Theorem lcfrlem31 40036

Description: Lemma for lcfr 40048. (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 2790	. . . . . 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 39573	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
9	4, 8	lduallmod 37615	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ LMod)
10		eqid 2736	. . . . . . . 8 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
11		eqid 2736	. . . . . . . 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 2736	. . . . . . . . 9 ⊢ (0g‘𝐷) = (0g‘𝐷)
21		eqid 2736	. . . . . . . . 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 40015	. . . . . . . 8 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (LFnl‘𝑈))
25	10, 4, 11, 8, 24	ldualelvbase 37589	. . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (Base‘𝐷))
26		eqid 2736	. . . . . . . . 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 40034	. . . . . . . . 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 40015	. . . . . . . . 9 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (LFnl‘𝑈))
38	10, 16, 17, 4, 26, 8, 36, 37	ldualvscl 37601	. . . . . . . 8 ⊢ (𝜑 → (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌)) ∈ (LFnl‘𝑈))
39	10, 4, 11, 8, 38	ldualelvbase 37589	. . . . . . 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 20381	. . . . . . 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 1371	. . . . . 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 231	. . . . 5 ⊢ (𝜑 → (𝐽‘𝑋) = (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌)))
44	43	fveq2d 6846	. . . 4 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑋)) = (𝐿‘(((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))))
45	5, 6, 7	dvhlvec 39572	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LVec)
46	16	lvecdrng 20566	. . . . . . . 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 40027	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
49	13, 28, 8, 48	lsatssv 37460	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ 𝑉)
50	49, 33	sseldd 3945	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
51	16, 17, 13, 10	lflcl 37526	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑌) ∈ (LFnl‘𝑈) ∧ 𝐼 ∈ 𝑉) → ((𝐽‘𝑌)‘𝐼) ∈ 𝑅)
52	8, 37, 50, 51	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ∈ 𝑅)
53	17, 32, 35	drnginvrn0 20206	. . . . . . 7 ⊢ ((𝑆 ∈ DivRing ∧ ((𝐽‘𝑌)‘𝐼) ∈ 𝑅 ∧ ((𝐽‘𝑌)‘𝐼) ≠ 𝑄) → (𝐹‘((𝐽‘𝑌)‘𝐼)) ≠ 𝑄)
54	47, 52, 34, 53	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → (𝐹‘((𝐽‘𝑌)‘𝐼)) ≠ 𝑄)
55		lcfrlem31.xi	. . . . . 6 ⊢ (𝜑 → ((𝐽‘𝑋)‘𝐼) ≠ 𝑄)
56		eqid 2736	. . . . . . 7 ⊢ (.r‘𝑆) = (.r‘𝑆)
57	17, 32, 35	drnginvrcl 20205	. . . . . . . 8 ⊢ ((𝑆 ∈ DivRing ∧ ((𝐽‘𝑌)‘𝐼) ∈ 𝑅 ∧ ((𝐽‘𝑌)‘𝐼) ≠ 𝑄) → (𝐹‘((𝐽‘𝑌)‘𝐼)) ∈ 𝑅)
58	47, 52, 34, 57	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → (𝐹‘((𝐽‘𝑌)‘𝐼)) ∈ 𝑅)
59	16, 17, 13, 10	lflcl 37526	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑋) ∈ (LFnl‘𝑈) ∧ 𝐼 ∈ 𝑉) → ((𝐽‘𝑋)‘𝐼) ∈ 𝑅)
60	8, 24, 50, 59	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → ((𝐽‘𝑋)‘𝐼) ∈ 𝑅)
61	17, 32, 56, 47, 58, 60	drngmulne0 20211	. . . . . 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 711	. . . . 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 37629	. . . 4 ⊢ (𝜑 → (𝐿‘(((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))) = (𝐿‘(𝐽‘𝑌)))
64	44, 63	eqtrd 2776	. . 3 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑋)) = (𝐿‘(𝐽‘𝑌)))
65	64	fveq2d 6846	. 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 40019	. 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 40019	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐽‘𝑌))) = (𝑁‘{𝑌}))
68	65, 66, 67	3eqtr3d 2784	1 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3073  {crab 3407   ∖ cdif 3907   ∩ cin 3909  {csn 4586  {cpr 4588   ↦ cmpt 5188  ‘cfv 6496  ℩crio 7312  (class class class)co 7357  Basecbs 17083  +_gcplusg 17133  ._rcmulr 17134  Scalarcsca 17136   ·_𝑠 cvsca 17137  0_gc0g 17321  -_gcsg 18750  inv_rcinvr 20100  DivRingcdr 20185  LModclmod 20322  LSpanclspn 20432  LVecclvec 20563  LSAtomsclsa 37436  LFnlclfn 37519  LKerclk 37547  LDualcld 37585  HLchlt 37812  LHypclh 38447  DVecHcdvh 39541  ocHcoch 39810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-riotaBAD 37415

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-undef 8204  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-0g 17323  df-mre 17466  df-mrc 17467  df-acs 17469  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-p1 18315  df-lat 18321  df-clat 18388  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-subg 18925  df-cntz 19097  df-oppg 19124  df-lsm 19418  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-oppr 20049  df-dvdsr 20070  df-unit 20071  df-invr 20101  df-dvr 20112  df-drng 20187  df-lmod 20324  df-lss 20393  df-lsp 20433  df-lvec 20564  df-lsatoms 37438  df-lshyp 37439  df-lcv 37481  df-lfl 37520  df-lkr 37548  df-ldual 37586  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-llines 37961  df-lplanes 37962  df-lvols 37963  df-lines 37964  df-psubsp 37966  df-pmap 37967  df-padd 38259  df-lhyp 38451  df-laut 38452  df-ldil 38567  df-ltrn 38568  df-trl 38622  df-tgrp 39206  df-tendo 39218  df-edring 39220  df-dveca 39466  df-disoa 39492  df-dvech 39542  df-dib 39602  df-dic 39636  df-dih 39692  df-doch 39811  df-djh 39858

This theorem is referenced by:  lcfrlem32  40037