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Theorem docaclN 35893
 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 2621 . . 3 (join‘𝐾) = (join‘𝐾)
2 eqid 2621 . . 3 (meet‘𝐾) = (meet‘𝐾)
3 eqid 2621 . . 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 35892 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ( 𝑋) = (𝐼‘((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)))
94, 6diaf11N 35818 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐼:dom 𝐼1-1-onto→ran 𝐼)
10 f1ofun 6096 . . . . 5 (𝐼:dom 𝐼1-1-onto→ran 𝐼 → Fun 𝐼)
119, 10syl 17 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → Fun 𝐼)
1211adantr 481 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → Fun 𝐼)
13 hllat 34130 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
1413ad2antrr 761 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝐾 ∈ Lat)
15 hlop 34129 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ OP)
1615ad2antrr 761 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝐾 ∈ OP)
17 simpl 473 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝐾 ∈ HL ∧ 𝑊𝐻))
18 ssrab2 3666 . . . . . . . . . . 11 {𝑧 ∈ ran 𝐼𝑋𝑧} ⊆ ran 𝐼
1918a1i 11 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → {𝑧 ∈ ran 𝐼𝑋𝑧} ⊆ ran 𝐼)
204, 5, 6dia1elN 35823 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑇 ∈ ran 𝐼)
2120anim1i 591 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝑇 ∈ ran 𝐼𝑋𝑇))
22 sseq2 3606 . . . . . . . . . . . . 13 (𝑧 = 𝑇 → (𝑋𝑧𝑋𝑇))
2322elrab 3346 . . . . . . . . . . . 12 (𝑇 ∈ {𝑧 ∈ ran 𝐼𝑋𝑧} ↔ (𝑇 ∈ ran 𝐼𝑋𝑇))
2421, 23sylibr 224 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝑇 ∈ {𝑧 ∈ ran 𝐼𝑋𝑧})
25 ne0i 3897 . . . . . . . . . . 11 (𝑇 ∈ {𝑧 ∈ ran 𝐼𝑋𝑧} → {𝑧 ∈ ran 𝐼𝑋𝑧} ≠ ∅)
2624, 25syl 17 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → {𝑧 ∈ ran 𝐼𝑋𝑧} ≠ ∅)
274, 6diaintclN 35827 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ({𝑧 ∈ ran 𝐼𝑋𝑧} ⊆ ran 𝐼 ∧ {𝑧 ∈ ran 𝐼𝑋𝑧} ≠ ∅)) → {𝑧 ∈ ran 𝐼𝑋𝑧} ∈ ran 𝐼)
2817, 19, 26, 27syl12anc 1321 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → {𝑧 ∈ ran 𝐼𝑋𝑧} ∈ ran 𝐼)
294, 6diacnvclN 35820 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ {𝑧 ∈ ran 𝐼𝑋𝑧} ∈ ran 𝐼) → (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}) ∈ dom 𝐼)
3028, 29syldan 487 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}) ∈ dom 𝐼)
31 eqid 2621 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
3231, 4, 6diadmclN 35806 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}) ∈ dom 𝐼) → (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}) ∈ (Base‘𝐾))
3330, 32syldan 487 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}) ∈ (Base‘𝐾))
3431, 3opoccl 33961 . . . . . . 7 ((𝐾 ∈ OP ∧ (𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ∈ (Base‘𝐾))
3516, 33, 34syl2anc 692 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ∈ (Base‘𝐾))
3631, 4lhpbase 34764 . . . . . . . 8 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
3736ad2antlr 762 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → 𝑊 ∈ (Base‘𝐾))
3831, 3opoccl 33961 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
3916, 37, 38syl2anc 692 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
4031, 1latjcl 16972 . . . . . 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))
4114, 35, 39, 40syl3anc 1323 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
4231, 2latmcl 16973 . . . . 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))
4314, 41, 37, 42syl3anc 1323 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
44 eqid 2621 . . . . . 6 (le‘𝐾) = (le‘𝐾)
4531, 44, 2latmle2 16998 . . . . 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)
4614, 41, 37, 45syl3anc 1323 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)
4731, 44, 4, 6diaeldm 35805 . . . . 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)))
4847adantr 481 . . . 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)))
4943, 46, 48mpbir2and 956 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼)
50 fvelrn 6308 . . 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)
5112, 49, 50syl2anc 692 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝐼‘((((oc‘𝐾)‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∈ ran 𝐼)
528, 51eqeltrd 2698 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ( 𝑋) ∈ ran 𝐼)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  {crab 2911   ⊆ wss 3555  ∅c0 3891  ∩ cint 4440   class class class wbr 4613  ◡ccnv 5073  dom cdm 5074  ran crn 5075  Fun wfun 5841  –1-1-onto→wf1o 5846  ‘cfv 5847  (class class class)co 6604  Basecbs 15781  lecple 15869  occoc 15870  joincjn 16865  meetcmee 16866  Latclat 16966  OPcops 33939  HLchlt 34117  LHypclh 34750  LTrncltrn 34867  DIsoAcdia 35797  ocAcocaN 35888 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-riotaBAD 33719 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-undef 7344  df-map 7804  df-preset 16849  df-poset 16867  df-plt 16879  df-lub 16895  df-glb 16896  df-join 16897  df-meet 16898  df-p0 16960  df-p1 16961  df-lat 16967  df-clat 17029  df-oposet 33943  df-ol 33945  df-oml 33946  df-covers 34033  df-ats 34034  df-atl 34065  df-cvlat 34089  df-hlat 34118  df-llines 34264  df-lplanes 34265  df-lvols 34266  df-lines 34267  df-psubsp 34269  df-pmap 34270  df-padd 34562  df-lhyp 34754  df-laut 34755  df-ldil 34870  df-ltrn 34871  df-trl 34926  df-disoa 35798  df-docaN 35889 This theorem is referenced by:  dvadiaN  35897  djaclN  35905
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