lclkrlem2a - Mathbox for Norm Megill

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Theorem lclkrlem2a 37661

Description: Lemma for lclkr 37687. Use lshpat 35210 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 2778	. 2 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
2		lclkrlem2a.p	. 2 ⊢ ⊕ = (LSSum‘𝑈)
3		eqid 2778	. 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 37263	. 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 37607	. 2 ⊢ (𝜑 → ( ⊥ ‘{𝐵}) ∈ (LSHyp‘𝑈))
14		lclkrlem2a.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
15	5, 6, 7	dvhlmod 37264	. . 3 ⊢ (𝜑 → 𝑈 ∈ LMod)
16		lclkrlem2a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
17	10, 14, 11, 4, 15, 16	lsatlspsn 35147	. 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝐴)
18		lclkrlem2a.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
19	10, 14, 11, 4, 15, 18	lsatlspsn 35147	. 2 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ 𝐴)
20		lclkrlem2a.e	. . 3 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌}))
21	16	eldifad 3804	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
22	21	snssd 4571	. . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
23	5, 6, 9, 10, 14, 7, 22	dochocsp 37533	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋})) = ( ⊥ ‘{𝑋}))
24	18	eldifad 3804	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
25	24	snssd 4571	. . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
26	5, 6, 9, 10, 14, 7, 25	dochocsp 37533	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑌})) = ( ⊥ ‘{𝑌}))
27	23, 26	eqeq12d 2793	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘(𝑁‘{𝑋})) = ( ⊥ ‘(𝑁‘{𝑌})) ↔ ( ⊥ ‘{𝑋}) = ( ⊥ ‘{𝑌})))
28		eqid 2778	. . . . . 6 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
29	5, 6, 10, 14, 28	dihlsprn 37485	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
30	7, 21, 29	syl2anc 579	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
31	5, 6, 10, 14, 28	dihlsprn 37485	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
32	7, 24, 31	syl2anc 579	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
33	5, 28, 9, 7, 30, 32	doch11 37527	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘(𝑁‘{𝑋})) = ( ⊥ ‘(𝑁‘{𝑌})) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
34	27, 33	bitr3d 273	. . . 4 ⊢ (𝜑 → (( ⊥ ‘{𝑋}) = ( ⊥ ‘{𝑌}) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
35	34	necon3bid 3013	. . 3 ⊢ (𝜑 → (( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌}) ↔ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})))
36	20, 35	mpbid 224	. 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
37		lclkrlem2a.d	. . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘{𝐵}))
38	12	eldifad 3804	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
39	38	snssd 4571	. . . . 5 ⊢ (𝜑 → {𝐵} ⊆ 𝑉)
40	5, 6, 10, 1, 9	dochlss 37508	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐵} ⊆ 𝑉) → ( ⊥ ‘{𝐵}) ∈ (LSubSp‘𝑈))
41	7, 39, 40	syl2anc 579	. . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝐵}) ∈ (LSubSp‘𝑈))
42	10, 1, 14, 15, 41, 21	lspsnel5 19390	. . 3 ⊢ (𝜑 → (𝑋 ∈ ( ⊥ ‘{𝐵}) ↔ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘{𝐵})))
43	37, 42	mtbid 316	. 2 ⊢ (𝜑 → ¬ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘{𝐵}))
44	1, 2, 3, 4, 8, 13, 17, 19, 36, 43	lshpat 35210	1 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∖ cdif 3789 ∩ cin 3791 ⊆ wss 3792 {csn 4398 ran crn 5356 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 0_gc0g 16486 LSSumclsm 18433 LSubSpclss 19324 LSpanclspn 19366 LSAtomsclsa 35128 LSHypclsh 35129 HLchlt 35504 LHypclh 36138 DVecHcdvh 37232 DIsoHcdih 37382 ocHcoch 37501

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-riotaBAD 35107

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-undef 7681 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-0g 16488 df-mre 16632 df-mrc 16633 df-acs 16635 df-proset 17314 df-poset 17332 df-plt 17344 df-lub 17360 df-glb 17361 df-join 17362 df-meet 17363 df-p0 17425 df-p1 17426 df-lat 17432 df-clat 17494 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-subg 17975 df-cntz 18133 df-oppg 18159 df-lsm 18435 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-dvr 19070 df-drng 19141 df-lmod 19257 df-lss 19325 df-lsp 19367 df-lvec 19498 df-lsatoms 35130 df-lshyp 35131 df-lcv 35173 df-oposet 35330 df-ol 35332 df-oml 35333 df-covers 35420 df-ats 35421 df-atl 35452 df-cvlat 35476 df-hlat 35505 df-llines 35652 df-lplanes 35653 df-lvols 35654 df-lines 35655 df-psubsp 35657 df-pmap 35658 df-padd 35950 df-lhyp 36142 df-laut 36143 df-ldil 36258 df-ltrn 36259 df-trl 36313 df-tgrp 36897 df-tendo 36909 df-edring 36911 df-dveca 37157 df-disoa 37183 df-dvech 37233 df-dib 37293 df-dic 37327 df-dih 37383 df-doch 37502 df-djh 37549

This theorem is referenced by: lclkrlem2b 37662

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