Theorem docaclN 41127

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 2736	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
2		eqid 2736	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
3		eqid 2736	. . 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 41126	. 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 41052	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼:dom 𝐼–1-1-onto→ran 𝐼)
10		f1ofun 6849	. . . . 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 39365	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
14	13	ad2antrr 726	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → 𝐾 ∈ Lat)
15		hlop 39364	. . . . . . . 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 4079	. . . . . . . . . . 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 41057	. . . . . . . . . . . . 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 4009	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑇 → (𝑋 ⊆ 𝑧 ↔ 𝑋 ⊆ 𝑇))
23	22	elrab 3691	. . . . . . . . . . . 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 4341	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ≠ ∅)
26	4, 6	diaintclN 41061	. . . . . . . . . 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 41054	. . . . . . . . 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 2736	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
31	30, 4, 6	diadmclN 41040	. . . . . . . 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 39196	. . . . . . 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 40001	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
36	35	ad2antlr 727	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → 𝑊 ∈ (Base‘𝐾))
37	30, 3	opoccl 39196	. . . . . . 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 18485	. . . . . 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 1372	. . . . 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 18486	. . . . 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 1372	. . . 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 2736	. . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾)
44	30, 43, 2	latmle2 18511	. . . . 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 1372	. . . 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 41039	. . . . 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 7095	. . 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 2840	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → ( ⊥ ‘𝑋) ∈ ran 𝐼)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ≠ wne 2939  {crab 3435   ⊆ wss 3950  ∅c0 4332  ∩ cint 4945   class class class wbr 5142  ^◡ccnv 5683  dom cdm 5684  ran crn 5685  Fun wfun 6554  –1-1-onto→wf1o 6559  ‘cfv 6560  (class class class)co 7432  Basecbs 17248  lecple 17305  occoc 17306  joincjn 18358  meetcmee 18359  Latclat 18477  OPcops 39174  HLchlt 39352  LHypclh 39987  LTrncltrn 40104  DIsoAcdia 41031  ocAcocaN 41122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-riotaBAD 38955

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-undef 8299  df-map 8869  df-proset 18341  df-poset 18360  df-plt 18376  df-lub 18392  df-glb 18393  df-join 18394  df-meet 18395  df-p0 18471  df-p1 18472  df-lat 18478  df-clat 18545  df-oposet 39178  df-ol 39180  df-oml 39181  df-covers 39268  df-ats 39269  df-atl 39300  df-cvlat 39324  df-hlat 39353  df-llines 39501  df-lplanes 39502  df-lvols 39503  df-lines 39504  df-psubsp 39506  df-pmap 39507  df-padd 39799  df-lhyp 39991  df-laut 39992  df-ldil 40107  df-ltrn 40108  df-trl 40162  df-disoa 41032  df-docaN 41123

This theorem is referenced by:  dvadiaN  41131  djaclN  41139