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 41782

Description: Lemma for lclkr 41793. 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 41370	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		lclkrlem2m.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
6		lclkrlem2m.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
7		eqid 2736	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
8		lclkrlem2n.n	. . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈)
9	6, 7, 8	lspsncl 20928	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
10	4, 5, 9	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
11	6, 7	lssss 20887	. . . 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 2785	. . . . . . 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 20819	. . . . . . . . . . 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 39390	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹)
26		eqid 2736	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
27	17, 26, 6, 20	lflcl 39324	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ (𝐸 + 𝐺) ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → ((𝐸 + 𝐺)‘𝑋) ∈ (Base‘𝑆))
28	4, 25, 16, 27	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑋) ∈ (Base‘𝑆))
29	2, 3, 1	dvhlvec 41369	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ LVec)
30	17	lvecdrng 21057	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ LVec → 𝑆 ∈ DivRing)
31	29, 30	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ DivRing)
32	17, 26, 6, 20	lflcl 39324	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ LMod ∧ (𝐸 + 𝐺) ∈ 𝐹 ∧ 𝑌 ∈ 𝑉) → ((𝐸 + 𝐺)‘𝑌) ∈ (Base‘𝑆))
33	4, 25, 5, 32	syl3anc 1373	. . . . . . . . . . 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 20686	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ DivRing ∧ ((𝐸 + 𝐺)‘𝑌) ∈ (Base‘𝑆) ∧ ((𝐸 + 𝐺)‘𝑌) ≠ 0 ) → (𝐼‘((𝐸 + 𝐺)‘𝑌)) ∈ (Base‘𝑆))
38	31, 33, 34, 37	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘((𝐸 + 𝐺)‘𝑌)) ∈ (Base‘𝑆))
39		lclkrlem2m.q	. . . . . . . . . . 11 ⊢ × = (._r‘𝑆)
40	26, 39	ringcl 20185	. . . . . . . . . 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 1373	. . . . . . . . 9 ⊢ (𝜑 → (((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) ∈ (Base‘𝑆))
42		lclkrlem2m.t	. . . . . . . . . 10 ⊢ · = ( ·_𝑠 ‘𝑈)
43	6, 17, 42, 26	lmodvscl 20829	. . . . . . . . 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 1373	. . . . . . . 8 ⊢ (𝜑 → ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌) ∈ 𝑉)
45		eqid 2736	. . . . . . . . 9 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
46		lclkrlem2m.m	. . . . . . . . 9 ⊢ − = (-_g‘𝑈)
47	6, 45, 46	lmodsubeq0 20872	. . . . . . . 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 1373	. . . . . . 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 4592	. . . . 5 ⊢ (𝜑 → {𝑋} = {((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)})
51	50	fveq2d 6838	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)}))
52	17, 26, 6, 42, 8	lspsnvsi 20955	. . . . 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 1373	. . . 4 ⊢ (𝜑 → (𝑁‘{((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)}) ⊆ (𝑁‘{𝑌}))
54	51, 53	eqsstrd 3968	. . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}))
55		lclkrlem2o.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
56	2, 3, 6, 55	dochss 41625	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑌}) ⊆ 𝑉 ∧ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌})) → ( ⊥ ‘(𝑁‘{𝑌})) ⊆ ( ⊥ ‘(𝑁‘{𝑋})))
57	1, 12, 54, 56	syl3anc 1373	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑌})) ⊆ ( ⊥ ‘(𝑁‘{𝑋})))
58	5	snssd 4765	. . 3 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
59	2, 3, 55, 6, 8, 1, 58	dochocsp 41639	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑌})) = ( ⊥ ‘{𝑌}))
60	16	snssd 4765	. . 3 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
61	2, 3, 55, 6, 8, 1, 60	dochocsp 41639	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋})) = ( ⊥ ‘{𝑋}))
62	57, 59, 61	3sstr3d 3988	1 ⊢ (𝜑 → ( ⊥ ‘{𝑌}) ⊆ ( ⊥ ‘{𝑋}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ⊆ wss 3901 {csn 4580 ‘cfv 6492 (class class class)co 7358 Basecbs 17136 +_gcplusg 17177 ._rcmulr 17178 Scalarcsca 17180 ·_𝑠 cvsca 17181 0_gc0g 17359 -_gcsg 18865 LSSumclsm 19563 Ringcrg 20168 inv_rcinvr 20323 DivRingcdr 20662 LModclmod 20811 LSubSpclss 20882 LSpanclspn 20922 LVecclvec 21054 LFnlclfn 39317 LKerclk 39345 LDualcld 39383 HLchlt 39610 LHypclh 40244 DVecHcdvh 41338 ocHcoch 41607

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-riotaBAD 39213

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-tpos 8168 df-undef 8215 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-n0 12402 df-z 12489 df-uz 12752 df-fz 13424 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-sca 17193 df-vsca 17194 df-0g 17361 df-proset 18217 df-poset 18236 df-plt 18251 df-lub 18267 df-glb 18268 df-join 18269 df-meet 18270 df-p0 18346 df-p1 18347 df-lat 18355 df-clat 18422 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19053 df-cntz 19246 df-lsm 19565 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-oppr 20273 df-dvdsr 20293 df-unit 20294 df-invr 20324 df-dvr 20337 df-drng 20664 df-lmod 20813 df-lss 20883 df-lsp 20923 df-lvec 21055 df-lsatoms 39236 df-lfl 39318 df-ldual 39384 df-oposet 39436 df-ol 39438 df-oml 39439 df-covers 39526 df-ats 39527 df-atl 39558 df-cvlat 39582 df-hlat 39611 df-llines 39758 df-lplanes 39759 df-lvols 39760 df-lines 39761 df-psubsp 39763 df-pmap 39764 df-padd 40056 df-lhyp 40248 df-laut 40249 df-ldil 40364 df-ltrn 40365 df-trl 40419 df-tendo 41015 df-edring 41017 df-disoa 41289 df-dvech 41339 df-dib 41399 df-dic 41433 df-dih 41489 df-doch 41608

This theorem is referenced by: lclkrlem2r 41784

W3C validator