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Theorem docaclN 38366
 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 ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ( 𝑋) ∈ ran 𝐼)

Proof of Theorem docaclN
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eqid 2824 . . 3 (join‘𝐾) = (join‘𝐾)
2 eqid 2824 . . 3 (meet‘𝐾) = (meet‘𝐾)
3 eqid 2824 . . 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)
81, 2, 3, 4, 5, 6, 7docavalN 38365 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ( 𝑋) = (𝐼‘((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)))
94, 6diaf11N 38291 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐼:dom 𝐼1-1-onto→ran 𝐼)
10 f1ofun 6609 . . . . 5 (𝐼:dom 𝐼1-1-onto→ran 𝐼 → Fun 𝐼)
119, 10syl 17 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → Fun 𝐼)
1211adantr 484 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → Fun 𝐼)
13 hllat 36605 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
1413ad2antrr 725 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝐾 ∈ Lat)
15 hlop 36604 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ OP)
1615ad2antrr 725 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝐾 ∈ OP)
17 simpl 486 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝐾 ∈ HL ∧ 𝑊𝐻))
18 ssrab2 4043 . . . . . . . . . . 11 {𝑧 ∈ ran 𝐼𝑋𝑧} ⊆ ran 𝐼
1918a1i 11 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → {𝑧 ∈ ran 𝐼𝑋𝑧} ⊆ ran 𝐼)
204, 5, 6dia1elN 38296 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑇 ∈ ran 𝐼)
2120anim1i 617 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝑇 ∈ ran 𝐼𝑋𝑇))
22 sseq2 3980 . . . . . . . . . . . . 13 (𝑧 = 𝑇 → (𝑋𝑧𝑋𝑇))
2322elrab 3667 . . . . . . . . . . . 12 (𝑇 ∈ {𝑧 ∈ ran 𝐼𝑋𝑧} ↔ (𝑇 ∈ ran 𝐼𝑋𝑇))
2421, 23sylibr 237 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝑇 ∈ {𝑧 ∈ ran 𝐼𝑋𝑧})
2524ne0d 4285 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → {𝑧 ∈ ran 𝐼𝑋𝑧} ≠ ∅)
264, 6diaintclN 38300 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ({𝑧 ∈ ran 𝐼𝑋𝑧} ⊆ ran 𝐼 ∧ {𝑧 ∈ ran 𝐼𝑋𝑧} ≠ ∅)) → {𝑧 ∈ ran 𝐼𝑋𝑧} ∈ ran 𝐼)
2717, 19, 25, 26syl12anc 835 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → {𝑧 ∈ ran 𝐼𝑋𝑧} ∈ ran 𝐼)
284, 6diacnvclN 38293 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ {𝑧 ∈ ran 𝐼𝑋𝑧} ∈ ran 𝐼) → (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}) ∈ dom 𝐼)
2927, 28syldan 594 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}) ∈ dom 𝐼)
30 eqid 2824 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
3130, 4, 6diadmclN 38279 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}) ∈ dom 𝐼) → (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}) ∈ (Base‘𝐾))
3229, 31syldan 594 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}) ∈ (Base‘𝐾))
3330, 3opoccl 36436 . . . . . . 7 ((𝐾 ∈ OP ∧ (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ∈ (Base‘𝐾))
3416, 32, 33syl2anc 587 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ∈ (Base‘𝐾))
3530, 4lhpbase 37240 . . . . . . . 8 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
3635ad2antlr 726 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝑊 ∈ (Base‘𝐾))
3730, 3opoccl 36436 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
3816, 36, 37syl2anc 587 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
3930, 1latjcl 17664 . . . . . 6 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ∈ (Base‘𝐾) ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾)) → (((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
4014, 34, 38, 39syl3anc 1368 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
4130, 2latmcl 17665 . . . . 5 ((𝐾 ∈ Lat ∧ (((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
4214, 40, 36, 41syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
43 eqid 2824 . . . . . 6 (le‘𝐾) = (le‘𝐾)
4430, 43, 2latmle2 17690 . . . . 5 ((𝐾 ∈ Lat ∧ (((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)
4514, 40, 36, 44syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)
4630, 43, 4, 6diaeldm 38278 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼 ↔ (((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)))
4746adantr 484 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼 ↔ (((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)))
4842, 45, 47mpbir2and 712 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼)
49 fvelrn 6836 . . 3 ((Fun 𝐼 ∧ ((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼) → (𝐼‘((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∈ ran 𝐼)
5012, 48, 49syl2anc 587 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝐼‘((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∈ ran 𝐼)
518, 50eqeltrd 2916 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ( 𝑋) ∈ ran 𝐼)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115   ≠ wne 3014  {crab 3137   ⊆ wss 3920  ∅c0 4277  ∩ cint 4863   class class class wbr 5053  ◡ccnv 5542  dom cdm 5543  ran crn 5544  Fun wfun 6338  –1-1-onto→wf1o 6343  ‘cfv 6344  (class class class)co 7150  Basecbs 16486  lecple 16575  occoc 16576  joincjn 17557  meetcmee 17558  Latclat 17658  OPcops 36414  HLchlt 36592  LHypclh 37226  LTrncltrn 37343  DIsoAcdia 38270  ocAcocaN 38361 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456  ax-riotaBAD 36195 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-int 4864  df-iun 4908  df-iin 4909  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7685  df-2nd 7686  df-undef 7936  df-map 8405  df-proset 17541  df-poset 17559  df-plt 17571  df-lub 17587  df-glb 17588  df-join 17589  df-meet 17590  df-p0 17652  df-p1 17653  df-lat 17659  df-clat 17721  df-oposet 36418  df-ol 36420  df-oml 36421  df-covers 36508  df-ats 36509  df-atl 36540  df-cvlat 36564  df-hlat 36593  df-llines 36740  df-lplanes 36741  df-lvols 36742  df-lines 36743  df-psubsp 36745  df-pmap 36746  df-padd 37038  df-lhyp 37230  df-laut 37231  df-ldil 37346  df-ltrn 37347  df-trl 37401  df-disoa 38271  df-docaN 38362 This theorem is referenced by:  dvadiaN  38370  djaclN  38378
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