lclkrlem2o - Mathbox for Norm Megill

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Theorem lclkrlem2o 40886

Description: Lemma for lclkr 40898. 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 2724	. . . 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 40474	. . . . . . 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 40884	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ 𝑉 ∧ ((𝐸 + 𝐺)‘𝐵) = 0 ))
24	23	simpld 494	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
25		lclkrlem2o.bn	. . . . 5 ⊢ (𝜑 → 𝐵 ≠ (0_g‘𝑈))
26		eldifsn 4783	. . . . 5 ⊢ (𝐵 ∈ (𝑉 ∖ {(0_g‘𝑈)}) ↔ (𝐵 ∈ 𝑉 ∧ 𝐵 ≠ (0_g‘𝑈)))
27	24, 25, 26	sylanbrc 582	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
28	1, 2, 3, 4, 5, 6, 27	dochnel 40758	. . 3 ⊢ (𝜑 → ¬ 𝐵 ∈ ( ⊥ ‘{𝐵}))
29	1, 3, 6	dvhlmod 40475	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
30	29	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ ( ⊥ ‘{𝐵}) ∧ 𝑌 ∈ ( ⊥ ‘{𝐵}))) → 𝑈 ∈ LMod)
31	24	snssd 4805	. . . . . . 7 ⊢ (𝜑 → {𝐵} ⊆ 𝑉)
32		eqid 2724	. . . . . . . 8 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
33	1, 3, 4, 32, 2	dochlss 40719	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐵} ⊆ 𝑉) → ( ⊥ ‘{𝐵}) ∈ (LSubSp‘𝑈))
34	6, 31, 33	syl2anc 583	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘{𝐵}) ∈ (LSubSp‘𝑈))
35	34	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ ( ⊥ ‘{𝐵}) ∧ 𝑌 ∈ ( ⊥ ‘{𝐵}))) → ( ⊥ ‘{𝐵}) ∈ (LSubSp‘𝑈))
36		simprl 768	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ ( ⊥ ‘{𝐵}) ∧ 𝑌 ∈ ( ⊥ ‘{𝐵}))) → 𝑋 ∈ ( ⊥ ‘{𝐵}))
37	8	lmodring 20706	. . . . . . . . 9 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Ring)
38	29, 37	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Ring)
39	13, 14, 15, 29, 18, 19	ldualvaddcl 38494	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹)
40		eqid 2724	. . . . . . . . . 10 ⊢ (Base‘𝑆) = (Base‘𝑆)
41	8, 40, 4, 13	lflcl 38428	. . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ (𝐸 + 𝐺) ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → ((𝐸 + 𝐺)‘𝑋) ∈ (Base‘𝑆))
42	29, 39, 16, 41	syl3anc 1368	. . . . . . . 8 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑋) ∈ (Base‘𝑆))
43	8	lvecdrng 20945	. . . . . . . . . 10 ⊢ (𝑈 ∈ LVec → 𝑆 ∈ DivRing)
44	20, 43	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ DivRing)
45	8, 40, 4, 13	lflcl 38428	. . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ (𝐸 + 𝐺) ∈ 𝐹 ∧ 𝑌 ∈ 𝑉) → ((𝐸 + 𝐺)‘𝑌) ∈ (Base‘𝑆))
46	29, 39, 17, 45	syl3anc 1368	. . . . . . . . 9 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) ∈ (Base‘𝑆))
47	40, 10, 11	drnginvrcl 20601	. . . . . . . . 9 ⊢ ((𝑆 ∈ DivRing ∧ ((𝐸 + 𝐺)‘𝑌) ∈ (Base‘𝑆) ∧ ((𝐸 + 𝐺)‘𝑌) ≠ 0 ) → (𝐼‘((𝐸 + 𝐺)‘𝑌)) ∈ (Base‘𝑆))
48	44, 46, 22, 47	syl3anc 1368	. . . . . . . 8 ⊢ (𝜑 → (𝐼‘((𝐸 + 𝐺)‘𝑌)) ∈ (Base‘𝑆))
49	40, 9	ringcl 20147	. . . . . . . 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 1368	. . . . . . 7 ⊢ (𝜑 → (((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) ∈ (Base‘𝑆))
51	50	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ ( ⊥ ‘{𝐵}) ∧ 𝑌 ∈ ( ⊥ ‘{𝐵}))) → (((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) ∈ (Base‘𝑆))
52		simprr 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ ( ⊥ ‘{𝐵}) ∧ 𝑌 ∈ ( ⊥ ‘{𝐵}))) → 𝑌 ∈ ( ⊥ ‘{𝐵}))
53	8, 7, 40, 32	lssvscl 20794	. . . . . 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 836	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ ( ⊥ ‘{𝐵}) ∧ 𝑌 ∈ ( ⊥ ‘{𝐵}))) → ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌) ∈ ( ⊥ ‘{𝐵}))
55	12, 32	lssvsubcl 20783	. . . . 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 836	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ ( ⊥ ‘{𝐵}) ∧ 𝑌 ∈ ( ⊥ ‘{𝐵}))) → (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ∈ ( ⊥ ‘{𝐵}))
57	21, 56	eqeltrid 2829	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ ( ⊥ ‘{𝐵}) ∧ 𝑌 ∈ ( ⊥ ‘{𝐵}))) → 𝐵 ∈ ( ⊥ ‘{𝐵}))
58	28, 57	mtand 813	. 2 ⊢ (𝜑 → ¬ (𝑋 ∈ ( ⊥ ‘{𝐵}) ∧ 𝑌 ∈ ( ⊥ ‘{𝐵})))
59		ianor 978	. 2 ⊢ (¬ (𝑋 ∈ ( ⊥ ‘{𝐵}) ∧ 𝑌 ∈ ( ⊥ ‘{𝐵})) ↔ (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
60	58, 59	sylib 217	1 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∖ cdif 3938 ⊆ wss 3941 {csn 4621 ‘cfv 6534 (class class class)co 7402 Basecbs 17145 +_gcplusg 17198 ._rcmulr 17199 Scalarcsca 17201 ·_𝑠 cvsca 17202 0_gc0g 17386 -_gcsg 18857 LSSumclsm 19546 Ringcrg 20130 inv_rcinvr 20281 DivRingcdr 20579 LModclmod 20698 LSubSpclss 20770 LSpanclspn 20810 LVecclvec 20942 LFnlclfn 38421 LKerclk 38449 LDualcld 38487 HLchlt 38714 LHypclh 39349 DVecHcdvh 40443 ocHcoch 40712

