lclkrlem2o - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2o		Structured version Visualization version GIF version

Theorem lclkrlem2o 39090

Description: Lemma for lclkr 39102. When 𝐵 is nonzero, the vectors 𝑋 and 𝑌 can't both belong to the hyperplane generated by 𝐵. (Contributed by NM, 17-Jan-2015.)

**Hypotheses**
Ref	Expression
lclkrlem2m.v	⊢ 𝑉 = (Base‘𝑈)
lclkrlem2m.t	⊢ · = ( ·_𝑠 ‘𝑈)
lclkrlem2m.s	⊢ 𝑆 = (Scalar‘𝑈)
lclkrlem2m.q	⊢ × = (._r‘𝑆)
lclkrlem2m.z	⊢ 0 = (0_g‘𝑆)
lclkrlem2m.i	⊢ 𝐼 = (inv_r‘𝑆)
lclkrlem2m.m	⊢ − = (-_g‘𝑈)
lclkrlem2m.f	⊢ 𝐹 = (LFnl‘𝑈)
lclkrlem2m.d	⊢ 𝐷 = (LDual‘𝑈)
lclkrlem2m.p	⊢ + = (+_g‘𝐷)
lclkrlem2m.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
lclkrlem2m.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
lclkrlem2m.e	⊢ (𝜑 → 𝐸 ∈ 𝐹)
lclkrlem2m.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
lclkrlem2n.n	⊢ 𝑁 = (LSpan‘𝑈)
lclkrlem2n.l	⊢ 𝐿 = (LKer‘𝑈)
lclkrlem2o.h	⊢ 𝐻 = (LHyp‘𝐾)
lclkrlem2o.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lclkrlem2o.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lclkrlem2o.a	⊢ ⊕ = (LSSum‘𝑈)
lclkrlem2o.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lclkrlem2o.b	⊢ 𝐵 = (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌))
lclkrlem2o.n	⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )
lclkrlem2o.bn	⊢ (𝜑 → 𝐵 ≠ (0_g‘𝑈))

