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Theorem docaclN 39983
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 = (LHypβ€˜πΎ)
docacl.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
docacl.i 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
docacl.n βŠ₯ = ((ocAβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
docaclN (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ ( βŠ₯ β€˜π‘‹) ∈ ran 𝐼)
Proof of Theorem docaclN
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . 3 (joinβ€˜πΎ) = (joinβ€˜πΎ)
2 eqid 2732 . . 3 (meetβ€˜πΎ) = (meetβ€˜πΎ)
3 eqid 2732 . . 3 (ocβ€˜πΎ) = (ocβ€˜πΎ)
4 docacl.h . . 3 𝐻 = (LHypβ€˜πΎ)
5 docacl.t . . 3 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
6 docacl.i . . 3 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
7 docacl.n . . 3 βŠ₯ = ((ocAβ€˜πΎ)β€˜π‘Š)
81, 2, 3, 4, 5, 6, 7docavalN 39982 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ ( βŠ₯ β€˜π‘‹) = (πΌβ€˜((((ocβ€˜πΎ)β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}))(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Š))(meetβ€˜πΎ)π‘Š)))
94, 6diaf11N 39908 . . . . 5 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐼:dom 𝐼–1-1-ontoβ†’ran 𝐼)
10 f1ofun 6832 . . . . 5 (𝐼:dom 𝐼–1-1-ontoβ†’ran 𝐼 β†’ Fun 𝐼)
119, 10syl 17 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ Fun 𝐼)
1211adantr 481 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ Fun 𝐼)
13 hllat 38221 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
1413ad2antrr 724 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ 𝐾 ∈ Lat)
15 hlop 38220 . . . . . . . 8 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
1615ad2antrr 724 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ 𝐾 ∈ OP)
17 simpl 483 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
18 ssrab2 4076 . . . . . . . . . . 11 {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧} βŠ† ran 𝐼
1918a1i 11 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧} βŠ† ran 𝐼)
204, 5, 6dia1elN 39913 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝑇 ∈ ran 𝐼)
2120anim1i 615 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ (𝑇 ∈ ran 𝐼 ∧ 𝑋 βŠ† 𝑇))
22 sseq2 4007 . . . . . . . . . . . . 13 (𝑧 = 𝑇 β†’ (𝑋 βŠ† 𝑧 ↔ 𝑋 βŠ† 𝑇))
2322elrab 3682 . . . . . . . . . . . 12 (𝑇 ∈ {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧} ↔ (𝑇 ∈ ran 𝐼 ∧ 𝑋 βŠ† 𝑇))
2421, 23sylibr 233 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ 𝑇 ∈ {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧})
2524ne0d 4334 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧} β‰  βˆ…)
264, 6diaintclN 39917 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ({𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧} βŠ† ran 𝐼 ∧ {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧} β‰  βˆ…)) β†’ ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧} ∈ ran 𝐼)
2717, 19, 25, 26syl12anc 835 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧} ∈ ran 𝐼)
284, 6diacnvclN 39910 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧} ∈ ran 𝐼) β†’ (β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}) ∈ dom 𝐼)
2927, 28syldan 591 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ (β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}) ∈ dom 𝐼)
30 eqid 2732 . . . . . . . . 9 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
3130, 4, 6diadmclN 39896 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}) ∈ dom 𝐼) β†’ (β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}) ∈ (Baseβ€˜πΎ))
3229, 31syldan 591 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ (β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}) ∈ (Baseβ€˜πΎ))
3330, 3opoccl 38052 . . . . . . 7 ((𝐾 ∈ OP ∧ (β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}) ∈ (Baseβ€˜πΎ)) β†’ ((ocβ€˜πΎ)β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧})) ∈ (Baseβ€˜πΎ))
3416, 32, 33syl2anc 584 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ ((ocβ€˜πΎ)β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧})) ∈ (Baseβ€˜πΎ))
3530, 4lhpbase 38857 . . . . . . . 8 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ (Baseβ€˜πΎ))
3635ad2antlr 725 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ π‘Š ∈ (Baseβ€˜πΎ))
3730, 3opoccl 38052 . . . . . . 7 ((𝐾 ∈ OP ∧ π‘Š ∈ (Baseβ€˜πΎ)) β†’ ((ocβ€˜πΎ)β€˜π‘Š) ∈ (Baseβ€˜πΎ))
3816, 36, 37syl2anc 584 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ ((ocβ€˜πΎ)β€˜π‘Š) ∈ (Baseβ€˜πΎ))
3930, 1latjcl 18388 . . . . . 6 ((𝐾 ∈ Lat ∧ ((ocβ€˜πΎ)β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧})) ∈ (Baseβ€˜πΎ) ∧ ((ocβ€˜πΎ)β€˜π‘Š) ∈ (Baseβ€˜πΎ)) β†’ (((ocβ€˜πΎ)β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}))(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Š)) ∈ (Baseβ€˜πΎ))
