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Theorem docaclN 40507
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 2726 . . 3 (joinβ€˜πΎ) = (joinβ€˜πΎ)
2 eqid 2726 . . 3 (meetβ€˜πΎ) = (meetβ€˜πΎ)
3 eqid 2726 . . 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 40506 . 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 40432 . . . . 5 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐼:dom 𝐼–1-1-ontoβ†’ran 𝐼)
10 f1ofun 6828 . . . . 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 480 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ Fun 𝐼)
13 hllat 38745 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
1413ad2antrr 723 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ 𝐾 ∈ Lat)
15 hlop 38744 . . . . . . . 8 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
1615ad2antrr 723 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ 𝐾 ∈ OP)
17 simpl 482 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
18 ssrab2 4072 . . . . . . . . . . 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 40437 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝑇 ∈ ran 𝐼)
2120anim1i 614 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ (𝑇 ∈ ran 𝐼 ∧ 𝑋 βŠ† 𝑇))
22 sseq2 4003 . . . . . . . . . . . . 13 (𝑧 = 𝑇 β†’ (𝑋 βŠ† 𝑧 ↔ 𝑋 βŠ† 𝑇))
2322elrab 3678 . . . . . . . . . . . 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 4330 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧} β‰  βˆ…)
264, 6diaintclN 40441 . . . . . . . . . 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 834 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧} ∈ ran 𝐼)
284, 6diacnvclN 40434 . . . . . . . . 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 590 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ (β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}) ∈ dom 𝐼)
30 eqid 2726 . . . . . . . . 9 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
3130, 4, 6diadmclN 40420 . . . . . . . 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 590 . . . . . . 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 38576 . . . . . . 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 583 . . . . . 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 39381 . . . . . . . 8 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ (Baseβ€˜πΎ))
3635ad2antlr 724 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ π‘Š ∈ (Baseβ€˜πΎ))
3730, 3opoccl 38576 . . . . . . 7 ((𝐾 ∈ OP ∧ π‘Š ∈ (Baseβ€˜πΎ)) β†’ ((ocβ€˜πΎ)β€˜π‘Š) ∈ (Baseβ€˜πΎ))
3816, 36, 37syl2anc 583 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ ((ocβ€˜πΎ)β€˜π‘Š) ∈ (Baseβ€˜πΎ))
3930, 1latjcl 18401 . . . . . 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 1368 . . . . 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 18402 . . . . 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 1368 . . . 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 2726 . . . . . 6 (leβ€˜πΎ) = (leβ€˜πΎ)
4430, 43, 2latmle2 18427 . . . . 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 1368 . . . 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 40419 . . . . 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 480 . . . 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 710 . . 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 7071 . . 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 583 . 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 2827 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 395   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  {crab 3426   βŠ† wss 3943  βˆ…c0 4317  βˆ© cint 4943   class class class wbr 5141  β—‘ccnv 5668  dom cdm 5669  ran crn 5670  Fun wfun 6530  β€“1-1-ontoβ†’wf1o 6535  β€˜cfv 6536  (class class class)co 7404  Basecbs 17150  lecple 17210  occoc 17211  joincjn 18273  meetcmee 18274  Latclat 18393  OPcops 38554  HLchlt 38732  LHypclh 39367  LTrncltrn 39484  DIsoAcdia 40411  ocAcocaN 40502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-riotaBAD 38335
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-undef 8256  df-map 8821  df-proset 18257  df-poset 18275  df-plt 18292  df-lub 18308  df-glb 18309  df-join 18310  df-meet 18311  df-p0 18387  df-p1 18388  df-lat 18394  df-clat 18461  df-oposet 38558  df-ol 38560  df-oml 38561  df-covers 38648  df-ats 38649  df-atl 38680  df-cvlat 38704  df-hlat 38733  df-llines 38881  df-lplanes 38882  df-lvols 38883  df-lines 38884  df-psubsp 38886  df-pmap 38887  df-padd 39179  df-lhyp 39371  df-laut 39372  df-ldil 39487  df-ltrn 39488  df-trl 39542  df-disoa 40412  df-docaN 40503
This theorem is referenced by:  dvadiaN  40511  djaclN  40519
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