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Theorem docaclN 41107
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 2735 . . 3 (join‘𝐾) = (join‘𝐾)
2 eqid 2735 . . 3 (meet‘𝐾) = (meet‘𝐾)
3 eqid 2735 . . 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 41106 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ( 𝑋) = (𝐼‘((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)))
94, 6diaf11N 41032 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐼:dom 𝐼1-1-onto→ran 𝐼)
10 f1ofun 6851 . . . . 5 (𝐼:dom 𝐼1-1-onto→ran 𝐼 → Fun 𝐼)
119, 10syl 17 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → Fun 𝐼)
1211adantr 480 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → Fun 𝐼)
13 hllat 39345 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
1413ad2antrr 726 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝐾 ∈ Lat)
15 hlop 39344 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ OP)
1615ad2antrr 726 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝐾 ∈ OP)
17 simpl 482 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝐾 ∈ HL ∧ 𝑊𝐻))
18 ssrab2 4090 . . . . . . . . . . 11 {𝑧 ∈ ran 𝐼𝑋𝑧} ⊆ ran 𝐼
1918a1i 11 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → {𝑧 ∈ ran 𝐼𝑋𝑧} ⊆ ran 𝐼)
204, 5, 6dia1elN 41037 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑇 ∈ ran 𝐼)
2120anim1i 615 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝑇 ∈ ran 𝐼𝑋𝑇))
22 sseq2 4022 . . . . . . . . . . . . 13 (𝑧 = 𝑇 → (𝑋𝑧𝑋𝑇))
2322elrab 3695 . . . . . . . . . . . 12 (𝑇 ∈ {𝑧 ∈ ran 𝐼𝑋𝑧} ↔ (𝑇 ∈ ran 𝐼𝑋𝑇))
2421, 23sylibr 234 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝑇 ∈ {𝑧 ∈ ran 𝐼𝑋𝑧})
2524ne0d 4348 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → {𝑧 ∈ ran 𝐼𝑋𝑧} ≠ ∅)
264, 6diaintclN 41041 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ({𝑧 ∈ ran 𝐼𝑋𝑧} ⊆ ran 𝐼 ∧ {𝑧 ∈ ran 𝐼𝑋𝑧} ≠ ∅)) → {𝑧 ∈ ran 𝐼𝑋𝑧} ∈ ran 𝐼)
2717, 19, 25, 26syl12anc 837 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → {𝑧 ∈ ran 𝐼𝑋𝑧} ∈ ran 𝐼)
284, 6diacnvclN 41034 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ {𝑧 ∈ ran 𝐼𝑋𝑧} ∈ ran 𝐼) → (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}) ∈ dom 𝐼)
2927, 28syldan 591 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}) ∈ dom 𝐼)
30 eqid 2735 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
3130, 4, 6diadmclN 41020 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}) ∈ dom 𝐼) → (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}) ∈ (Base‘𝐾))
3229, 31syldan 591 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}) ∈ (Base‘𝐾))
3330, 3opoccl 39176 . . . . . . 7 ((𝐾 ∈ OP ∧ (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ∈ (Base‘𝐾))
3416, 32, 33syl2anc 584 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ∈ (Base‘𝐾))
3530, 4lhpbase 39981 . . . . . . . 8 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
3635ad2antlr 727 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝑊 ∈ (Base‘𝐾))
3730, 3opoccl 39176 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
3816, 36, 37syl2anc 584 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
3930, 1latjcl 18497 . . . . . 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 1370 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
4130, 2latmcl 18498 . . . . 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 1370 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
43 eqid 2735 . . . . . 6 (le‘𝐾) = (le‘𝐾)
4430, 43, 2latmle2 18523 . . . . 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 1370 . . . 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 41019 . . . . 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 480 . . . 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 713 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼)
49 fvelrn 7096 . . 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 584 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝐼‘((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∈ ran 𝐼)
518, 50eqeltrd 2839 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ( 𝑋) ∈ ran 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  {crab 3433  wss 3963  c0 4339   cint 4951   class class class wbr 5148  ccnv 5688  dom cdm 5689  ran crn 5690  Fun wfun 6557  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  occoc 17306  joincjn 18369  meetcmee 18370  Latclat 18489  OPcops 39154  HLchlt 39332  LHypclh 39967  LTrncltrn 40084  DIsoAcdia 41011  ocAcocaN 41102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-riotaBAD 38935
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-undef 8297  df-map 8867  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-p1 18484  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-llines 39481  df-lplanes 39482  df-lvols 39483  df-lines 39484  df-psubsp 39486  df-pmap 39487  df-padd 39779  df-lhyp 39971  df-laut 39972  df-ldil 40087  df-ltrn 40088  df-trl 40142  df-disoa 41012  df-docaN 41103
This theorem is referenced by:  dvadiaN  41111  djaclN  41119
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