Theorem docaclN 39138

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 2738	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
2		eqid 2738	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
3		eqid 2738	. . 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 39137	. 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 39063	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼:dom 𝐼–1-1-onto→ran 𝐼)
10		f1ofun 6718	. . . . 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 481	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → Fun 𝐼)
13		hllat 37377	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
14	13	ad2antrr 723	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → 𝐾 ∈ Lat)
15		hlop 37376	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
16	15	ad2antrr 723	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → 𝐾 ∈ OP)
17		simpl 483	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
18		ssrab2 4013	. . . . . . . . . . 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 39068	. . . . . . . . . . . . 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 3947	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑇 → (𝑋 ⊆ 𝑧 ↔ 𝑋 ⊆ 𝑇))
23	22	elrab 3624	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ↔ (𝑇 ∈ ran 𝐼 ∧ 𝑋 ⊆ 𝑇))
24	21, 23	sylibr 233	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → 𝑇 ∈ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})
25	24	ne0d 4269	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ≠ ∅)
26	4, 6	diaintclN 39072	. . . . . . . . . 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 834	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ∈ ran 𝐼)
28	4, 6	diacnvclN 39065	. . . . . . . . 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 2738	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
31	30, 4, 6	diadmclN 39051	. . . . . . . 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 37208	. . . . . . 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 38012	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
36	35	ad2antlr 724	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → 𝑊 ∈ (Base‘𝐾))
37	30, 3	opoccl 37208	. . . . . . 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 18157	. . . . . 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 1370	. . . . 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 18158	. . . . 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 1370	. . . 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 2738	. . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾)
44	30, 43, 2	latmle2 18183	. . . . 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 1370	. . . 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 39050	. . . . 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 481	. . . 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 710	. . 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 6954	. . 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 2839	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → ( ⊥ ‘𝑋) ∈ ran 𝐼)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  {crab 3068   ⊆ wss 3887  ∅c0 4256  ∩ cint 4879   class class class wbr 5074  ^◡ccnv 5588  dom cdm 5589  ran crn 5590  Fun wfun 6427  –1-1-onto→wf1o 6432  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  lecple 16969  occoc 16970  joincjn 18029  meetcmee 18030  Latclat 18149  OPcops 37186  HLchlt 37364  LHypclh 37998  LTrncltrn 38115  DIsoAcdia 39042  ocAcocaN 39133

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-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-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-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  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-id 5489  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-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-1st 7831  df-2nd 7832  df-undef 8089  df-map 8617  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-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-disoa 39043  df-docaN 39134

This theorem is referenced by:  dvadiaN  39142  djaclN  39150