lclkrlem2p - Mathbox for Norm Megill

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Theorem lclkrlem2p 41479

Description: Lemma for lclkr 41490. When 𝐵 is zero, 𝑋 and 𝑌 must colinear, so their orthocomplements must be comparable. (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 )
lclkrlem2p.bn	⊢ (𝜑 → 𝐵 = (0_g‘𝑈))

**Assertion**
Ref	Expression
lclkrlem2p	⊢ (𝜑 → ( ⊥ ‘{𝑌}) ⊆ ( ⊥ ‘{𝑋}))

**Proof of Theorem lclkrlem2p**
Step	Hyp	Ref	Expression
1		lclkrlem2o.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		lclkrlem2o.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
3		lclkrlem2o.u	. . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4	2, 3, 1	dvhlmod 41067	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		lclkrlem2m.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
6		lclkrlem2m.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
7		eqid 2740	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
8		lclkrlem2n.n	. . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈)
9	6, 7, 8	lspsncl 20998	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
10	4, 5, 9	syl2anc 583	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
11	6, 7	lssss 20957	. . . 4 ⊢ ((𝑁‘{𝑌}) ∈ (LSubSp‘𝑈) → (𝑁‘{𝑌}) ⊆ 𝑉)
12	10, 11	syl 17	. . 3 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ 𝑉)
13		lclkrlem2o.b	. . . . . . . 8 ⊢ 𝐵 = (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌))
14		lclkrlem2p.bn	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (0_g‘𝑈))
15	13, 14	eqtr3id 2794	. . . . . . 7 ⊢ (𝜑 → (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0_g‘𝑈))
16		lclkrlem2m.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
17		lclkrlem2m.s	. . . . . . . . . . . 12 ⊢ 𝑆 = (Scalar‘𝑈)
18	17	lmodring 20888	. . . . . . . . . . 11 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Ring)
19	4, 18	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ Ring)
20		lclkrlem2m.f	. . . . . . . . . . . 12 ⊢ 𝐹 = (LFnl‘𝑈)
21		lclkrlem2m.d	. . . . . . . . . . . 12 ⊢ 𝐷 = (LDual‘𝑈)
22		lclkrlem2m.p	. . . . . . . . . . . 12 ⊢ + = (+_g‘𝐷)
23		lclkrlem2m.e	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ 𝐹)
24		lclkrlem2m.g	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
25	20, 21, 22, 4, 23, 24	ldualvaddcl 39086	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹)
26		eqid 2740	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
27	17, 26, 6, 20	lflcl 39020	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ (𝐸 + 𝐺) ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → ((𝐸 + 𝐺)‘𝑋) ∈ (Base‘𝑆))
28	4, 25, 16, 27	syl3anc 1371	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑋) ∈ (Base‘𝑆))
29	2, 3, 1	dvhlvec 41066	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ LVec)
30	17	lvecdrng 21127	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ LVec → 𝑆 ∈ DivRing)
31	29, 30	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ DivRing)
32	17, 26, 6, 20	lflcl 39020	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ LMod ∧ (𝐸 + 𝐺) ∈ 𝐹 ∧ 𝑌 ∈ 𝑉) → ((𝐸 + 𝐺)‘𝑌) ∈ (Base‘𝑆))
33	4, 25, 5, 32	syl3anc 1371	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) ∈ (Base‘𝑆))
34		lclkrlem2o.n	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )
35		lclkrlem2m.z	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘𝑆)
36		lclkrlem2m.i	. . . . . . . . . . . 12 ⊢ 𝐼 = (inv_r‘𝑆)
37	26, 35, 36	drnginvrcl 20775	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ DivRing ∧ ((𝐸 + 𝐺)‘𝑌) ∈ (Base‘𝑆) ∧ ((𝐸 + 𝐺)‘𝑌) ≠ 0 ) → (𝐼‘((𝐸 + 𝐺)‘𝑌)) ∈ (Base‘𝑆))
38	31, 33, 34, 37	syl3anc 1371	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘((𝐸 + 𝐺)‘𝑌)) ∈ (Base‘𝑆))
39		lclkrlem2m.q	. . . . . . . . . . 11 ⊢ × = (._r‘𝑆)
40	26, 39	ringcl 20277	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Ring ∧ ((𝐸 + 𝐺)‘𝑋) ∈ (Base‘𝑆) ∧ (𝐼‘((𝐸 + 𝐺)‘𝑌)) ∈ (Base‘𝑆)) → (((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) ∈ (Base‘𝑆))
41	19, 28, 38, 40	syl3anc 1371	. . . . . . . . 9 ⊢ (𝜑 → (((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) ∈ (Base‘𝑆))
42		lclkrlem2m.t	. . . . . . . . . 10 ⊢ · = ( ·_𝑠 ‘𝑈)
43	6, 17, 42, 26	lmodvscl 20898	. . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ (((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) ∈ (Base‘𝑆) ∧ 𝑌 ∈ 𝑉) → ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌) ∈ 𝑉)
44	4, 41, 5, 43	syl3anc 1371	. . . . . . . 8 ⊢ (𝜑 → ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌) ∈ 𝑉)
45		eqid 2740	. . . . . . . . 9 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
46		lclkrlem2m.m	. . . . . . . . 9 ⊢ − = (-_g‘𝑈)
47	6, 45, 46	lmodsubeq0 20941	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌) ∈ 𝑉) → ((𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0_g‘𝑈) ↔ 𝑋 = ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)))
48	4, 16, 44, 47	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → ((𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0_g‘𝑈) ↔ 𝑋 = ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)))
49	15, 48	mpbid 232	. . . . . 6 ⊢ (𝜑 → 𝑋 = ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌))
50	49	sneqd 4660	. . . . 5 ⊢ (𝜑 → {𝑋} = {((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)})
51	50	fveq2d 6924	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)}))
52	17, 26, 6, 42, 8	lspsnvsi 21025	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) ∈ (Base‘𝑆) ∧ 𝑌 ∈ 𝑉) → (𝑁‘{((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)}) ⊆ (𝑁‘{𝑌}))
53	4, 41, 5, 52	syl3anc 1371	. . . 4 ⊢ (𝜑 → (𝑁‘{((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)}) ⊆ (𝑁‘{𝑌}))
54	51, 53	eqsstrd 4047	. . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}))
55		lclkrlem2o.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
56	2, 3, 6, 55	dochss 41322	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑌}) ⊆ 𝑉 ∧ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌})) → ( ⊥ ‘(𝑁‘{𝑌})) ⊆ ( ⊥ ‘(𝑁‘{𝑋})))
57	1, 12, 54, 56	syl3anc 1371	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑌})) ⊆ ( ⊥ ‘(𝑁‘{𝑋})))
58	5	snssd 4834	. . 3 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
59	2, 3, 55, 6, 8, 1, 58	dochocsp 41336	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑌})) = ( ⊥ ‘{𝑌}))
60	16	snssd 4834	. . 3 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
61	2, 3, 55, 6, 8, 1, 60	dochocsp 41336	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋})) = ( ⊥ ‘{𝑋}))
62	57, 59, 61	3sstr3d 4055	1 ⊢ (𝜑 → ( ⊥ ‘{𝑌}) ⊆ ( ⊥ ‘{𝑋}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ⊆ wss 3976 {csn 4648 ‘cfv 6573 (class class class)co 7448 Basecbs 17258 +_gcplusg 17311 ._rcmulr 17312 Scalarcsca 17314 ·_𝑠 cvsca 17315 0_gc0g 17499 -_gcsg 18975 LSSumclsm 19676 Ringcrg 20260 inv_rcinvr 20413 DivRingcdr 20751 LModclmod 20880 LSubSpclss 20952 LSpanclspn 20992 LVecclvec 21124 LFnlclfn 39013 LKerclk 39041 LDualcld 39079 HLchlt 39306 LHypclh 39941 DVecHcdvh 41035 ocHcoch 41304