**Assertion**
Ref	Expression
lclkrlem2o	⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))

**Proof of Theorem lclkrlem2o**
Step	Hyp	Ref	Expression
1		lclkrlem2o.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		lclkrlem2o.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
3		lclkrlem2o.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		lclkrlem2m.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
5		eqid 2759	. . . 4 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
6		lclkrlem2o.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
7		lclkrlem2m.t	. . . . . . 7 ⊢ · = ( ·_𝑠 ‘𝑈)
8		lclkrlem2m.s	. . . . . . 7 ⊢ 𝑆 = (Scalar‘𝑈)
9		lclkrlem2m.q	. . . . . . 7 ⊢ × = (._r‘𝑆)
10		lclkrlem2m.z	. . . . . . 7 ⊢ 0 = (0_g‘𝑆)
11		lclkrlem2m.i	. . . . . . 7 ⊢ 𝐼 = (inv_r‘𝑆)
12		lclkrlem2m.m	. . . . . . 7 ⊢ − = (-_g‘𝑈)
13		lclkrlem2m.f	. . . . . . 7 ⊢ 𝐹 = (LFnl‘𝑈)
14		lclkrlem2m.d	. . . . . . 7 ⊢ 𝐷 = (LDual‘𝑈)
15		lclkrlem2m.p	. . . . . . 7 ⊢ + = (+_g‘𝐷)
16		lclkrlem2m.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
17		lclkrlem2m.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
18		lclkrlem2m.e	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝐹)
19		lclkrlem2m.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
20	1, 3, 6	dvhlvec 38678	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec)
21		lclkrlem2o.b	. . . . . . 7 ⊢ 𝐵 = (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌))
22		lclkrlem2o.n	. . . . . . 7 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )
23	4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22	lclkrlem2m 39088	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ 𝑉 ∧ ((𝐸 + 𝐺)‘𝐵) = 0 ))
24	23	simpld 499	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
25		lclkrlem2o.bn	. . . . 5 ⊢ (𝜑 → 𝐵 ≠ (0_g‘𝑈))
26		eldifsn 4678	. . . . 5 ⊢ (𝐵 ∈ (𝑉 ∖ {(0_g‘𝑈)}) ↔ (𝐵 ∈ 𝑉 ∧ 𝐵 ≠ (0_g‘𝑈)))
27	24, 25, 26	sylanbrc 587	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
28	1, 2, 3, 4, 5, 6, 27	dochnel 38962	. . 3 ⊢ (𝜑 → ¬ 𝐵 ∈ ( ⊥ ‘{𝐵}))
29	1, 3, 6	dvhlmod 38679	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
30	29	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ ( ⊥ ‘{𝐵}) ∧ 𝑌 ∈ ( ⊥ ‘{𝐵}))) → 𝑈 ∈ LMod)
31	24	snssd 4700	. . . . . . 7 ⊢ (𝜑 → {𝐵} ⊆ 𝑉)
32		eqid 2759	. . . . . . . 8 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
33	1, 3, 4, 32, 2	dochlss 38923	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐵} ⊆ 𝑉) → ( ⊥ ‘{𝐵}) ∈ (LSubSp‘𝑈))
34	6, 31, 33	syl2anc 588	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘{𝐵}) ∈ (LSubSp‘𝑈))
35	34	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ ( ⊥ ‘{𝐵}) ∧ 𝑌 ∈ ( ⊥ ‘{𝐵}))) → ( ⊥ ‘{𝐵}) ∈ (LSubSp‘𝑈))
36		simprl 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ ( ⊥ ‘{𝐵}) ∧ 𝑌 ∈ ( ⊥ ‘{𝐵}))) → 𝑋 ∈ ( ⊥ ‘{𝐵}))
37	8	lmodring 19703	. . . . . . . . 9 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Ring)
38	29, 37	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Ring)
39	13, 14, 15, 29, 18, 19	ldualvaddcl 36699	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹)
40		eqid 2759	. . . . . . . . . 10 ⊢ (Base‘𝑆) = (Base‘𝑆)
