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Theorem docaclN 37742
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 2771 . . 3 (join‘𝐾) = (join‘𝐾)
2 eqid 2771 . . 3 (meet‘𝐾) = (meet‘𝐾)
3 eqid 2771 . . 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 37741 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ( 𝑋) = (𝐼‘((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)))
94, 6diaf11N 37667 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐼:dom 𝐼1-1-onto→ran 𝐼)
10 f1ofun 6443 . . . . 5 (𝐼:dom 𝐼1-1-onto→ran 𝐼 → Fun 𝐼)
119, 10syl 17 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → Fun 𝐼)
1211adantr 473 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → Fun 𝐼)
13 hllat 35981 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
1413ad2antrr 714 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝐾 ∈ Lat)
15 hlop 35980 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ OP)
1615ad2antrr 714 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝐾 ∈ OP)
17 simpl 475 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝐾 ∈ HL ∧ 𝑊𝐻))
18 ssrab2 3939 . . . . . . . . . . 11 {𝑧 ∈ ran 𝐼𝑋𝑧} ⊆ ran 𝐼
1918a1i 11 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → {𝑧 ∈ ran 𝐼𝑋𝑧} ⊆ ran 𝐼)
204, 5, 6dia1elN 37672 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑇 ∈ ran 𝐼)
2120anim1i 606 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝑇 ∈ ran 𝐼𝑋𝑇))
22 sseq2 3876 . . . . . . . . . . . . 13 (𝑧 = 𝑇 → (𝑋𝑧𝑋𝑇))
2322elrab 3588 . . . . . . . . . . . 12 (𝑇 ∈ {𝑧 ∈ ran 𝐼𝑋𝑧} ↔ (𝑇 ∈ ran 𝐼𝑋𝑇))
2421, 23sylibr 226 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝑇 ∈ {𝑧 ∈ ran 𝐼𝑋𝑧})
2524ne0d 4181 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → {𝑧 ∈ ran 𝐼𝑋𝑧} ≠ ∅)
264, 6diaintclN 37676 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ({𝑧 ∈ ran 𝐼𝑋𝑧} ⊆ ran 𝐼 ∧ {𝑧 ∈ ran 𝐼𝑋𝑧} ≠ ∅)) → {𝑧 ∈ ran 𝐼𝑋𝑧} ∈ ran 𝐼)
2717, 19, 25, 26syl12anc 825 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → {𝑧 ∈ ran 𝐼𝑋𝑧} ∈ ran 𝐼)
284, 6diacnvclN 37669 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ {𝑧 ∈ ran 𝐼𝑋𝑧} ∈ ran 𝐼) → (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}) ∈ dom 𝐼)
2927, 28syldan 583 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}) ∈ dom 𝐼)
30 eqid 2771 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
3130, 4, 6diadmclN 37655 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}) ∈ dom 𝐼) → (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}) ∈ (Base‘𝐾))
3229, 31syldan 583 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}) ∈ (Base‘𝐾))
3330, 3opoccl 35812 . . . . . . 7 ((𝐾 ∈ OP ∧ (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ∈ (Base‘𝐾))
3416, 32, 33syl2anc 576 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ∈ (Base‘𝐾))
3530, 4lhpbase 36616 . . . . . . . 8 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
3635ad2antlr 715 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝑊 ∈ (Base‘𝐾))
3730, 3opoccl 35812 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
3816, 36, 37syl2anc 576 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
3930, 1latjcl 17531 . . . . . 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 1352 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
4130, 2latmcl 17532 . . . . 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 1352 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
43 eqid 2771 . . . . . 6 (le‘𝐾) = (le‘𝐾)
4430, 43, 2latmle2 17557 . . . . 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 1352 . . . 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 37654 . . . . 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 473 . . . 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 701 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼)
49 fvelrn 6667 . . 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 576 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝐼‘((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∈ ran 𝐼)
518, 50eqeltrd 2859 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ( 𝑋) ∈ ran 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1508  wcel 2051  wne 2960  {crab 3085  wss 3822  c0 4172   cint 4745   class class class wbr 4925  ccnv 5402  dom cdm 5403  ran crn 5404  Fun wfun 6179  1-1-ontowf1o 6184  cfv 6185  (class class class)co 6974  Basecbs 16337  lecple 16426  occoc 16427  joincjn 17424  meetcmee 17425  Latclat 17525  OPcops 35790  HLchlt 35968  LHypclh 36602  LTrncltrn 36719  DIsoAcdia 37646  ocAcocaN 37737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-riotaBAD 35571
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rmo 3089  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-iin 4791  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-1st 7499  df-2nd 7500  df-undef 7740  df-map 8206  df-proset 17408  df-poset 17426  df-plt 17438  df-lub 17454  df-glb 17455  df-join 17456  df-meet 17457  df-p0 17519  df-p1 17520  df-lat 17526  df-clat 17588  df-oposet 35794  df-ol 35796  df-oml 35797  df-covers 35884  df-ats 35885  df-atl 35916  df-cvlat 35940  df-hlat 35969  df-llines 36116  df-lplanes 36117  df-lvols 36118  df-lines 36119  df-psubsp 36121  df-pmap 36122  df-padd 36414  df-lhyp 36606  df-laut 36607  df-ldil 36722  df-ltrn 36723  df-trl 36777  df-disoa 37647  df-docaN 37738
This theorem is referenced by:  dvadiaN  37746  djaclN  37754
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