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 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-riotaBAD 38317

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-undef 8254 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-n0 12471 df-z 12557 df-uz 12821 df-fz 13483 df-struct 17081 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-mulr 17212 df-sca 17214 df-vsca 17215 df-0g 17388 df-proset 18252 df-poset 18270 df-plt 18287 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-p0 18382 df-p1 18383 df-lat 18389 df-clat 18456 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-subg 19042 df-cntz 19225 df-lsm 19548 df-cmn 19694 df-abl 19695 df-mgp 20032 df-rng 20050 df-ur 20079 df-ring 20132 df-oppr 20228 df-dvdsr 20251 df-unit 20252 df-invr 20282 df-dvr 20295 df-drng 20581 df-lmod 20700 df-lss 20771 df-lsp 20811 df-lvec 20943 df-lsatoms 38340 df-lfl 38422 df-ldual 38488 df-oposet 38540 df-ol 38542 df-oml 38543 df-covers 38630 df-ats 38631 df-atl 38662 df-cvlat 38686 df-hlat 38715 df-llines 38863 df-lplanes 38864 df-lvols 38865 df-lines 38866 df-psubsp 38868 df-pmap 38869 df-padd 39161 df-lhyp 39353 df-laut 39354 df-ldil 39469 df-ltrn 39470 df-trl 39524 df-tendo 40120 df-edring 40122 df-disoa 40394 df-dvech 40444 df-dib 40504 df-dic 40538 df-dih 40594 df-doch 40713

This theorem is referenced by: lclkrlem2q 40888

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