Theorem lcfrlem31 42066

Description: Lemma for lcfr 42078. (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 2789	. . . . . 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 41603	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
9	4, 8	lduallmod 39646	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ LMod)
10		eqid 2740	. . . . . . . 8 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
11		eqid 2740	. . . . . . . 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 2740	. . . . . . . . 9 ⊢ (0g‘𝐷) = (0g‘𝐷)
21		eqid 2740	. . . . . . . . 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 42045	. . . . . . . 8 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (LFnl‘𝑈))
25	10, 4, 11, 8, 24	ldualelvbase 39620	. . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (Base‘𝐷))
26		eqid 2740	. . . . . . . . 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 42064	. . . . . . . . 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 42045	. . . . . . . . 9 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (LFnl‘𝑈))
38	10, 16, 17, 4, 26, 8, 36, 37	ldualvscl 39632	. . . . . . . 8 ⊢ (𝜑 → (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌)) ∈ (LFnl‘𝑈))
39	10, 4, 11, 8, 38	ldualelvbase 39620	. . . . . . 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 20918	. . . . . . 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 1379	. . . . . 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 233	. . . . 5 ⊢ (𝜑 → (𝐽‘𝑋) = (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌)))
44	43	fveq2d 6838	. . . 4 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑋)) = (𝐿‘(((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))))
45	5, 6, 7	dvhlvec 41602	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LVec)
46	16	lvecdrng 21102	. . . . . . . 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 42057	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
49	13, 28, 8, 48	lsatssv 39491	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ 𝑉)
50	49, 33	sseldd 3923	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
51	16, 17, 13, 10	lflcl 39557	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑌) ∈ (LFnl‘𝑈) ∧ 𝐼 ∈ 𝑉) → ((𝐽‘𝑌)‘𝐼) ∈ 𝑅)
52	8, 37, 50, 51	syl3anc 1379	. . . . . . 7 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ∈ 𝑅)
53	17, 32, 35	drnginvrn0 20733	. . . . . . 7 ⊢ ((𝑆 ∈ DivRing ∧ ((𝐽‘𝑌)‘𝐼) ∈ 𝑅 ∧ ((𝐽‘𝑌)‘𝐼) ≠ 𝑄) → (𝐹‘((𝐽‘𝑌)‘𝐼)) ≠ 𝑄)
54	47, 52, 34, 53	syl3anc 1379	. . . . . 6 ⊢ (𝜑 → (𝐹‘((𝐽‘𝑌)‘𝐼)) ≠ 𝑄)
55		lcfrlem31.xi	. . . . . 6 ⊢ (𝜑 → ((𝐽‘𝑋)‘𝐼) ≠ 𝑄)
56		eqid 2740	. . . . . . 7 ⊢ (.r‘𝑆) = (.r‘𝑆)
57	17, 32, 35	drnginvrcl 20732	. . . . . . . 8 ⊢ ((𝑆 ∈ DivRing ∧ ((𝐽‘𝑌)‘𝐼) ∈ 𝑅 ∧ ((𝐽‘𝑌)‘𝐼) ≠ 𝑄) → (𝐹‘((𝐽‘𝑌)‘𝐼)) ∈ 𝑅)
58	47, 52, 34, 57	syl3anc 1379	. . . . . . 7 ⊢ (𝜑 → (𝐹‘((𝐽‘𝑌)‘𝐼)) ∈ 𝑅)
59	16, 17, 13, 10	lflcl 39557	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑋) ∈ (LFnl‘𝑈) ∧ 𝐼 ∈ 𝑉) → ((𝐽‘𝑋)‘𝐼) ∈ 𝑅)
60	8, 24, 50, 59	syl3anc 1379	. . . . . . 7 ⊢ (𝜑 → ((𝐽‘𝑋)‘𝐼) ∈ 𝑅)
61	17, 32, 56, 47, 58, 60	drngmulne0 20741	. . . . . 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 719	. . . . 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 39660	. . . 4 ⊢ (𝜑 → (𝐿‘(((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))) = (𝐿‘(𝐽‘𝑌)))
64	44, 63	eqtrd 2775	. . 3 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑋)) = (𝐿‘(𝐽‘𝑌)))
65	64	fveq2d 6838	. 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 42049	. 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 42049	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐽‘𝑌))) = (𝑁‘{𝑌}))
68	65, 66, 67	3eqtr3d 2783	1 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∃wrex 3064  {crab 3392   ∖ cdif 3887   ∩ cin 3889  {csn 4562  {cpr 4564   ↦ cmpt 5160  ‘cfv 6492  ℩crio 7319  (class class class)co 7363  Basecbs 17177  +_gcplusg 17218  ._rcmulr 17219  Scalarcsca 17221   ·_𝑠 cvsca 17222  0_gc0g 17400  -_gcsg 18909  inv_rcinvr 20365  DivRingcdr 20708  LModclmod 20857  LSpanclspn 20968  LVecclvec 21099  LSAtomsclsa 39467  LFnlclfn 39550  LKerclk 39578  LDualcld 39616  HLchlt 39843  LHypclh 40477  DVecHcdvh 41571  ocHcoch 41840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113  ax-riotaBAD 39446

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-iin 4931  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-of 7627  df-om 7814  df-1st 7938  df-2nd 7939  df-tpos 8173  df-undef 8220  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-er 8640  df-map 8772  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-3 12243  df-4 12244  df-5 12245  df-6 12246  df-n0 12436  df-z 12523  df-uz 12787  df-fz 13460  df-struct 17115  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17178  df-ress 17199  df-plusg 17231  df-mulr 17232  df-sca 17234  df-vsca 17235  df-0g 17402  df-mre 17546  df-mrc 17547  df-acs 17549  df-proset 18258  df-poset 18277  df-plt 18292  df-lub 18308  df-glb 18309  df-join 18310  df-meet 18311  df-p0 18387  df-p1 18388  df-lat 18396  df-clat 18463  df-mgm 18606  df-sgrp 18685  df-mnd 18701  df-submnd 18750  df-grp 18910  df-minusg 18911  df-sbg 18912  df-subg 19097  df-cntz 19290  df-oppg 19319  df-lsm 19609  df-cmn 19755  df-abl 19756  df-mgp 20120  df-rng 20132  df-ur 20161  df-ring 20214  df-oppr 20315  df-dvdsr 20335  df-unit 20336  df-invr 20366  df-dvr 20379  df-nzr 20492  df-rlreg 20673  df-domn 20674  df-drng 20710  df-lmod 20859  df-lss 20929  df-lsp 20969  df-lvec 21100  df-lsatoms 39469  df-lshyp 39470  df-lcv 39512  df-lfl 39551  df-lkr 39579  df-ldual 39617  df-oposet 39669  df-ol 39671  df-oml 39672  df-covers 39759  df-ats 39760  df-atl 39791  df-cvlat 39815  df-hlat 39844  df-llines 39991  df-lplanes 39992  df-lvols 39993  df-lines 39994  df-psubsp 39996  df-pmap 39997  df-padd 40289  df-lhyp 40481  df-laut 40482  df-ldil 40597  df-ltrn 40598  df-trl 40652  df-tgrp 41236  df-tendo 41248  df-edring 41250  df-dveca 41496  df-disoa 41522  df-dvech 41572  df-dib 41632  df-dic 41666  df-dih 41722  df-doch 41841  df-djh 41888

This theorem is referenced by:  lcfrlem32  42067