Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  docaclN Structured version   Visualization version   GIF version

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 ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ( 𝑋) ∈ ran 𝐼)

Proof of Theorem docaclN
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef 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)
81, 2, 3, 4, 5, 6, 7docavalN 41126 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ( 𝑋) = (𝐼‘((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)))
94, 6diaf11N 41052 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐼:dom 𝐼1-1-onto→ran 𝐼)
10 f1ofun 6849 . . . . 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 39365 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
1413ad2antrr 726 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝐾 ∈ Lat)
15 hlop 39364 . . . . . . . 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 4079 . . . . . . . . . . 11 {𝑧 ∈ ran 𝐼𝑋𝑧} ⊆ ran 𝐼
1918a1i 11 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → {𝑧 ∈ ran 𝐼𝑋𝑧} ⊆ ran 𝐼)
204, 5, 6dia1elN 41057 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑇 ∈ ran 𝐼)
2120anim1i 615 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝑇 ∈ ran 𝐼𝑋𝑇))
22 sseq2 4009 . . . . . . . . . . . . 13 (𝑧 = 𝑇 → (𝑋𝑧𝑋𝑇))
2322elrab 3691 . . . . . . . . . . . 12 (𝑇 ∈ {𝑧 ∈ ran 𝐼𝑋𝑧} ↔ (𝑇 ∈ ran 𝐼𝑋𝑇))
2421, 23sylibr 234 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝑇 ∈ {𝑧 ∈ ran 𝐼𝑋𝑧})
2524ne0d 4341 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → {𝑧 ∈ ran 𝐼𝑋𝑧} ≠ ∅)
264, 6diaintclN 41061 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ({𝑧 ∈ ran 𝐼𝑋𝑧} ⊆ ran 𝐼 ∧ {𝑧 ∈ ran 𝐼𝑋𝑧} ≠ ∅)) → {𝑧 ∈ ran 𝐼𝑋𝑧} ∈ ran 𝐼)
2717, 19, 25, 26syl12anc 836 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → {𝑧 ∈ ran 𝐼𝑋𝑧} ∈ ran 𝐼)
284, 6diacnvclN 41054 . . . . . . . . 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 2736 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
3130, 4, 6diadmclN 41040 . . . . . . . 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 39196 . . . . . . 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 40001 . . . . . . . 8 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
3635ad2antlr 727 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝑊 ∈ (Base‘𝐾))
3730, 3opoccl 39196 . . . . . . 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 18485 . . . . . 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 1372 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
4130, 2latmcl 18486 . . . . 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 1372 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
43 eqid 2736 . . . . . 6 (le‘𝐾) = (le‘𝐾)
4430, 43, 2latmle2 18511 . . . . 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 1372 . . . 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 41039 . . . . 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 7095 . . 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 2840 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ( 𝑋) ∈ ran 𝐼)
Colors of variables: wff setvar class
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-ontowf1o 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
  Copyright terms: Public domain W3C validator