4014, 34, 38, 39syl3anc 1371 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ (((ocβ€˜πΎ)β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}))(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Š)) ∈ (Baseβ€˜πΎ))
4130, 2latmcl 18389 . . . . 5 ((𝐾 ∈ Lat ∧ (((ocβ€˜πΎ)β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}))(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Š)) ∈ (Baseβ€˜πΎ) ∧ π‘Š ∈ (Baseβ€˜πΎ)) β†’ ((((ocβ€˜πΎ)β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}))(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Š))(meetβ€˜πΎ)π‘Š) ∈ (Baseβ€˜πΎ))
4214, 40, 36, 41syl3anc 1371 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ ((((ocβ€˜πΎ)β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}))(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Š))(meetβ€˜πΎ)π‘Š) ∈ (Baseβ€˜πΎ))
43 eqid 2732 . . . . . 6 (leβ€˜πΎ) = (leβ€˜πΎ)
4430, 43, 2latmle2 18414 . . . . 5 ((𝐾 ∈ Lat ∧ (((ocβ€˜πΎ)β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}))(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Š)) ∈ (Baseβ€˜πΎ) ∧ π‘Š ∈ (Baseβ€˜πΎ)) β†’ ((((ocβ€˜πΎ)β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}))(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Š))(meetβ€˜πΎ)π‘Š)(leβ€˜πΎ)π‘Š)
4514, 40, 36, 44syl3anc 1371 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ ((((ocβ€˜πΎ)β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}))(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Š))(meetβ€˜πΎ)π‘Š)(leβ€˜πΎ)π‘Š)
4630, 43, 4, 6diaeldm 39895 . . . . 5 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (((((ocβ€˜πΎ)β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}))(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Š))(meetβ€˜πΎ)π‘Š) ∈ dom 𝐼 ↔ (((((ocβ€˜πΎ)β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}))(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Š))(meetβ€˜πΎ)π‘Š) ∈ (Baseβ€˜πΎ) ∧ ((((ocβ€˜πΎ)β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}))(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Š))(meetβ€˜πΎ)π‘Š)(leβ€˜πΎ)π‘Š)))
4746adantr 481 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ (((((ocβ€˜πΎ)β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}))(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Š))(meetβ€˜πΎ)π‘Š) ∈ dom 𝐼 ↔ (((((ocβ€˜πΎ)β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}))(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Š))(meetβ€˜πΎ)π‘Š) ∈ (Baseβ€˜πΎ) ∧ ((((ocβ€˜πΎ)β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}))(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Š))(meetβ€˜πΎ)π‘Š)(leβ€˜πΎ)π‘Š)))
4842, 45, 47mpbir2and 711 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ ((((ocβ€˜πΎ)β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}))(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Š))(meetβ€˜πΎ)π‘Š) ∈ dom 𝐼)
49 fvelrn 7075 . . 3 ((Fun 𝐼 ∧ ((((ocβ€˜πΎ)β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}))(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Š))(meetβ€˜πΎ)π‘Š) ∈ dom 𝐼) β†’ (πΌβ€˜((((ocβ€˜πΎ)β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}))(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Š))(meetβ€˜πΎ)π‘Š)) ∈ ran 𝐼)
5012, 48, 49syl2anc 584 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ (πΌβ€˜((((ocβ€˜πΎ)β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}))(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜π‘Š))(meetβ€˜πΎ)π‘Š)) ∈ ran 𝐼)
518, 50eqeltrd 2833 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ ( βŠ₯ β€˜π‘‹) ∈ ran 𝐼)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  {crab 3432   βŠ† wss 3947  βˆ…c0 4321  βˆ© cint 4949   class class class wbr 5147  β—‘ccnv 5674  dom cdm 5675  ran crn 5676  Fun wfun 6534  β€“1-1-ontoβ†’wf1o 6539  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  lecple 17200  occoc 17201  joincjn 18260  meetcmee 18261  Latclat 18380  OPcops 38030  HLchlt 38208  LHypclh 38843  LTrncltrn 38960  DIsoAcdia 39887  ocAcocaN 39978
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-riotaBAD 37811
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-undef 8254  df-map 8818  df-proset 18244  df-poset 18262  df-plt 18279  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p0 18374  df-p1 18375  df-lat 18381  df-clat 18448  df-oposet 38034  df-ol 38036  df-oml 38037  df-covers 38124  df-ats 38125  df-atl 38156  df-cvlat 38180  df-hlat 38209  df-llines 38357  df-lplanes 38358  df-lvols 38359  df-lines 38360  df-psubsp 38362  df-pmap 38363  df-padd 38655  df-lhyp 38847  df-laut 38848  df-ldil 38963  df-ltrn 38964  df-trl 39018  df-disoa 39888  df-docaN 39979
This theorem is referenced by:  dvadiaN  39987  djaclN  39995
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