dochpolN - Mathbox for Norm Megill

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Theorem dochpolN 39241

Description: The subspace orthocomplement for the DVecH vector space is a polarity. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
dochpol.h	⊢ 𝐻 = (LHyp‘𝐾)
dochpol.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
dochpol.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dochpol.p	⊢ 𝑃 = (LPol‘𝑈)
dochpol.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))

**Assertion**
Ref	Expression
dochpolN	⊢ (𝜑 → ⊥ ∈ 𝑃)

Proof of Theorem dochpolN Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. 2 ⊢ (Base‘𝑈) = (Base‘𝑈)
2		eqid 2737	. 2 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
3		eqid 2737	. 2 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
4		eqid 2737	. 2 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
5		eqid 2737	. 2 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈)
6		dochpol.p	. 2 ⊢ 𝑃 = (LPol‘𝑈)
7		dochpol.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
8	7	fvexi 6731	. . 3 ⊢ 𝑈 ∈ V
9	8	a1i 11	. 2 ⊢ (𝜑 → 𝑈 ∈ V)
10		dochpol.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
11		eqid 2737	. . . 4 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
12		dochpol.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
13		dochpol.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
14	10, 11, 7, 1, 12, 13	dochfN 39107	. . 3 ⊢ (𝜑 → ⊥ :𝒫 (Base‘𝑈)⟶ran ((DIsoH‘𝐾)‘𝑊))
15	10, 7, 11, 2	dihsslss 39027	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ran ((DIsoH‘𝐾)‘𝑊) ⊆ (LSubSp‘𝑈))
16	13, 15	syl 17	. . 3 ⊢ (𝜑 → ran ((DIsoH‘𝐾)‘𝑊) ⊆ (LSubSp‘𝑈))
17	14, 16	fssd 6563	. 2 ⊢ (𝜑 → ⊥ :𝒫 (Base‘𝑈)⟶(LSubSp‘𝑈))
18	10, 7, 12, 1, 3	doch1 39110	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘(Base‘𝑈)) = {(0_g‘𝑈)})
19	13, 18	syl 17	. 2 ⊢ (𝜑 → ( ⊥ ‘(Base‘𝑈)) = {(0_g‘𝑈)})
20	13	adantr 484	. . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ (Base‘𝑈) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
21		simpr2 1197	. . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ (Base‘𝑈) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦)) → 𝑦 ⊆ (Base‘𝑈))
22		simpr3 1198	. . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ (Base‘𝑈) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦)) → 𝑥 ⊆ 𝑦)
23	10, 7, 1, 12	dochss 39116	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥))
24	20, 21, 22, 23	syl3anc 1373	. 2 ⊢ ((𝜑 ∧ (𝑥 ⊆ (Base‘𝑈) ∧ 𝑦 ⊆ (Base‘𝑈) ∧ 𝑥 ⊆ 𝑦)) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥))
25	13	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
26		simpr 488	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → 𝑥 ∈ (LSAtoms‘𝑈))
27	10, 7, 12, 4, 5, 25, 26	dochsatshp 39202	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → ( ⊥ ‘𝑥) ∈ (LSHyp‘𝑈))
28	10, 7, 11, 4	dih1dimat 39081	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊))
29	25, 26, 28	syl2anc 587	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊))
30	10, 11, 12	dochoc 39118	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)
31	25, 29, 30	syl2anc 587	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (LSAtoms‘𝑈)) → ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)
32	1, 2, 3, 4, 5, 6, 9, 17, 19, 24, 27, 31	islpoldN 39235	1 ⊢ (𝜑 → ⊥ ∈ 𝑃)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 Vcvv 3408 ⊆ wss 3866 𝒫 cpw 4513 {csn 4541 ran crn 5552 ‘cfv 6380 Basecbs 16760 0_gc0g 16944 LSubSpclss 19968 LSAtomsclsa 36725 LSHypclsh 36726 HLchlt 37101 LHypclh 37735 DVecHcdvh 38829 DIsoHcdih 38979 ocHcoch 39098 LPolclpoN 39231

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-riotaBAD 36704

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-tpos 7968 df-undef 8015 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-n0 12091 df-z 12177 df-uz 12439 df-fz 13096 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-sca 16818 df-vsca 16819 df-0g 16946 df-proset 17802 df-poset 17820 df-plt 17836 df-lub 17852 df-glb 17853 df-join 17854 df-meet 17855 df-p0 17931 df-p1 17932 df-lat 17938 df-clat 18005 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-grp 18368 df-minusg 18369 df-sbg 18370 df-subg 18540 df-cntz 18711 df-lsm 19025 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-ring 19564 df-oppr 19641 df-dvdsr 19659 df-unit 19660 df-invr 19690 df-dvr 19701 df-drng 19769 df-lmod 19901 df-lss 19969 df-lsp 20009 df-lvec 20140 df-lsatoms 36727 df-lshyp 36728 df-oposet 36927 df-ol 36929 df-oml 36930 df-covers 37017 df-ats 37018 df-atl 37049 df-cvlat 37073 df-hlat 37102 df-llines 37249 df-lplanes 37250 df-lvols 37251 df-lines 37252 df-psubsp 37254 df-pmap 37255 df-padd 37547 df-lhyp 37739 df-laut 37740 df-ldil 37855 df-ltrn 37856 df-trl 37910 df-tgrp 38494 df-tendo 38506 df-edring 38508 df-dveca 38754 df-disoa 38780 df-dvech 38830 df-dib 38890 df-dic 38924 df-dih 38980 df-doch 39099 df-djh 39146 df-lpolN 39232

This theorem is referenced by: (None)

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