lclkrlem2a - Mathbox for Norm Megill

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Theorem lclkrlem2a 41546

Description: Lemma for lclkr 41572. Use lshpat 39095 to show that the intersection of a hyperplane with a noncomparable sum of atoms is an atom. (Contributed by NM, 16-Jan-2015.)

**Hypotheses**
Ref	Expression
lclkrlem2a.h	⊢ 𝐻 = (LHyp‘𝐾)
lclkrlem2a.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lclkrlem2a.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lclkrlem2a.v	⊢ 𝑉 = (Base‘𝑈)
lclkrlem2a.z	⊢ 0 = (0_g‘𝑈)
lclkrlem2a.p	⊢ ⊕ = (LSSum‘𝑈)
lclkrlem2a.n	⊢ 𝑁 = (LSpan‘𝑈)
lclkrlem2a.a	⊢ 𝐴 = (LSAtoms‘𝑈)
lclkrlem2a.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lclkrlem2a.b	⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))
lclkrlem2a.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
lclkrlem2a.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
lclkrlem2a.e	⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌}))
lclkrlem2a.d	⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘{𝐵}))

**Assertion**
Ref	Expression
lclkrlem2a	⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴)

**Proof of Theorem lclkrlem2a**
Step	Hyp	Ref	Expression
1		eqid 2731	. 2 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
2		lclkrlem2a.p	. 2 ⊢ ⊕ = (LSSum‘𝑈)
3		eqid 2731	. 2 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈)
4		lclkrlem2a.a	. 2 ⊢ 𝐴 = (LSAtoms‘𝑈)
5		lclkrlem2a.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
6		lclkrlem2a.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		lclkrlem2a.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
8	5, 6, 7	dvhlvec 41148	. 2 ⊢ (𝜑 → 𝑈 ∈ LVec)
9		lclkrlem2a.o	. . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
10		lclkrlem2a.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
11		lclkrlem2a.z	. . 3 ⊢ 0 = (0_g‘𝑈)
12		lclkrlem2a.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))
13	5, 9, 6, 10, 11, 3, 7, 12	dochsnshp 41492	. 2 ⊢ (𝜑 → ( ⊥ ‘{𝐵}) ∈ (LSHyp‘𝑈))
14		lclkrlem2a.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
15	5, 6, 7	dvhlmod 41149	. . 3 ⊢ (𝜑 → 𝑈 ∈ LMod)
16		lclkrlem2a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
17	10, 14, 11, 4, 15, 16	lsatlspsn 39032	. 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝐴)
18		lclkrlem2a.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
19	10, 14, 11, 4, 15, 18	lsatlspsn 39032	. 2 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ 𝐴)
20		lclkrlem2a.e	. . 3 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌}))
21	16	eldifad 3909	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
22	21	snssd 4756	. . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
23	5, 6, 9, 10, 14, 7, 22	dochocsp 41418	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋})) = ( ⊥ ‘{𝑋}))
24	18	eldifad 3909	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
25	24	snssd 4756	. . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
26	5, 6, 9, 10, 14, 7, 25	dochocsp 41418	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑌})) = ( ⊥ ‘{𝑌}))
27	23, 26	eqeq12d 2747	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘(𝑁‘{𝑋})) = ( ⊥ ‘(𝑁‘{𝑌})) ↔ ( ⊥ ‘{𝑋}) = ( ⊥ ‘{𝑌})))
28		eqid 2731	. . . . . 6 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
29	5, 6, 10, 14, 28	dihlsprn 41370	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
30	7, 21, 29	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
31	5, 6, 10, 14, 28	dihlsprn 41370	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
32	7, 24, 31	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
33	5, 28, 9, 7, 30, 32	doch11 41412	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘(𝑁‘{𝑋})) = ( ⊥ ‘(𝑁‘{𝑌})) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
34	27, 33	bitr3d 281	. . . 4 ⊢ (𝜑 → (( ⊥ ‘{𝑋}) = ( ⊥ ‘{𝑌}) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
35	34	necon3bid 2972	. . 3 ⊢ (𝜑 → (( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌}) ↔ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})))
36	20, 35	mpbid 232	. 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
37		lclkrlem2a.d	. . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘{𝐵}))
38	12	eldifad 3909	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
39	38	snssd 4756	. . . . 5 ⊢ (𝜑 → {𝐵} ⊆ 𝑉)
40	5, 6, 10, 1, 9	dochlss 41393	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐵} ⊆ 𝑉) → ( ⊥ ‘{𝐵}) ∈ (LSubSp‘𝑈))
41	7, 39, 40	syl2anc 584	. . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝐵}) ∈ (LSubSp‘𝑈))
42	10, 1, 14, 15, 41, 21	ellspsn5b 20923	. . 3 ⊢ (𝜑 → (𝑋 ∈ ( ⊥ ‘{𝐵}) ↔ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘{𝐵})))
43	37, 42	mtbid 324	. 2 ⊢ (𝜑 → ¬ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘{𝐵}))
44	1, 2, 3, 4, 8, 13, 17, 19, 36, 43	lshpat 39095	1 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∖ cdif 3894 ∩ cin 3896 ⊆ wss 3897 {csn 4571 ran crn 5612 ‘cfv 6476 (class class class)co 7341 Basecbs 17115 0_gc0g 17338 LSSumclsm 19541 LSubSpclss 20859 LSpanclspn 20899 LSAtomsclsa 39013 LSHypclsh 39014 HLchlt 39389 LHypclh 40023 DVecHcdvh 41117 DIsoHcdih 41267 ocHcoch 41386

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-riotaBAD 38992

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-iin 4939 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-tpos 8151 df-undef 8198 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-er 8617 df-map 8747 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-n0 12377 df-z 12464 df-uz 12728 df-fz 13403 df-struct 17053 df-sets 17070 df-slot 17088 df-ndx 17100 df-base 17116 df-ress 17137 df-plusg 17169 df-mulr 17170 df-sca 17172 df-vsca 17173 df-0g 17340 df-mre 17483 df-mrc 17484 df-acs 17486 df-proset 18195 df-poset 18214 df-plt 18229 df-lub 18245 df-glb 18246 df-join 18247 df-meet 18248 df-p0 18324 df-p1 18325 df-lat 18333 df-clat 18400 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-grp 18844 df-minusg 18845 df-sbg 18846 df-subg 19031 df-cntz 19224 df-oppg 19253 df-lsm 19543 df-cmn 19689 df-abl 19690 df-mgp 20054 df-rng 20066 df-ur 20095 df-ring 20148 df-oppr 20250 df-dvdsr 20270 df-unit 20271 df-invr 20301 df-dvr 20314 df-drng 20641 df-lmod 20790 df-lss 20860 df-lsp 20900 df-lvec 21032 df-lsatoms 39015 df-lshyp 39016 df-lcv 39058 df-oposet 39215 df-ol 39217 df-oml 39218 df-covers 39305 df-ats 39306 df-atl 39337 df-cvlat 39361 df-hlat 39390 df-llines 39537 df-lplanes 39538 df-lvols 39539 df-lines 39540 df-psubsp 39542 df-pmap 39543 df-padd 39835 df-lhyp 40027 df-laut 40028 df-ldil 40143 df-ltrn 40144 df-trl 40198 df-tgrp 40782 df-tendo 40794 df-edring 40796 df-dveca 41042 df-disoa 41068 df-dvech 41118 df-dib 41178 df-dic 41212 df-dih 41268 df-doch 41387 df-djh 41434

This theorem is referenced by: lclkrlem2b 41547

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