Theorem lcfrlem31 41552

Description: Lemma for lcfr 41564. (Contributed by NM, 10-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)
lcfrlem28.jn	⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄)
lcfrlem29.i	⊢ 𝐹 = (invr‘𝑆)
lcfrlem30.m	⊢ − = (-g‘𝐷)
lcfrlem30.c	⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌)))
lcfrlem31.xi	⊢ (𝜑 → ((𝐽‘𝑋)‘𝐼) ≠ 𝑄)
lcfrlem31.cn	⊢ (𝜑 → 𝐶 = (0g‘𝐷))

Assertion

Ref	Expression
lcfrlem31	⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{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)   𝐶(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)   𝐹(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)   − (𝑥,𝑤,𝑣,𝑘)   𝑁(𝑥,𝑤,𝑣,𝑘)   𝑉(𝑤,𝑘)   𝑊(𝑥,𝑤,𝑣,𝑘)   0 (𝑤,𝑣,𝑘)

Proof of Theorem lcfrlem31

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lcfrlem30.c	. . . . . . 7 ⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌)))
2		lcfrlem31.cn	. . . . . . 7 ⊢ (𝜑 → 𝐶 = (0g‘𝐷))
3	1, 2	eqtr3id 2778	. . . . . 6 ⊢ (𝜑 → ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))) = (0g‘𝐷))
4		lcfrlem25.d	. . . . . . . 8 ⊢ 𝐷 = (LDual‘𝑈)
5		lcfrlem17.h	. . . . . . . . 9 ⊢ 𝐻 = (LHyp‘𝐾)
6		lcfrlem17.u	. . . . . . . . 9 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		lcfrlem17.k	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
8	5, 6, 7	dvhlmod 41089	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
9	4, 8	lduallmod 39131	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ LMod)
10		eqid 2729	. . . . . . . 8 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
11		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
12		lcfrlem17.o	. . . . . . . . 9 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
13		lcfrlem17.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑈)
14		lcfrlem17.p	. . . . . . . . 9 ⊢ + = (+g‘𝑈)
15		lcfrlem24.t	. . . . . . . . 9 ⊢ · = ( ·𝑠 ‘𝑈)
16		lcfrlem24.s	. . . . . . . . 9 ⊢ 𝑆 = (Scalar‘𝑈)
17		lcfrlem24.r	. . . . . . . . 9 ⊢ 𝑅 = (Base‘𝑆)
18		lcfrlem17.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝑈)
19		lcfrlem24.l	. . . . . . . . 9 ⊢ 𝐿 = (LKer‘𝑈)
20		eqid 2729	. . . . . . . . 9 ⊢ (0g‘𝐷) = (0g‘𝐷)
21		eqid 2729	. . . . . . . . 9 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
22		lcfrlem24.j	. . . . . . . . 9 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
23		lcfrlem17.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
24	5, 12, 6, 13, 14, 15, 16, 17, 18, 10, 19, 4, 20, 21, 22, 7, 23	lcfrlem10 41531	. . . . . . . 8 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (LFnl‘𝑈))
25	10, 4, 11, 8, 24	ldualelvbase 39105	. . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (Base‘𝐷))
26		eqid 2729	. . . . . . . . 9 ⊢ ( ·𝑠 ‘𝐷) = ( ·𝑠 ‘𝐷)
27		lcfrlem17.n	. . . . . . . . . 10 ⊢ 𝑁 = (LSpan‘𝑈)
28		lcfrlem17.a	. . . . . . . . . 10 ⊢ 𝐴 = (LSAtoms‘𝑈)
29		lcfrlem17.y	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
30		lcfrlem17.ne	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
31		lcfrlem22.b	. . . . . . . . . 10 ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))
32		lcfrlem24.q	. . . . . . . . . 10 ⊢ 𝑄 = (0g‘𝑆)
33		lcfrlem24.ib	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
34		lcfrlem28.jn	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄)
35		lcfrlem29.i	. . . . . . . . . 10 ⊢ 𝐹 = (invr‘𝑆)
36	5, 12, 6, 13, 14, 18, 27, 28, 7, 23, 29, 30, 31, 15, 16, 32, 17, 22, 33, 19, 4, 34, 35	lcfrlem29 41550	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼)) ∈ 𝑅)
37	5, 12, 6, 13, 14, 15, 16, 17, 18, 10, 19, 4, 20, 21, 22, 7, 29	lcfrlem10 41531	. . . . . . . . 9 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (LFnl‘𝑈))
38	10, 16, 17, 4, 26, 8, 36, 37	ldualvscl 39117	. . . . . . . 8 ⊢ (𝜑 → (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌)) ∈ (LFnl‘𝑈))
39	10, 4, 11, 8, 38	ldualelvbase 39105	. . . . . . 7 ⊢ (𝜑 → (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌)) ∈ (Base‘𝐷))
