}))
73		eldifsni 4815	. . . . 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 39118	. . . 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 38944	. 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 1537   ∈ wcel 2108   ≠ wne 2946  ∃wrex 3076  {crab 3443   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976  {csn 4648  {cpr 4650   ↦ cmpt 5249  ‘cfv 6573  ℩crio 7403  (class class class)co 7448  Basecbs 17258  +_gcplusg 17311  Scalarcsca 17314   ·_𝑠 cvsca 17315  0_gc0g 17499  SubGrpcsubg 19160  LSSumclsm 19676  LModclmod 20880  LSubSpclss 20952  LSpanclspn 20992  LSAtomsclsa 38930  LSHypclsh 38931  LFnlclfn 39013  LKerclk 39041  LDualcld 39079  HLchlt 39306  LHypclh 39941  DVecHcdvh 41035  ocHcoch 41304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-riotaBAD 38909

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-om 7904  df-1st 8030  df-2nd 8031  df-tpos 8267  df-undef 8314  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-0g 17501  df-mre 17644  df-mrc 17645  df-acs 17647  df-proset 18365  df-poset 18383  df-plt 18400  df-lub 18416  df-glb 18417  df-join 18418  df-meet 18419  df-p0 18495  df-p1 18496  df-lat 18502  df-clat 18569  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-submnd 18819  df-grp 18976  df-minusg 18977  df-sbg 18978  df-subg 19163  df-cntz 19357  df-oppg 19386  df-lsm 19678  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-ring 20262  df-oppr 20360  df-dvdsr 20383  df-unit 20384  df-invr 20414  df-dvr 20427  df-drng 20753  df-lmod 20882  df-lss 20953  df-lsp 20993  df-lvec 21125  df-lsatoms 38932  df-lshyp 38933  df-lcv 38975  df-lfl 39014  df-lkr 39042  df-ldual 39080  df-oposet 39132  df-ol 39134  df-oml 39135  df-covers 39222  df-ats 39223  df-atl 39254  df-cvlat 39278  df-hlat 39307  df-llines 39455  df-lplanes 39456  df-lvols 39457  df-lines 39458  df-psubsp 39460  df-pmap 39461  df-padd 39753  df-lhyp 39945  df-laut 39946  df-ldil 40061  df-ltrn 40062  df-trl 40116  df-tgrp 40700  df-tendo 40712  df-edring 40714  df-dveca 40960  df-disoa 40986  df-dvech 41036  df-dib 41096  df-dic 41130  df-dih 41186  df-doch 41305  df-djh 41352

This theorem is referenced by:  lcfrlem26  41525