Theorem docaclN 41246

Description: Closure of subspace orthocomplement for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
docacl.h	⊢ 𝐻 = (LHyp‘𝐾)
docacl.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
docacl.i	⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
docacl.n	⊢ ⊥ = ((ocA‘𝐾)‘𝑊)

Assertion

Ref	Expression
docaclN	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → ( ⊥ ‘𝑋) ∈ ran 𝐼)

Proof of Theorem docaclN

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2733	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
2		eqid 2733	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
3		eqid 2733	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
4		docacl.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
5		docacl.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
6		docacl.i	. . 3 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
7		docacl.n	. . 3 ⊢ ⊥ = ((ocA‘𝐾)‘𝑊)
8	1, 2, 3, 4, 5, 6, 7	docavalN 41245	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → ( ⊥ ‘𝑋) = (𝐼‘((((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)))
9	4, 6	diaf11N 41171	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼:dom 𝐼–1-1-onto→ran 𝐼)
10		f1ofun 6772	. . . . 5 ⊢ (𝐼:dom 𝐼–1-1-onto→ran 𝐼 → Fun 𝐼)
11	9, 10	syl 17	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Fun 𝐼)
12	11	adantr 480	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → Fun 𝐼)
13		hllat 39485	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
14	13	ad2antrr 726	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → 𝐾 ∈ Lat)
15		hlop 39484	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
16	15	ad2antrr 726	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → 𝐾 ∈ OP)
17		simpl 482	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
18		ssrab2 4029	. . . . . . . . . . 11 ⊢ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ⊆ ran 𝐼
19	18	a1i 11	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ⊆ ran 𝐼)
20	4, 5, 6	dia1elN 41176	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑇 ∈ ran 𝐼)
21	20	anim1i 615	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → (𝑇 ∈ ran 𝐼 ∧ 𝑋 ⊆ 𝑇))
22		sseq2 3957	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑇 → (𝑋 ⊆ 𝑧 ↔ 𝑋 ⊆ 𝑇))
23	22	elrab 3643	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ↔ (𝑇 ∈ ran 𝐼 ∧ 𝑋 ⊆ 𝑇))
24	21, 23	sylibr 234	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → 𝑇 ∈ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})
25	24	ne0d 4291	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ≠ ∅)
26	4, 6	diaintclN 41180	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ({𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ⊆ ran 𝐼 ∧ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ≠ ∅)) → ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ∈ ran 𝐼)
27	17, 19, 25, 26	syl12anc 836	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ∈ ran 𝐼)
28	4, 6	diacnvclN 41173	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ∈ ran 𝐼) → (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) ∈ dom 𝐼)
29	27, 28	syldan 591	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) ∈ dom 𝐼)
30		eqid 2733	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
31	30, 4, 6	diadmclN 41159	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) ∈ dom 𝐼) → (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) ∈ (Base‘𝐾))
32	29, 31	syldan 591	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) ∈ (Base‘𝐾))
33	30, 3	opoccl 39316	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) ∈ (Base‘𝐾))
34	16, 32, 33	syl2anc 584	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → ((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) ∈ (Base‘𝐾))
35	30, 4	lhpbase 40120	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
36	35	ad2antlr 727	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → 𝑊 ∈ (Base‘𝐾))
37	30, 3	opoccl 39316	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
38	16, 36, 37	syl2anc 584	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
39	30, 1	latjcl 18349	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) ∈ (Base‘𝐾) ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾)) → (((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
40	14, 34, 38, 39	syl3anc 1373	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → (((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
41	30, 2	latmcl 18350	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
42	14, 40, 36, 41	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → ((((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
43		eqid 2733	. . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾)
44	30, 43, 2	latmle2 18375	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)
45	14, 40, 36, 44	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → ((((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)
46	30, 43, 4, 6	diaeldm 41158	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼 ↔ (((((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)))
47	46	adantr 480	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → (((((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼 ↔ (((((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)))
48	42, 45, 47	mpbir2and 713	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → ((((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼)
49		fvelrn 7017	. . 3 ⊢ ((Fun 𝐼 ∧ ((((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼) → (𝐼‘((((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∈ ran 𝐼)
50	12, 48, 49	syl2anc 584	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → (𝐼‘((((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∈ ran 𝐼)
51	8, 50	eqeltrd 2833	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → ( ⊥ ‘𝑋) ∈ ran 𝐼)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  {crab 3396   ⊆ wss 3898  ∅c0 4282  ∩ cint 4899   class class class wbr 5095  ^◡ccnv 5620  dom cdm 5621  ran crn 5622  Fun wfun 6482  –1-1-onto→wf1o 6487  ‘cfv 6488  (class class class)co 7354  Basecbs 17124  lecple 17172  occoc 17173  joincjn 18221  meetcmee 18222  Latclat 18341  OPcops 39294  HLchlt 39472  LHypclh 40106  LTrncltrn 40223  DIsoAcdia 41150  ocAcocaN 41241

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676  ax-riotaBAD 39075

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-iin 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7311  df-ov 7357  df-oprab 7358  df-mpo 7359  df-1st 7929  df-2nd 7930  df-undef 8211  df-map 8760  df-proset 18204  df-poset 18223  df-plt 18238  df-lub 18254  df-glb 18255  df-join 18256  df-meet 18257  df-p0 18333  df-p1 18334  df-lat 18342  df-clat 18409  df-oposet 39298  df-ol 39300  df-oml 39301  df-covers 39388  df-ats 39389  df-atl 39420  df-cvlat 39444  df-hlat 39473  df-llines 39620  df-lplanes 39621  df-lvols 39622  df-lines 39623  df-psubsp 39625  df-pmap 39626  df-padd 39918  df-lhyp 40110  df-laut 40111  df-ldil 40226  df-ltrn 40227  df-trl 40281  df-disoa 41151  df-docaN 41242

This theorem is referenced by:  dvadiaN  41250  djaclN  41258