40		lcfrlem30.m	. . . . . . . 8 ⊢ − = (-g‘𝐷)
41	11, 20, 40	lmodsubeq0 20842	. . . . . . 7 ⊢ ((𝐷 ∈ LMod ∧ (𝐽‘𝑋) ∈ (Base‘𝐷) ∧ (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌)) ∈ (Base‘𝐷)) → (((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))) = (0g‘𝐷) ↔ (𝐽‘𝑋) = (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))))
42	9, 25, 39, 41	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → (((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))) = (0g‘𝐷) ↔ (𝐽‘𝑋) = (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))))
43	3, 42	mpbid 232	. . . . 5 ⊢ (𝜑 → (𝐽‘𝑋) = (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌)))
44	43	fveq2d 6830	. . . 4 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑋)) = (𝐿‘(((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))))
45	5, 6, 7	dvhlvec 41088	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LVec)
46	16	lvecdrng 21027	. . . . . . . 8 ⊢ (𝑈 ∈ LVec → 𝑆 ∈ DivRing)
47	45, 46	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ DivRing)
48	5, 12, 6, 13, 14, 18, 27, 28, 7, 23, 29, 30, 31	lcfrlem22 41543	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
49	13, 28, 8, 48	lsatssv 38976	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ 𝑉)
50	49, 33	sseldd 3938	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
51	16, 17, 13, 10	lflcl 39042	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑌) ∈ (LFnl‘𝑈) ∧ 𝐼 ∈ 𝑉) → ((𝐽‘𝑌)‘𝐼) ∈ 𝑅)
52	8, 37, 50, 51	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ∈ 𝑅)
53	17, 32, 35	drnginvrn0 20657	. . . . . . 7 ⊢ ((𝑆 ∈ DivRing ∧ ((𝐽‘𝑌)‘𝐼) ∈ 𝑅 ∧ ((𝐽‘𝑌)‘𝐼) ≠ 𝑄) → (𝐹‘((𝐽‘𝑌)‘𝐼)) ≠ 𝑄)
54	47, 52, 34, 53	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → (𝐹‘((𝐽‘𝑌)‘𝐼)) ≠ 𝑄)
55		lcfrlem31.xi	. . . . . 6 ⊢ (𝜑 → ((𝐽‘𝑋)‘𝐼) ≠ 𝑄)
56		eqid 2729	. . . . . . 7 ⊢ (.r‘𝑆) = (.r‘𝑆)
57	17, 32, 35	drnginvrcl 20656	. . . . . . . 8 ⊢ ((𝑆 ∈ DivRing ∧ ((𝐽‘𝑌)‘𝐼) ∈ 𝑅 ∧ ((𝐽‘𝑌)‘𝐼) ≠ 𝑄) → (𝐹‘((𝐽‘𝑌)‘𝐼)) ∈ 𝑅)
58	47, 52, 34, 57	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → (𝐹‘((𝐽‘𝑌)‘𝐼)) ∈ 𝑅)
59	16, 17, 13, 10	lflcl 39042	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑋) ∈ (LFnl‘𝑈) ∧ 𝐼 ∈ 𝑉) → ((𝐽‘𝑋)‘𝐼) ∈ 𝑅)
60	8, 24, 50, 59	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → ((𝐽‘𝑋)‘𝐼) ∈ 𝑅)
61	17, 32, 56, 47, 58, 60	drngmulne0 20665	. . . . . 6 ⊢ (𝜑 → (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼)) ≠ 𝑄 ↔ ((𝐹‘((𝐽‘𝑌)‘𝐼)) ≠ 𝑄 ∧ ((𝐽‘𝑋)‘𝐼) ≠ 𝑄)))
62	54, 55, 61	mpbir2and 713	. . . . 5 ⊢ (𝜑 → ((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼)) ≠ 𝑄)
63	16, 17, 32, 10, 19, 4, 26, 45, 37, 36, 62	ldualkrsc 39145	. . . 4 ⊢ (𝜑 → (𝐿‘(((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))) = (𝐿‘(𝐽‘𝑌)))
64	44, 63	eqtrd 2764	. . 3 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑋)) = (𝐿‘(𝐽‘𝑌)))
65	64	fveq2d 6830	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐽‘𝑋))) = ( ⊥ ‘(𝐿‘(𝐽‘𝑌))))
66	5, 12, 6, 13, 14, 15, 16, 17, 18, 10, 19, 4, 20, 21, 22, 7, 23, 27	lcfrlem14 41535	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐽‘𝑋))) = (𝑁‘{𝑋}))
67	5, 12, 6, 13, 14, 15, 16, 17, 18, 10, 19, 4, 20, 21, 22, 7, 29, 27	lcfrlem14 41535	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐽‘𝑌))) = (𝑁‘{𝑌}))
68	65, 66, 67	3eqtr3d 2772	1 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053  {crab 3396   ∖ cdif 3902   ∩ cin 3904  {csn 4579  {cpr 4581   ↦ cmpt 5176  ‘cfv 6486  ℩crio 7309  (class class class)co 7353  Basecbs 17138  +_gcplusg 17179  ._rcmulr 17180  Scalarcsca 17182   ·_𝑠 cvsca 17183  0_gc0g 17361  -_gcsg 18832  inv_rcinvr 20290  DivRingcdr 20632  LModclmod 20781  LSpanclspn 20892  LVecclvec 21024  LSAtomsclsa 38952  LFnlclfn 39035  LKerclk 39063  LDualcld 39101  HLchlt 39328  LHypclh 39963  DVecHcdvh 41057  ocHcoch 41326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-riotaBAD 38931