41	8, 40, 4, 13	lflcl 36633	. . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ (𝐸 + 𝐺) ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → ((𝐸 + 𝐺)‘𝑋) ∈ (Base‘𝑆))
42	29, 39, 16, 41	syl3anc 1369	. . . . . . . 8 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑋) ∈ (Base‘𝑆))
43	8	lvecdrng 19938	. . . . . . . . . 10 ⊢ (𝑈 ∈ LVec → 𝑆 ∈ DivRing)
44	20, 43	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ DivRing)
45	8, 40, 4, 13	lflcl 36633	. . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ (𝐸 + 𝐺) ∈ 𝐹 ∧ 𝑌 ∈ 𝑉) → ((𝐸 + 𝐺)‘𝑌) ∈ (Base‘𝑆))
46	29, 39, 17, 45	syl3anc 1369	. . . . . . . . 9 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) ∈ (Base‘𝑆))
47	40, 10, 11	drnginvrcl 19580	. . . . . . . . 9 ⊢ ((𝑆 ∈ DivRing ∧ ((𝐸 + 𝐺)‘𝑌) ∈ (Base‘𝑆) ∧ ((𝐸 + 𝐺)‘𝑌) ≠ 0 ) → (𝐼‘((𝐸 + 𝐺)‘𝑌)) ∈ (Base‘𝑆))
48	44, 46, 22, 47	syl3anc 1369	. . . . . . . 8 ⊢ (𝜑 → (𝐼‘((𝐸 + 𝐺)‘𝑌)) ∈ (Base‘𝑆))
49	40, 9	ringcl 19375	. . . . . . . 8 ⊢ ((𝑆 ∈ Ring ∧ ((𝐸 + 𝐺)‘𝑋) ∈ (Base‘𝑆) ∧ (𝐼‘((𝐸 + 𝐺)‘𝑌)) ∈ (Base‘𝑆)) → (((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) ∈ (Base‘𝑆))
50	38, 42, 48, 49	syl3anc 1369	. . . . . . 7 ⊢ (𝜑 → (((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) ∈ (Base‘𝑆))
51	50	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ ( ⊥ ‘{𝐵}) ∧ 𝑌 ∈ ( ⊥ ‘{𝐵}))) → (((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) ∈ (Base‘𝑆))
52		simprr 773	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ ( ⊥ ‘{𝐵}) ∧ 𝑌 ∈ ( ⊥ ‘{𝐵}))) → 𝑌 ∈ ( ⊥ ‘{𝐵}))
53	8, 7, 40, 32	lssvscl 19788	. . . . . 6 ⊢ (((𝑈 ∈ LMod ∧ ( ⊥ ‘{𝐵}) ∈ (LSubSp‘𝑈)) ∧ ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) ∈ (Base‘𝑆) ∧ 𝑌 ∈ ( ⊥ ‘{𝐵}))) → ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌) ∈ ( ⊥ ‘{𝐵}))
54	30, 35, 51, 52, 53	syl22anc 838	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ ( ⊥ ‘{𝐵}) ∧ 𝑌 ∈ ( ⊥ ‘{𝐵}))) → ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌) ∈ ( ⊥ ‘{𝐵}))
55	12, 32	lssvsubcl 19776	. . . . 5 ⊢ (((𝑈 ∈ LMod ∧ ( ⊥ ‘{𝐵}) ∈ (LSubSp‘𝑈)) ∧ (𝑋 ∈ ( ⊥ ‘{𝐵}) ∧ ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌) ∈ ( ⊥ ‘{𝐵}))) → (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ∈ ( ⊥ ‘{𝐵}))
56	30, 35, 36, 54, 55	syl22anc 838	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ ( ⊥ ‘{𝐵}) ∧ 𝑌 ∈ ( ⊥ ‘{𝐵}))) → (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ∈ ( ⊥ ‘{𝐵}))
57	21, 56	eqeltrid 2857	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ ( ⊥ ‘{𝐵}) ∧ 𝑌 ∈ ( ⊥ ‘{𝐵}))) → 𝐵 ∈ ( ⊥ ‘{𝐵}))
58	28, 57	mtand 816	. 2 ⊢ (𝜑 → ¬ (𝑋 ∈ ( ⊥ ‘{𝐵}) ∧ 𝑌 ∈ ( ⊥ ‘{𝐵})))
59		ianor 980	. 2 ⊢ (¬ (𝑋 ∈ ( ⊥ ‘{𝐵}) ∧ 𝑌 ∈ ( ⊥ ‘{𝐵})) ↔ (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
60	58, 59	sylib 221	1 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 ∨ wo 845 = wceq 1539 ∈ wcel 2112 ≠ wne 2952 ∖ cdif 3856 ⊆ wss 3859 {csn 4523 ‘cfv 6336 (class class class)co 7151 Basecbs 16534 +_gcplusg 16616 ._rcmulr 16617 Scalarcsca 16619 ·_𝑠 cvsca 16620 0_gc0g 16764 -_gcsg 18164 LSSumclsm 18819 Ringcrg 19358 inv_rcinvr 19485 DivRingcdr 19563 LModclmod 19695 LSubSpclss 19764 LSpanclspn 19804 LVecclvec 19935 LFnlclfn 36626 LKerclk 36654 LDualcld 36692 HLchlt 36919 LHypclh 37553 DVecHcdvh 38647 ocHcoch 38916

