Theorem docaclN 42447

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 2763	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
2		eqid 2763	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
3		eqid 2763	. . 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 42446	. 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 42372	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼:dom 𝐼⟶1-1ontoran 𝐼)
10		f1ofun 6856	. . . . 5 ⊢ (𝐼:dom 𝐼⟶1-1ontoran 𝐼 → Fun 𝐼)
11	9, 10	syl 18	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Fun 𝐼)
12	11	adantr 486	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → Fun 𝐼)
13		hllat 40686	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
14	13	ad2antrr 739	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → 𝐾 ∈ Lat)
15		hlop 40685	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
16	15	ad2antrr 739	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → 𝐾 ∈ OP)
17		simpl 488	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
18		ssrab2 4008	. . . . . . . . . . 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 42377	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑇 ∈ ran 𝐼)
21	20	anim1i 627	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → (𝑇 ∈ ran 𝐼 ∧ 𝑋 ⊆ 𝑇))
22		sseq2 3939	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑇 → (𝑋 ⊆ 𝑧 ↔ 𝑋 ⊆ 𝑇))
23	22	elrab 3648	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ↔ (𝑇 ∈ ran 𝐼 ∧ 𝑋 ⊆ 𝑇))
24	21, 23	sylibr 237	. . . . . . . . . . 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 42381	. . . . . . . . . 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 850	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ∈ ran 𝐼)
28	4, 6	diacnvclN 42374	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ∈ ran 𝐼) → (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) ∈ dom 𝐼)
29	27, 28	syldan 603	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) ∈ dom 𝐼)
30		eqid 2763	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
31	30, 4, 6	diadmclN 42360	. . . . . . . 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 603	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) ∈ (Base‘𝐾))
33	30, 3	opoccl 40517	. . . . . . 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 596	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → ((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) ∈ (Base‘𝐾))
35	30, 4	lhpbase 41321	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
36	35	ad2antlr 740	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → 𝑊 ∈ (Base‘𝐾))
37	30, 3	opoccl 40517	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
38	16, 36, 37	syl2anc 596	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
39	30, 1	latjcl 18663	. . . . . 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 1399	. . . . 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 18664	. . . . 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 1399	. . . 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 2763	. . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾)
44	30, 43, 2	latmle2 18689	. . . . 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 1399	. . . 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 42359	. . . . 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 486	. . . 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 726	. . 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 7108	. . 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 596	. 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 2864	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → ( ⊥ ‘𝑋) ∈ ran 𝐼)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  {crab 3416   ⊆ wss 3900  ∅c0 4281  ∩ cint 4912   class class class wbr 5108  ^◡ccnv 5657  dom cdm 5658  ran crn 5659  Fun wfun 6560  ⟶1-1ontowf1o 6565  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  lecple 17479  occoc 17480  joincjn 18535  meetcmee 18536  Latclat 18655  OPcops 40495  HLchlt 40673  LHypclh 41307  LTrncltrn 41424  DIsoAcdia 42351  ocAcocaN 42442

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-riotaBAD 40276

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-undef 8328  df-map 8887  df-proset 18518  df-poset 18537  df-plt 18552  df-lub 18568  df-glb 18569  df-join 18570  df-meet 18571  df-p0 18647  df-p1 18648  df-lat 18656  df-clat 18723  df-oposet 40499  df-ol 40501  df-oml 40502  df-covers 40589  df-ats 40590  df-atl 40621  df-cvlat 40645  df-hlat 40674  df-llines 40821  df-lplanes 40822  df-lvols 40823  df-lines 40824  df-psubsp 40826  df-pmap 40827  df-padd 41119  df-lhyp 41311  df-laut 41312  df-ldil 41427  df-ltrn 41428  df-trl 41482  df-disoa 42352  df-docaN 42443

This theorem is used by:  dvadiaN  42451  djaclN  42459