dochshpsat - Mathbox for Norm Megill

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Theorem dochshpsat 39030

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 488	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
2		dochshpsat.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑌)
3	2	adantr 484	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → 𝑋 ∈ 𝑌)
4	1, 3	eqeltrd 2852	. . 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 2758	. . . . 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 38686	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
13	8, 10, 12, 2	lshplss 36557	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (LSubSp‘𝑈))
14		eqid 2758	. . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈)
15	14, 8	lssss 19776	. . . . . . 7 ⊢ (𝑋 ∈ (LSubSp‘𝑈) → 𝑋 ⊆ (Base‘𝑈))
16	13, 15	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝑈))
17	5, 7, 14, 8, 6	dochlss 38930	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑋) ∈ (LSubSp‘𝑈))
18	11, 16, 17	syl2anc 587	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ (LSubSp‘𝑈))
19	5, 6, 7, 8, 9, 10, 11, 18	dochsatshpb 39028	. . . 4 ⊢ (𝜑 → (( ⊥ ‘𝑋) ∈ 𝐴 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) ∈ 𝑌))
20	19	adantr 484	. . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (( ⊥ ‘𝑋) ∈ 𝐴 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) ∈ 𝑌))
21	4, 20	mpbird 260	. 2 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘𝑋) ∈ 𝐴)
22		eqid 2758	. . . . . 6 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
23	12	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → 𝑈 ∈ LMod)
24		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ( ⊥ ‘𝑋) ∈ 𝐴)
25	22, 9, 23, 24	lsatn0 36575	. . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ( ⊥ ‘𝑋) ≠ {(0_g‘𝑈)})
26	25	neneqd 2956	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ¬ ( ⊥ ‘𝑋) = {(0_g‘𝑈)})
27	11	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
28	5, 7, 6, 14, 22	doch0 38934	. . . . . . 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 2769	. . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘{(0_g‘𝑈)}) ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈)))
31		eqid 2758	. . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
32	5, 31, 7, 14, 6	dochcl 38929	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
33	11, 16, 32	syl2anc 587	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
34	5, 31, 7, 22	dih0rn 38860	. . . . . . . 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 38949	. . . . . 6 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘{(0_g‘𝑈)}) ↔ ( ⊥ ‘𝑋) = {(0_g‘𝑈)}))
37	36	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘{(0_g‘𝑈)}) ↔ ( ⊥ ‘𝑋) = {(0_g‘𝑈)}))
38	30, 37	bitr3d 284	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈) ↔ ( ⊥ ‘𝑋) = {(0_g‘𝑈)}))
39	26, 38	mtbird 328	. . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ¬ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈))
40	5, 6, 7, 14, 10, 11, 2	dochshpncl 38960	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) ≠ 𝑋 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈)))
41	40	necon1bbid 2990	. . . 4 ⊢ (𝜑 → (¬ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈) ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))
42	41	adantr 484	. . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → (¬ ( ⊥ ‘( ⊥ ‘𝑋)) = (Base‘𝑈) ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))
43	39, 42	mpbid 235	. 2 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑋) ∈ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
44	21, 43	impbida 800	1 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ↔ ( ⊥ ‘𝑋) ∈ 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ⊆ wss 3858 {csn 4522 ran crn 5525 ‘cfv 6335 Basecbs 16541 0_gc0g 16771 LModclmod 19702 LSubSpclss 19771 LSAtomsclsa 36550 LSHypclsh 36551 HLchlt 36926 LHypclh 37560 DVecHcdvh 38654 DIsoHcdih 38804 ocHcoch 38923

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5156 ax-sep 5169 ax-nul 5176 ax-pow 5234 ax-pr 5298 ax-un 7459 ax-cnex 10631 ax-resscn 10632 ax-1cn 10633 ax-icn 10634 ax-addcl 10635 ax-addrcl 10636 ax-mulcl 10637 ax-mulrcl 10638 ax-mulcom 10639 ax-addass 10640 ax-mulass 10641 ax-distr 10642 ax-i2m1 10643 ax-1ne0 10644 ax-1rid 10645 ax-rnegex 10646 ax-rrecex 10647 ax-cnre 10648 ax-pre-lttri 10649 ax-pre-lttrn 10650 ax-pre-ltadd 10651 ax-pre-mulgt0 10652 ax-riotaBAD 36529

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3697 df-csb 3806 df-dif 3861 df-un 3863 df-in 3865 df-ss 3875 df-pss 3877 df-nul 4226 df-if 4421 df-pw 4496 df-sn 4523 df-pr 4525 df-tp 4527 df-op 4529 df-uni 4799 df-int 4839 df-iun 4885 df-iin 4886 df-br 5033 df-opab 5095 df-mpt 5113 df-tr 5139 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5483 df-we 5485 df-xp 5530 df-rel 5531 df-cnv 5532 df-co 5533 df-dm 5534 df-rn 5535 df-res 5536 df-ima 5537 df-pred 6126 df-ord 6172 df-on 6173 df-lim 6174 df-suc 6175 df-iota 6294 df-fun 6337 df-fn 6338 df-f 6339 df-f1 6340 df-fo 6341 df-f1o 6342 df-fv 6343 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7580 df-1st 7693 df-2nd 7694 df-tpos 7902 df-undef 7949 df-wrecs 7957 df-recs 8018 df-rdg 8056 df-1o 8112 df-er 8299 df-map 8418 df-en 8528 df-dom 8529 df-sdom 8530 df-fin 8531 df-pnf 10715 df-mnf 10716 df-xr 10717 df-ltxr 10718 df-le 10719 df-sub 10910 df-neg 10911 df-nn 11675 df-2 11737 df-3 11738 df-4 11739 df-5 11740 df-6 11741 df-n0 11935 df-z 12021 df-uz 12283 df-fz 12940 df-struct 16543 df-ndx 16544 df-slot 16545 df-base 16547 df-sets 16548 df-ress 16549 df-plusg 16636 df-mulr 16637 df-sca 16639 df-vsca 16640 df-0g 16773 df-proset 17604 df-poset 17622 df-plt 17634 df-lub 17650 df-glb 17651 df-join 17652 df-meet 17653 df-p0 17715 df-p1 17716 df-lat 17722 df-clat 17784 df-mgm 17918 df-sgrp 17967 df-mnd 17978 df-submnd 18023 df-grp 18172 df-minusg 18173 df-sbg 18174 df-subg 18343 df-cntz 18514 df-lsm 18828 df-cmn 18975 df-abl 18976 df-mgp 19308 df-ur 19320 df-ring 19367 df-oppr 19444 df-dvdsr 19462 df-unit 19463 df-invr 19493 df-dvr 19504 df-drng 19572 df-lmod 19704 df-lss 19772 df-lsp 19812 df-lvec 19943 df-lsatoms 36552 df-lshyp 36553 df-oposet 36752 df-ol 36754 df-oml 36755 df-covers 36842 df-ats 36843 df-atl 36874 df-cvlat 36898 df-hlat 36927 df-llines 37074 df-lplanes 37075 df-lvols 37076 df-lines 37077 df-psubsp 37079 df-pmap 37080 df-padd 37372 df-lhyp 37564 df-laut 37565 df-ldil 37680 df-ltrn 37681 df-trl 37735 df-tgrp 38319 df-tendo 38331 df-edring 38333 df-dveca 38579 df-disoa 38605 df-dvech 38655 df-dib 38715 df-dic 38749 df-dih 38805 df-doch 38924 df-djh 38971

This theorem is referenced by: mapdordlem2 39213

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