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5157 ax-sep 5170 ax-nul 5177 ax-pow 5235 ax-pr 5299 ax-un 7460 ax-cnex 10624 ax-resscn 10625 ax-1cn 10626 ax-icn 10627 ax-addcl 10628 ax-addrcl 10629 ax-mulcl 10630 ax-mulrcl 10631 ax-mulcom 10632 ax-addass 10633 ax-mulass 10634 ax-distr 10635 ax-i2m1 10636 ax-1ne0 10637 ax-1rid 10638 ax-rnegex 10639 ax-rrecex 10640 ax-cnre 10641 ax-pre-lttri 10642 ax-pre-lttrn 10643 ax-pre-ltadd 10644 ax-pre-mulgt0 10645 ax-riotaBAD 36522

This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2902 df-ne 2953 df-nel 3057 df-ral 3076 df-rex 3077 df-reu 3078 df-rmo 3079 df-rab 3080 df-v 3412 df-sbc 3698 df-csb 3807 df-dif 3862 df-un 3864 df-in 3866 df-ss 3876 df-pss 3878 df-nul 4227 df-if 4422 df-pw 4497 df-sn 4524 df-pr 4526 df-tp 4528 df-op 4530 df-uni 4800 df-int 4840 df-iun 4886 df-iin 4887 df-br 5034 df-opab 5096 df-mpt 5114 df-tr 5140 df-id 5431 df-eprel 5436 df-po 5444 df-so 5445 df-fr 5484 df-we 5486 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6127 df-ord 6173 df-on 6174 df-lim 6175 df-suc 6176 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7109 df-ov 7154 df-oprab 7155 df-mpo 7156 df-of 7406 df-om 7581 df-1st 7694 df-2nd 7695 df-tpos 7903 df-undef 7950 df-wrecs 7958 df-recs 8019 df-rdg 8057 df-1o 8113 df-oadd 8117 df-er 8300 df-map 8419 df-en 8529 df-dom 8530 df-sdom 8531 df-fin 8532 df-pnf 10708 df-mnf 10709 df-xr 10710 df-ltxr 10711 df-le 10712 df-sub 10903 df-neg 10904 df-nn 11668 df-2 11730 df-3 11731 df-4 11732 df-5 11733 df-6 11734 df-n0 11928 df-z 12014 df-uz 12276 df-fz 12933 df-struct 16536 df-ndx 16537 df-slot 16538 df-base 16540 df-sets 16541 df-ress 16542 df-plusg 16629 df-mulr 16630 df-sca 16632 df-vsca 16633 df-0g 16766 df-proset 17597 df-poset 17615 df-plt 17627 df-lub 17643 df-glb 17644 df-join 17645 df-meet 17646 df-p0 17708 df-p1 17709 df-lat 17715 df-clat 17777 df-mgm 17911 df-sgrp 17960 df-mnd 17971 df-submnd 18016 df-grp 18165 df-minusg 18166 df-sbg 18167 df-subg 18336 df-cntz 18507 df-lsm 18821 df-cmn 18968 df-abl 18969 df-mgp 19301 df-ur 19313 df-ring 19360 df-oppr 19437 df-dvdsr 19455 df-unit 19456 df-invr 19486 df-dvr 19497 df-drng 19565 df-lmod 19697 df-lss 19765 df-lsp 19805 df-lvec 19936 df-lsatoms 36545 df-lfl 36627 df-ldual 36693 df-oposet 36745 df-ol 36747 df-oml 36748 df-covers 36835 df-ats 36836 df-atl 36867 df-cvlat 36891 df-hlat 36920 df-llines 37067 df-lplanes 37068 df-lvols 37069 df-lines 37070 df-psubsp 37072 df-pmap 37073 df-padd 37365 df-lhyp 37557 df-laut 37558 df-ldil 37673 df-ltrn 37674 df-trl 37728 df-tendo 38324 df-edring 38326 df-disoa 38598 df-dvech 38648 df-dib 38708 df-dic 38742 df-dih 38798 df-doch 38917

This theorem is referenced by: lclkrlem2q 39092

W3C validator