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-riotaBAD 38909

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-tpos 8267 df-undef 8314 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-n0 12554 df-z 12640 df-uz 12904 df-fz 13568 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-0g 17501 df-proset 18365 df-poset 18383 df-plt 18400 df-lub 18416 df-glb 18417 df-join 18418 df-meet 18419 df-p0 18495 df-p1 18496 df-lat 18502 df-clat 18569 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-subg 19163 df-cntz 19357 df-lsm 19678 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-dvr 20427 df-drng 20753 df-lmod 20882 df-lss 20953 df-lsp 20993 df-lvec 21125 df-lsatoms 38932 df-lfl 39014 df-ldual 39080 df-oposet 39132 df-ol 39134 df-oml 39135 df-covers 39222 df-ats 39223 df-atl 39254 df-cvlat 39278 df-hlat 39307 df-llines 39455 df-lplanes 39456 df-lvols 39457 df-lines 39458 df-psubsp 39460 df-pmap 39461 df-padd 39753 df-lhyp 39945 df-laut 39946 df-ldil 40061 df-ltrn 40062 df-trl 40116 df-tendo 40712 df-edring 40714 df-disoa 40986 df-dvech 41036 df-dib 41096 df-dic 41130 df-dih 41186 df-doch 41305

This theorem is referenced by: lclkrlem2r 41481

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