lclkrlem2a - Mathbox for Norm Megill

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Theorem lclkrlem2a 39521

Description: Lemma for lclkr 39547. Use lshpat 37070 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 2738	. 2 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
2		lclkrlem2a.p	. 2 ⊢ ⊕ = (LSSum‘𝑈)
3		eqid 2738	. 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 39123	. 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 39467	. 2 ⊢ (𝜑 → ( ⊥ ‘{𝐵}) ∈ (LSHyp‘𝑈))
14		lclkrlem2a.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
15	5, 6, 7	dvhlmod 39124	. . 3 ⊢ (𝜑 → 𝑈 ∈ LMod)
16		lclkrlem2a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
17	10, 14, 11, 4, 15, 16	lsatlspsn 37007	. 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝐴)
18		lclkrlem2a.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
19	10, 14, 11, 4, 15, 18	lsatlspsn 37007	. 2 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ 𝐴)
20		lclkrlem2a.e	. . 3 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌}))
21	16	eldifad 3899	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
22	21	snssd 4742	. . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
23	5, 6, 9, 10, 14, 7, 22	dochocsp 39393	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋})) = ( ⊥ ‘{𝑋}))
24	18	eldifad 3899	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
25	24	snssd 4742	. . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
26	5, 6, 9, 10, 14, 7, 25	dochocsp 39393	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑌})) = ( ⊥ ‘{𝑌}))
27	23, 26	eqeq12d 2754	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘(𝑁‘{𝑋})) = ( ⊥ ‘(𝑁‘{𝑌})) ↔ ( ⊥ ‘{𝑋}) = ( ⊥ ‘{𝑌})))
28		eqid 2738	. . . . . 6 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
29	5, 6, 10, 14, 28	dihlsprn 39345	. . . . . . 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 39345	. . . . . . 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 39387	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘(𝑁‘{𝑋})) = ( ⊥ ‘(𝑁‘{𝑌})) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
34	27, 33	bitr3d 280	. . . 4 ⊢ (𝜑 → (( ⊥ ‘{𝑋}) = ( ⊥ ‘{𝑌}) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
35	34	necon3bid 2988	. . 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 3899	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
39	38	snssd 4742	. . . . 5 ⊢ (𝜑 → {𝐵} ⊆ 𝑉)
40	5, 6, 10, 1, 9	dochlss 39368	. . . . 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	lspsnel5 20257	. . 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 37070	1 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3884 ∩ cin 3886 ⊆ wss 3887 {csn 4561 ran crn 5590 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 0_gc0g 17150 LSSumclsm 19239 LSubSpclss 20193 LSpanclspn 20233 LSAtomsclsa 36988 LSHypclsh 36989 HLchlt 37364 LHypclh 37998 DVecHcdvh 39092 DIsoHcdih 39242 ocHcoch 39361

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-riotaBAD 36967

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-undef 8089 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-0g 17152 df-mre 17295 df-mrc 17296 df-acs 17298 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-p1 18144 df-lat 18150 df-clat 18217 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-cntz 18923 df-oppg 18950 df-lsm 19241 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-drng 19993 df-lmod 20125 df-lss 20194 df-lsp 20234 df-lvec 20365 df-lsatoms 36990 df-lshyp 36991 df-lcv 37033 df-oposet 37190 df-ol 37192 df-oml 37193 df-covers 37280 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-llines 37512 df-lplanes 37513 df-lvols 37514 df-lines 37515 df-psubsp 37517 df-pmap 37518 df-padd 37810 df-lhyp 38002 df-laut 38003 df-ldil 38118 df-ltrn 38119 df-trl 38173 df-tgrp 38757 df-tendo 38769 df-edring 38771 df-dveca 39017 df-disoa 39043 df-dvech 39093 df-dib 39153 df-dic 39187 df-dih 39243 df-doch 39362 df-djh 39409

This theorem is referenced by: lclkrlem2b 39522

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