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Theorem docaclN 40629
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 2728 . . 3 (joinβ€˜πΎ) = (joinβ€˜πΎ)
2 eqid 2728 . . 3 (meetβ€˜πΎ) = (meetβ€˜πΎ)
3 eqid 2728 . . 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 40628 . 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 40554 . . . . 5 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐼:dom 𝐼–1-1-ontoβ†’ran 𝐼)
10 f1ofun 6846 . . . . 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 479 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ Fun 𝐼)
13 hllat 38867 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
1413ad2antrr 724 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ 𝐾 ∈ Lat)
15 hlop 38866 . . . . . . . 8 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
1615ad2antrr 724 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ 𝐾 ∈ OP)
17 simpl 481 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
18 ssrab2 4077 . . . . . . . . . . 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 40559 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝑇 ∈ ran 𝐼)
2120anim1i 613 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ (𝑇 ∈ ran 𝐼 ∧ 𝑋 βŠ† 𝑇))
22 sseq2 4008 . . . . . . . . . . . . 13 (𝑧 = 𝑇 β†’ (𝑋 βŠ† 𝑧 ↔ 𝑋 βŠ† 𝑇))
2322elrab 3684 . . . . . . . . . . . 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 4339 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧} β‰  βˆ…)
264, 6diaintclN 40563 . . . . . . . . . 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 40556 . . . . . . . . 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 589 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ (β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}) ∈ dom 𝐼)
30 eqid 2728 . . . . . . . . 9 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
3130, 4, 6diadmclN 40542 . . . . . . . 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 589 . . . . . . 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 38698 . . . . . . 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 582 . . . . . 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 39503 . . . . . . . 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 38698 . . . . . . 7 ((𝐾 ∈ OP ∧ π‘Š ∈ (Baseβ€˜πΎ)) β†’ ((ocβ€˜πΎ)β€˜π‘Š) ∈ (Baseβ€˜πΎ))
3816, 36, 37syl2anc 582 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ ((ocβ€˜πΎ)β€˜π‘Š) ∈ (Baseβ€˜πΎ))
3930, 1latjcl 18438 . . . . . 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 18439 . . . . 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 2728 . . . . . 6 (leβ€˜πΎ) = (leβ€˜πΎ)
4430, 43, 2latmle2 18464 . . . . 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 40541 . . . . 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 479 . . . 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 7091 . . 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 582 . 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 2829 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 394   = wceq 1533   ∈ wcel 2098   β‰  wne 2937  {crab 3430   βŠ† wss 3949  βˆ…c0 4326  βˆ© cint 4953   class class class wbr 5152  β—‘ccnv 5681  dom cdm 5682  ran crn 5683  Fun wfun 6547  β€“1-1-ontoβ†’wf1o 6552  β€˜cfv 6553  (class class class)co 7426  Basecbs 17187  lecple 17247  occoc 17248  joincjn 18310  meetcmee 18311  Latclat 18430  OPcops 38676  HLchlt 38854  LHypclh 39489  LTrncltrn 39606  DIsoAcdia 40533  ocAcocaN 40624
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-riotaBAD 38457
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-undef 8285  df-map 8853  df-proset 18294  df-poset 18312  df-plt 18329  df-lub 18345  df-glb 18346  df-join 18347  df-meet 18348  df-p0 18424  df-p1 18425  df-lat 18431  df-clat 18498  df-oposet 38680  df-ol 38682  df-oml 38683  df-covers 38770  df-ats 38771  df-atl 38802  df-cvlat 38826  df-hlat 38855  df-llines 39003  df-lplanes 39004  df-lvols 39005  df-lines 39006  df-psubsp 39008  df-pmap 39009  df-padd 39301  df-lhyp 39493  df-laut 39494  df-ldil 39609  df-ltrn 39610  df-trl 39664  df-disoa 40534  df-docaN 40625
This theorem is referenced by:  dvadiaN  40633  djaclN  40641
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