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-iin 4947  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-of 7617  df-om 7807  df-1st 7931  df-2nd 7932  df-tpos 8166  df-undef 8213  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-3 12210  df-4 12211  df-5 12212  df-6 12213  df-n0 12403  df-z 12490  df-uz 12754  df-fz 13429  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-plusg 17192  df-mulr 17193  df-sca 17195  df-vsca 17196  df-0g 17363  df-mre 17506  df-mrc 17507  df-acs 17509  df-proset 18218  df-poset 18237  df-plt 18252  df-lub 18268  df-glb 18269  df-join 18270  df-meet 18271  df-p0 18347  df-p1 18348  df-lat 18356  df-clat 18423  df-mgm 18532  df-sgrp 18611  df-mnd 18627  df-submnd 18676  df-grp 18833  df-minusg 18834  df-sbg 18835  df-subg 19020  df-cntz 19214  df-oppg 19243  df-lsm 19533  df-cmn 19679  df-abl 19680  df-mgp 20044  df-rng 20056  df-ur 20085  df-ring 20138  df-oppr 20240  df-dvdsr 20260  df-unit 20261  df-invr 20291  df-dvr 20304  df-nzr 20416  df-rlreg 20597  df-domn 20598  df-drng 20634  df-lmod 20783  df-lss 20853  df-lsp 20893  df-lvec 21025  df-lsatoms 38954  df-lshyp 38955  df-lcv 38997  df-lfl 39036  df-lkr 39064  df-ldual 39102  df-oposet 39154  df-ol 39156  df-oml 39157  df-covers 39244  df-ats 39245  df-atl 39276  df-cvlat 39300  df-hlat 39329  df-llines 39477  df-lplanes 39478  df-lvols 39479  df-lines 39480  df-psubsp 39482  df-pmap 39483  df-padd 39775  df-lhyp 39967  df-laut 39968  df-ldil 40083  df-ltrn 40084  df-trl 40138  df-tgrp 40722  df-tendo 40734  df-edring 40736  df-dveca 40982  df-disoa 41008  df-dvech 41058  df-dib 41118  df-dic 41152  df-dih 41208  df-doch 41327  df-djh 41374

This theorem is referenced by:  lcfrlem32  41553