lclkrlem2p - Mathbox for Norm Megill

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Theorem lclkrlem2p 41694

Description: Lemma for lclkr 41705. 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 41282	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		lclkrlem2m.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
6		lclkrlem2m.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
7		eqid 2733	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
8		lclkrlem2n.n	. . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈)
9	6, 7, 8	lspsncl 20919	. . . . 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 20878	. . . 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 2782	. . . . . . 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 20810	. . . . . . . . . . 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 39302	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹)
26		eqid 2733	. . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆)
27	17, 26, 6, 20	lflcl 39236	. . . . . . . . . . 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 41281	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ LVec)
30	17	lvecdrng 21048	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ LVec → 𝑆 ∈ DivRing)
31	29, 30	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ DivRing)
32	17, 26, 6, 20	lflcl 39236	. . . . . . . . . . . 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 20677	. . . . . . . . . . 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 20176	. . . . . . . . . 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 20820	. . . . . . . . 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 2733	. . . . . . . . 9 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
46		lclkrlem2m.m	. . . . . . . . 9 ⊢ − = (-_g‘𝑈)
47	6, 45, 46	lmodsubeq0 20863	. . . . . . . 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 4589	. . . . 5 ⊢ (𝜑 → {𝑋} = {((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)})
51	50	fveq2d 6835	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)}))
52	17, 26, 6, 42, 8	lspsnvsi 20946	. . . . 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 3965	. . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}))
55		lclkrlem2o.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
56	2, 3, 6, 55	dochss 41537	. . 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 4762	. . 3 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
59	2, 3, 55, 6, 8, 1, 58	dochocsp 41551	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑌})) = ( ⊥ ‘{𝑌}))
60	16	snssd 4762	. . 3 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
61	2, 3, 55, 6, 8, 1, 60	dochocsp 41551	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋})) = ( ⊥ ‘{𝑋}))
62	57, 59, 61	3sstr3d 3985	1 ⊢ (𝜑 → ( ⊥ ‘{𝑌}) ⊆ ( ⊥ ‘{𝑋}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2929 ⊆ wss 3898 {csn 4577 ‘cfv 6489 (class class class)co 7355 Basecbs 17127 +_gcplusg 17168 ._rcmulr 17169 Scalarcsca 17171 ·_𝑠 cvsca 17172 0_gc0g 17350 -_gcsg 18856 LSSumclsm 19554 Ringcrg 20159 inv_rcinvr 20314 DivRingcdr 20653 LModclmod 20802 LSubSpclss 20873 LSpanclspn 20913 LVecclvec 21045 LFnlclfn 39229 LKerclk 39257 LDualcld 39295 HLchlt 39522 LHypclh 40156 DVecHcdvh 41250 ocHcoch 41519

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 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-riotaBAD 39125

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 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-om 7806 df-1st 7930 df-2nd 7931 df-tpos 8165 df-undef 8212 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-er 8631 df-map 8761 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-nn 12137 df-2 12199 df-3 12200 df-4 12201 df-5 12202 df-6 12203 df-n0 12393 df-z 12480 df-uz 12743 df-fz 13415 df-struct 17065 df-sets 17082 df-slot 17100 df-ndx 17112 df-base 17128 df-ress 17149 df-plusg 17181 df-mulr 17182 df-sca 17184 df-vsca 17185 df-0g 17352 df-proset 18208 df-poset 18227 df-plt 18242 df-lub 18258 df-glb 18259 df-join 18260 df-meet 18261 df-p0 18337 df-p1 18338 df-lat 18346 df-clat 18413 df-mgm 18556 df-sgrp 18635 df-mnd 18651 df-submnd 18700 df-grp 18857 df-minusg 18858 df-sbg 18859 df-subg 19044 df-cntz 19237 df-lsm 19556 df-cmn 19702 df-abl 19703 df-mgp 20067 df-rng 20079 df-ur 20108 df-ring 20161 df-oppr 20264 df-dvdsr 20284 df-unit 20285 df-invr 20315 df-dvr 20328 df-drng 20655 df-lmod 20804 df-lss 20874 df-lsp 20914 df-lvec 21046 df-lsatoms 39148 df-lfl 39230 df-ldual 39296 df-oposet 39348 df-ol 39350 df-oml 39351 df-covers 39438 df-ats 39439 df-atl 39470 df-cvlat 39494 df-hlat 39523 df-llines 39670 df-lplanes 39671 df-lvols 39672 df-lines 39673 df-psubsp 39675 df-pmap 39676 df-padd 39968 df-lhyp 40160 df-laut 40161 df-ldil 40276 df-ltrn 40277 df-trl 40331 df-tendo 40927 df-edring 40929 df-disoa 41201 df-dvech 41251 df-dib 41311 df-dic 41345 df-dih 41401 df-doch 41520

This theorem is referenced by: lclkrlem2r 41696

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