dochshpsat - Mathbox for Norm Megill

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Theorem dochshpsat 38035

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 477	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
2		dochshpsat.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑌)
3	2	adantr 473	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → 𝑋 ∈ 𝑌)
4	1, 3	eqeltrd 2866	. . 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 2778	. . . . 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 37691	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
13	8, 10, 12, 2	lshplss 35562	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (LSubSp‘𝑈))
14		eqid 2778	. . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈)
15	14, 8	lssss 19433	. . . . . . 7 ⊢ (𝑋 ∈ (LSubSp‘𝑈) → 𝑋 ⊆ (Base‘𝑈))
16	13, 15	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝑈))
17	5, 7, 14, 8, 6	dochlss 37935	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑋) ∈ (LSubSp‘𝑈))
18	11, 16, 17	syl2anc 576	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ (LSubSp‘𝑈))
19	5, 6, 7, 8, 9, 10, 11, 18	dochsatshpb 38033	. . . 4 ⊢ (𝜑 → (( ⊥ ‘𝑋) ∈ 𝐴 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) ∈ 𝑌))
20	19	adantr 473	. . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (( ⊥ ‘𝑋) ∈ 𝐴 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) ∈ 𝑌))
21	4, 20	mpbird 249	. 2 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘𝑋) ∈ 𝐴)
22		eqid 2778	. . . . . 6 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
23	12	adantr 473	. . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → 𝑈 ∈ LMod)
24		simpr 477	. . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ( ⊥ ‘𝑋) ∈ 𝐴)
25	22, 9, 23, 24	lsatn0 35580	. . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ( ⊥ ‘𝑋) ≠ {(0_g‘𝑈)})
26	25	neneqd 2972	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ¬ ( ⊥ ‘𝑋) = {(0_g‘𝑈)})
27	11	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
28	5, 7, 6, 14, 22	doch0 37939	. . . . . . 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 2788	. . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘{(0_g‘𝑈)}) ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈)))
31		eqid 2778	. . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
32	5, 31, 7, 14, 6	dochcl 37934	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
33	11, 16, 32	syl2anc 576	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
34	5, 31, 7, 22	dih0rn 37865	. . . . . . . 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 37954	. . . . . 6 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘{(0_g‘𝑈)}) ↔ ( ⊥ ‘𝑋) = {(0_g‘𝑈)}))
37	36	adantr 473	. . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘{(0_g‘𝑈)}) ↔ ( ⊥ ‘𝑋) = {(0_g‘𝑈)}))
38	30, 37	bitr3d 273	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈) ↔ ( ⊥ ‘𝑋) = {(0_g‘𝑈)}))
39	26, 38	mtbird 317	. . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ¬ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈))
40	5, 6, 7, 14, 10, 11, 2	dochshpncl 37965	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) ≠ 𝑋 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈)))
41	40	necon1bbid 3006	. . . 4 ⊢ (𝜑 → (¬ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈) ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))
42	41	adantr 473	. . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (¬ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈) ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))
43	39, 42	mpbid 224	. 2 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
44	21, 43	impbida 788	1 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ↔ ( ⊥ ‘𝑋) ∈ 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ⊆ wss 3831 {csn 4442 ran crn 5409 ‘cfv 6190 Basecbs 16342 0_gc0g 16572 LModclmod 19359 LSubSpclss 19428 LSAtomsclsa 35555 LSHypclsh 35556 HLchlt 35931 LHypclh 36565 DVecHcdvh 37659 DIsoHcdih 37809 ocHcoch 37928

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 ax-riotaBAD 35534

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-iin 4796 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-1st 7503 df-2nd 7504 df-tpos 7697 df-undef 7744 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-oadd 7911 df-er 8091 df-map 8210 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-nn 11442 df-2 11506 df-3 11507 df-4 11508 df-5 11509 df-6 11510 df-n0 11711 df-z 11797 df-uz 12062 df-fz 12712 df-struct 16344 df-ndx 16345 df-slot 16346 df-base 16348 df-sets 16349 df-ress 16350 df-plusg 16437 df-mulr 16438 df-sca 16440 df-vsca 16441 df-0g 16574 df-proset 17399 df-poset 17417 df-plt 17429 df-lub 17445 df-glb 17446 df-join 17447 df-meet 17448 df-p0 17510 df-p1 17511 df-lat 17517 df-clat 17579 df-mgm 17713 df-sgrp 17755 df-mnd 17766 df-submnd 17807 df-grp 17897 df-minusg 17898 df-sbg 17899 df-subg 18063 df-cntz 18221 df-lsm 18525 df-cmn 18671 df-abl 18672 df-mgp 18966 df-ur 18978 df-ring 19025 df-oppr 19099 df-dvdsr 19117 df-unit 19118 df-invr 19148 df-dvr 19159 df-drng 19230 df-lmod 19361 df-lss 19429 df-lsp 19469 df-lvec 19600 df-lsatoms 35557 df-lshyp 35558 df-oposet 35757 df-ol 35759 df-oml 35760 df-covers 35847 df-ats 35848 df-atl 35879 df-cvlat 35903 df-hlat 35932 df-llines 36079 df-lplanes 36080 df-lvols 36081 df-lines 36082 df-psubsp 36084 df-pmap 36085 df-padd 36377 df-lhyp 36569 df-laut 36570 df-ldil 36685 df-ltrn 36686 df-trl 36740 df-tgrp 37324 df-tendo 37336 df-edring 37338 df-dveca 37584 df-disoa 37610 df-dvech 37660 df-dib 37720 df-dic 37754 df-dih 37810 df-doch 37929 df-djh 37976

This theorem is referenced by: mapdordlem2 38218

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