dochshpsat - Mathbox for Norm Megill

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Theorem dochshpsat 39468

Description: A hyperplane is closed iff its orthocomplement is an atom. (Contributed by NM, 29-Oct-2014.)

**Hypotheses**
Ref	Expression
dochshpsat.h	⊢ 𝐻 = (LHyp‘𝐾)
dochshpsat.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
dochshpsat.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dochshpsat.a	⊢ 𝐴 = (LSAtoms‘𝑈)
dochshpsat.y	⊢ 𝑌 = (LSHyp‘𝑈)
dochshpsat.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dochshpsat.x	⊢ (𝜑 → 𝑋 ∈ 𝑌)

**Assertion**
Ref	Expression
dochshpsat	⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ↔ ( ⊥ ‘𝑋) ∈ 𝐴))

**Proof of Theorem dochshpsat**
Step	Hyp	Ref	Expression
1		simpr 485	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
2		dochshpsat.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑌)
3	2	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → 𝑋 ∈ 𝑌)
4	1, 3	eqeltrd 2839	. . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ 𝑌)
5		dochshpsat.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
6		dochshpsat.o	. . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
7		dochshpsat.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
8		eqid 2738	. . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
9		dochshpsat.a	. . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑈)
10		dochshpsat.y	. . . . 5 ⊢ 𝑌 = (LSHyp‘𝑈)
11		dochshpsat.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12	5, 7, 11	dvhlmod 39124	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
13	8, 10, 12, 2	lshplss 36995	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (LSubSp‘𝑈))
14		eqid 2738	. . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈)
15	14, 8	lssss 20198	. . . . . . 7 ⊢ (𝑋 ∈ (LSubSp‘𝑈) → 𝑋 ⊆ (Base‘𝑈))
16	13, 15	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝑈))
17	5, 7, 14, 8, 6	dochlss 39368	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑋) ∈ (LSubSp‘𝑈))
18	11, 16, 17	syl2anc 584	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ (LSubSp‘𝑈))
19	5, 6, 7, 8, 9, 10, 11, 18	dochsatshpb 39466	. . . 4 ⊢ (𝜑 → (( ⊥ ‘𝑋) ∈ 𝐴 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) ∈ 𝑌))
20	19	adantr 481	. . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (( ⊥ ‘𝑋) ∈ 𝐴 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) ∈ 𝑌))
21	4, 20	mpbird 256	. 2 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘𝑋) ∈ 𝐴)
22		eqid 2738	. . . . . 6 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
23	12	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → 𝑈 ∈ LMod)
24		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ( ⊥ ‘𝑋) ∈ 𝐴)
25	22, 9, 23, 24	lsatn0 37013	. . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ( ⊥ ‘𝑋) ≠ {(0_g‘𝑈)})
26	25	neneqd 2948	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ¬ ( ⊥ ‘𝑋) = {(0_g‘𝑈)})
27	11	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
28	5, 7, 6, 14, 22	doch0 39372	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{(0_g‘𝑈)}) = (Base‘𝑈))
29	27, 28	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ( ⊥ ‘{(0_g‘𝑈)}) = (Base‘𝑈))
30	29	eqeq2d 2749	. . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘{(0_g‘𝑈)}) ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈)))
31		eqid 2738	. . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
32	5, 31, 7, 14, 6	dochcl 39367	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
33	11, 16, 32	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
34	5, 31, 7, 22	dih0rn 39298	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → {(0_g‘𝑈)} ∈ ran ((DIsoH‘𝐾)‘𝑊))
35	11, 34	syl 17	. . . . . . 7 ⊢ (𝜑 → {(0_g‘𝑈)} ∈ ran ((DIsoH‘𝐾)‘𝑊))
36	5, 31, 6, 11, 33, 35	doch11 39387	. . . . . 6 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘{(0_g‘𝑈)}) ↔ ( ⊥ ‘𝑋) = {(0_g‘𝑈)}))
37	36	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘{(0_g‘𝑈)}) ↔ ( ⊥ ‘𝑋) = {(0_g‘𝑈)}))
38	30, 37	bitr3d 280	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈) ↔ ( ⊥ ‘𝑋) = {(0_g‘𝑈)}))
39	26, 38	mtbird 325	. . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ¬ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈))
40	5, 6, 7, 14, 10, 11, 2	dochshpncl 39398	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) ≠ 𝑋 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈)))
41	40	necon1bbid 2983	. . . 4 ⊢ (𝜑 → (¬ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈) ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))
42	41	adantr 481	. . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (¬ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈) ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))
43	39, 42	mpbid 231	. 2 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
44	21, 43	impbida 798	1 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ↔ ( ⊥ ‘𝑋) ∈ 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ⊆ wss 3887 {csn 4561 ran crn 5590 ‘cfv 6433 Basecbs 16912 0_gc0g 17150 LModclmod 20123 LSubSpclss 20193 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-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-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-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: mapdordlem2 39651

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