lclkrlem2a - Mathbox for Norm Megill

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Theorem lclkrlem2a 40868

Description: Lemma for lclkr 40894. Use lshpat 38416 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 2724	. 2 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
2		lclkrlem2a.p	. 2 ⊢ ⊕ = (LSSum‘𝑈)
3		eqid 2724	. 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 40470	. 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 40814	. 2 ⊢ (𝜑 → ( ⊥ ‘{𝐵}) ∈ (LSHyp‘𝑈))
14		lclkrlem2a.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
15	5, 6, 7	dvhlmod 40471	. . 3 ⊢ (𝜑 → 𝑈 ∈ LMod)
16		lclkrlem2a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
17	10, 14, 11, 4, 15, 16	lsatlspsn 38353	. 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝐴)
18		lclkrlem2a.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
19	10, 14, 11, 4, 15, 18	lsatlspsn 38353	. 2 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ 𝐴)
20		lclkrlem2a.e	. . 3 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌}))
21	16	eldifad 3952	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
22	21	snssd 4804	. . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
23	5, 6, 9, 10, 14, 7, 22	dochocsp 40740	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋})) = ( ⊥ ‘{𝑋}))
24	18	eldifad 3952	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
25	24	snssd 4804	. . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
26	5, 6, 9, 10, 14, 7, 25	dochocsp 40740	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑌})) = ( ⊥ ‘{𝑌}))
27	23, 26	eqeq12d 2740	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘(𝑁‘{𝑋})) = ( ⊥ ‘(𝑁‘{𝑌})) ↔ ( ⊥ ‘{𝑋}) = ( ⊥ ‘{𝑌})))
28		eqid 2724	. . . . . 6 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
29	5, 6, 10, 14, 28	dihlsprn 40692	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
30	7, 21, 29	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
31	5, 6, 10, 14, 28	dihlsprn 40692	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
32	7, 24, 31	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
33	5, 28, 9, 7, 30, 32	doch11 40734	. . . . 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 2977	. . 3 ⊢ (𝜑 → (( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌}) ↔ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})))
36	20, 35	mpbid 231	. 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
37		lclkrlem2a.d	. . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘{𝐵}))
38	12	eldifad 3952	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
39	38	snssd 4804	. . . . 5 ⊢ (𝜑 → {𝐵} ⊆ 𝑉)
40	5, 6, 10, 1, 9	dochlss 40715	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐵} ⊆ 𝑉) → ( ⊥ ‘{𝐵}) ∈ (LSubSp‘𝑈))
41	7, 39, 40	syl2anc 583	. . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝐵}) ∈ (LSubSp‘𝑈))
42	10, 1, 14, 15, 41, 21	lspsnel5 20832	. . 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 38416	1 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∖ cdif 3937 ∩ cin 3939 ⊆ wss 3940 {csn 4620 ran crn 5667 ‘cfv 6533 (class class class)co 7401 Basecbs 17143 0_gc0g 17384 LSSumclsm 19544 LSubSpclss 20768 LSpanclspn 20808 LSAtomsclsa 38334 LSHypclsh 38335 HLchlt 38710 LHypclh 39345 DVecHcdvh 40439 DIsoHcdih 40589 ocHcoch 40708

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-riotaBAD 38313

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-tpos 8206 df-undef 8253 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-0g 17386 df-mre 17529 df-mrc 17530 df-acs 17532 df-proset 18250 df-poset 18268 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18387 df-clat 18454 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18704 df-grp 18856 df-minusg 18857 df-sbg 18858 df-subg 19040 df-cntz 19223 df-oppg 19252 df-lsm 19546 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-dvr 20293 df-drng 20579 df-lmod 20698 df-lss 20769 df-lsp 20809 df-lvec 20941 df-lsatoms 38336 df-lshyp 38337 df-lcv 38379 df-oposet 38536 df-ol 38538 df-oml 38539 df-covers 38626 df-ats 38627 df-atl 38658 df-cvlat 38682 df-hlat 38711 df-llines 38859 df-lplanes 38860 df-lvols 38861 df-lines 38862 df-psubsp 38864 df-pmap 38865 df-padd 39157 df-lhyp 39349 df-laut 39350 df-ldil 39465 df-ltrn 39466 df-trl 39520 df-tgrp 40104 df-tendo 40116 df-edring 40118 df-dveca 40364 df-disoa 40390 df-dvech 40440 df-dib 40500 df-dic 40534 df-dih 40590 df-doch 40709 df-djh 40756

This theorem is referenced by: lclkrlem2b 40869

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