lclkrlem2p - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2p		Structured version Visualization version GIF version

Theorem lclkrlem2p 42329

Description: Lemma for lclkr 42340. 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 41917	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		lclkrlem2m.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
6		lclkrlem2m.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
7		eqid 2765	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
8		lclkrlem2n.n	. . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈)
9	6, 7, 8	lspsncl 21128	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
10	4, 5, 9	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
11	6, 7	lssss 21087	. . . 4 ⊢ ((𝑁‘{𝑌}) ∈ (LSubSp‘𝑈) → (𝑁‘{𝑌}) ⊆ 𝑉)
12	10, 11	syl 18	. . 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 2814	. . . . . . 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 21019	. . . . . . . . . . 11 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Ring)
19	4, 18	syl 18	. . . . . . . . . 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 39937	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹)
26		eqid 2765	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
27	17, 26, 6, 20	lflcl 39871	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ (𝐸 + 𝐺) ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → ((𝐸 + 𝐺)‘𝑋) ∈ (Base‘𝑆))
28	4, 25, 16, 27	syl3anc 1398	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑋) ∈ (Base‘𝑆))
29	2, 3, 1	dvhlvec 41916	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ LVec)
30	17	lvecdrng 21256	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ LVec → 𝑆 ∈ DivRing)
31	29, 30	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ DivRing)
32	17, 26, 6, 20	lflcl 39871	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ LMod ∧ (𝐸 + 𝐺) ∈ 𝐹 ∧ 𝑌 ∈ 𝑉) → ((𝐸 + 𝐺)‘𝑌) ∈ (Base‘𝑆))
33	4, 25, 5, 32	syl3anc 1398	. . . . . . . . . . 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 20887	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ DivRing ∧ ((𝐸 + 𝐺)‘𝑌) ∈ (Base‘𝑆) ∧ ((𝐸 + 𝐺)‘𝑌) ≠ 0 ) → (𝐼‘((𝐸 + 𝐺)‘𝑌)) ∈ (Base‘𝑆))
38	31, 33, 34, 37	syl3anc 1398	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘((𝐸 + 𝐺)‘𝑌)) ∈ (Base‘𝑆))
39		lclkrlem2m.q	. . . . . . . . . . 11 ⊢ × = (._r‘𝑆)
40	26, 39	ringcl 20356	. . . . . . . . . 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 1398	. . . . . . . . 9 ⊢ (𝜑 → (((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) ∈ (Base‘𝑆))
42		lclkrlem2m.t	. . . . . . . . . 10 ⊢ · = ( ·_𝑠 ‘𝑈)
43	6, 17, 42, 26	lmodvscl 21029	. . . . . . . . 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 1398	. . . . . . . 8 ⊢ (𝜑 → ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌) ∈ 𝑉)
45		eqid 2765	. . . . . . . . 9 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
46		lclkrlem2m.m	. . . . . . . . 9 ⊢ − = (-_g‘𝑈)
47	6, 45, 46	lmodsubeq0 21072	. . . . . . . 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 1398	. . . . . . 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 235	. . . . . 6 ⊢ (𝜑 → 𝑋 = ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌))
50	49	sneqd 4603	. . . . 5 ⊢ (𝜑 → {𝑋} = {((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)})
51	50	fveq2d 6889	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)}))
52	17, 26, 6, 42, 8	lspsnvsi 21155	. . . . 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 1398	. . . 4 ⊢ (𝜑 → (𝑁‘{((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)}) ⊆ (𝑁‘{𝑌}))
54	51, 53	eqsstrd 3972	. . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}))
55		lclkrlem2o.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
56	2, 3, 6, 55	dochss 42172	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑌}) ⊆ 𝑉 ∧ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌})) → ( ⊥ ‘(𝑁‘{𝑌})) ⊆ ( ⊥ ‘(𝑁‘{𝑋})))
57	1, 12, 54, 56	syl3anc 1398	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑌})) ⊆ ( ⊥ ‘(𝑁‘{𝑋})))
58	5	snssd 4754	. . 3 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
59	2, 3, 55, 6, 8, 1, 58	dochocsp 42186	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑌})) = ( ⊥ ‘{𝑌}))
60	16	snssd 4754	. . 3 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
61	2, 3, 55, 6, 8, 1, 60	dochocsp 42186	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋})) = ( ⊥ ‘{𝑋}))
62	57, 59, 61	3sstr3d 3992	1 ⊢ (𝜑 → ( ⊥ ‘{𝑌}) ⊆ ( ⊥ ‘{𝑋}))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ⊆ wss 3906 {csn 4591 ‘cfv 6540 (class class class)co 7416 Basecbs 17287 +_gcplusg 17328 ._rcmulr 17329 Scalarcsca 17331 ·_𝑠 cvsca 17332 0_gc0g 17510 -_gcsg 19026 LSSumclsm 19728 Ringcrg 20339 inv_rcinvr 20495 DivRingcdr 20857 LModclmod 21011 LSubSpclss 21082 LSpanclspn 21122 LVecclvec 21253 LFnlclfn 39864 LKerclk 39892 LDualcld 39930 HLchlt 40157 LHypclh 40791 DVecHcdvh 41885 ocHcoch 42154

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188 ax-riotaBAD 39760

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7680 df-om 7865 df-1st 7988 df-2nd 7989 df-tpos 8224 df-undef 8271 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-er 8696 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-nn 12245 df-2 12314 df-3 12315 df-4 12316 df-5 12317 df-6 12318 df-n0 12516 df-z 12603 df-uz 12875 df-fz 13548 df-struct 17225 df-sets 17242 df-slot 17260 df-ndx 17272 df-base 17288 df-ress 17309 df-plusg 17341 df-mulr 17342 df-sca 17344 df-vsca 17345 df-0g 17512 df-proset 18368 df-poset 18387 df-plt 18402 df-lub 18418 df-glb 18419 df-join 18420 df-meet 18421 df-p0 18497 df-p1 18498 df-lat 18506 df-clat 18573 df-mgm 18716 df-sgrp 18799 df-mnd 18815 df-submnd 18866 df-grp 19027 df-minusg 19028 df-sbg 19029 df-subg 19213 df-cntz 19411 df-lsm 19730 df-cmn 19876 df-abl 19877 df-mgp 20241 df-rng 20255 df-ur 20288 df-ring 20341 df-oppr 20445 df-dvdsr 20465 df-unit 20466 df-invr 20496 df-dvr 20509 df-drng 20859 df-lmod 21013 df-lss 21083 df-lsp 21123 df-lvec 21254 df-lsatoms 39783 df-lfl 39865 df-ldual 39931 df-oposet 39983 df-ol 39985 df-oml 39986 df-covers 40073 df-ats 40074 df-atl 40105 df-cvlat 40129 df-hlat 40158 df-llines 40305 df-lplanes 40306 df-lvols 40307 df-lines 40308 df-psubsp 40310 df-pmap 40311 df-padd 40603 df-lhyp 40795 df-laut 40796 df-ldil 40911 df-ltrn 40912 df-trl 40966 df-tendo 41562 df-edring 41564 df-disoa 41836 df-dvech 41886 df-dib 41946 df-dic 41980 df-dih 42036 df-doch 42155

This theorem is used by: lclkrlem2r 42331

W3C validator