Theorem lcfrlem31 41078

Description: Lemma for lcfr 41090. (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 2782	. . . . . 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 40615	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
9	4, 8	lduallmod 38657	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ LMod)
10		eqid 2728	. . . . . . . 8 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
11		eqid 2728	. . . . . . . 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 2728	. . . . . . . . 9 ⊢ (0g‘𝐷) = (0g‘𝐷)
21		eqid 2728	. . . . . . . . 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 41057	. . . . . . . 8 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (LFnl‘𝑈))
25	10, 4, 11, 8, 24	ldualelvbase 38631	. . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (Base‘𝐷))
26		eqid 2728	. . . . . . . . 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 41076	. . . . . . . . 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 41057	. . . . . . . . 9 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (LFnl‘𝑈))
38	10, 16, 17, 4, 26, 8, 36, 37	ldualvscl 38643	. . . . . . . 8 ⊢ (𝜑 → (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌)) ∈ (LFnl‘𝑈))
39	10, 4, 11, 8, 38	ldualelvbase 38631	. . . . . . 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 20811	. . . . . . 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 1368	. . . . . 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 6906	. . . 4 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑋)) = (𝐿‘(((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))))
45	5, 6, 7	dvhlvec 40614	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LVec)
46	16	lvecdrng 20997	. . . . . . . 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 41069	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
49	13, 28, 8, 48	lsatssv 38502	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ 𝑉)
50	49, 33	sseldd 3983	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
51	16, 17, 13, 10	lflcl 38568	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑌) ∈ (LFnl‘𝑈) ∧ 𝐼 ∈ 𝑉) → ((𝐽‘𝑌)‘𝐼) ∈ 𝑅)
52	8, 37, 50, 51	syl3anc 1368	. . . . . . 7 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ∈ 𝑅)
53	17, 32, 35	drnginvrn0 20654	. . . . . . 7 ⊢ ((𝑆 ∈ DivRing ∧ ((𝐽‘𝑌)‘𝐼) ∈ 𝑅 ∧ ((𝐽‘𝑌)‘𝐼) ≠ 𝑄) → (𝐹‘((𝐽‘𝑌)‘𝐼)) ≠ 𝑄)
54	47, 52, 34, 53	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → (𝐹‘((𝐽‘𝑌)‘𝐼)) ≠ 𝑄)
55		lcfrlem31.xi	. . . . . 6 ⊢ (𝜑 → ((𝐽‘𝑋)‘𝐼) ≠ 𝑄)
56		eqid 2728	. . . . . . 7 ⊢ (.r‘𝑆) = (.r‘𝑆)
57	17, 32, 35	drnginvrcl 20653	. . . . . . . 8 ⊢ ((𝑆 ∈ DivRing ∧ ((𝐽‘𝑌)‘𝐼) ∈ 𝑅 ∧ ((𝐽‘𝑌)‘𝐼) ≠ 𝑄) → (𝐹‘((𝐽‘𝑌)‘𝐼)) ∈ 𝑅)
58	47, 52, 34, 57	syl3anc 1368	. . . . . . 7 ⊢ (𝜑 → (𝐹‘((𝐽‘𝑌)‘𝐼)) ∈ 𝑅)
59	16, 17, 13, 10	lflcl 38568	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑋) ∈ (LFnl‘𝑈) ∧ 𝐼 ∈ 𝑉) → ((𝐽‘𝑋)‘𝐼) ∈ 𝑅)
60	8, 24, 50, 59	syl3anc 1368	. . . . . . 7 ⊢ (𝜑 → ((𝐽‘𝑋)‘𝐼) ∈ 𝑅)
61	17, 32, 56, 47, 58, 60	drngmulne0 20661	. . . . . 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 38671	. . . 4 ⊢ (𝜑 → (𝐿‘(((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌))) = (𝐿‘(𝐽‘𝑌)))
64	44, 63	eqtrd 2768	. . 3 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑋)) = (𝐿‘(𝐽‘𝑌)))
65	64	fveq2d 6906	. 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 41061	. 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 41061	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐽‘𝑌))) = (𝑁‘{𝑌}))
68	65, 66, 67	3eqtr3d 2776	1 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ≠ wne 2937  ∃wrex 3067  {crab 3430   ∖ cdif 3946   ∩ cin 3948  {csn 4632  {cpr 4634   ↦ cmpt 5235  ‘cfv 6553  ℩crio 7381  (class class class)co 7426  Basecbs 17187  +_gcplusg 17240  ._rcmulr 17241  Scalarcsca 17243   ·_𝑠 cvsca 17244  0_gc0g 17428  -_gcsg 18899  inv_rcinvr 20333  DivRingcdr 20631  LModclmod 20750  LSpanclspn 20862  LVecclvec 20994  LSAtomsclsa 38478  LFnlclfn 38561  LKerclk 38589  LDualcld 38627  HLchlt 38854  LHypclh 39489  DVecHcdvh 40583  ocHcoch 40852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-riotaBAD 38457

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-of 7691  df-om 7877  df-1st 7999  df-2nd 8000  df-tpos 8238  df-undef 8285  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-map 8853  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-n0 12511  df-z 12597  df-uz 12861  df-fz 13525  df-struct 17123  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-plusg 17253  df-mulr 17254  df-sca 17256  df-vsca 17257  df-0g 17430  df-mre 17573  df-mrc 17574  df-acs 17576  df-proset 18294  df-poset 18312  df-plt 18329  df-lub 18345  df-glb 18346  df-join 18347  df-meet 18348  df-p0 18424  df-p1 18425  df-lat 18431  df-clat 18498  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-submnd 18748  df-grp 18900  df-minusg 18901  df-sbg 18902  df-subg 19085  df-cntz 19275  df-oppg 19304  df-lsm 19598  df-cmn 19744  df-abl 19745  df-mgp 20082  df-rng 20100  df-ur 20129  df-ring 20182  df-oppr 20280  df-dvdsr 20303  df-unit 20304  df-invr 20334  df-dvr 20347  df-drng 20633  df-lmod 20752  df-lss 20823  df-lsp 20863  df-lvec 20995  df-lsatoms 38480  df-lshyp 38481  df-lcv 38523  df-lfl 38562  df-lkr 38590  df-ldual 38628  df-oposet 38680  df-ol 38682  df-oml 38683  df-covers 38770  df-ats 38771  df-atl 38802  df-cvlat 38826  df-hlat 38855  df-llines 39003  df-lplanes 39004  df-lvols 39005  df-lines 39006  df-psubsp 39008  df-pmap 39009  df-padd 39301  df-lhyp 39493  df-laut 39494  df-ldil 39609  df-ltrn 39610  df-trl 39664  df-tgrp 40248  df-tendo 40260  df-edring 40262  df-dveca 40508  df-disoa 40534  df-dvech 40584  df-dib 40644  df-dic 40678  df-dih 40734  df-doch 40853  df-djh 40900

This theorem is referenced by:  lcfrlem32  41079

Otomatik - 173.255.232.114 CloudFlare DNS Türk Telekom DNS Google DNS Open DNS

OSZAR »