Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  docaclN Unicode version

Theorem docaclN 31314
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  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  (  ._|_  `  X )  e.  ran  I )

Proof of Theorem docaclN
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( join `  K )  =  (
join `  K )
2 eqid 2283 . . 3  |-  ( meet `  K )  =  (
meet `  K )
3 eqid 2283 . . 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 31313 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  (  ._|_  `  X )  =  ( I `  ( ( ( ( oc `  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W ) ) )
94, 6diaf11N 31239 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I : dom  I -1-1-onto-> ran  I )
10 f1ofun 5474 . . . . 5  |-  ( I : dom  I -1-1-onto-> ran  I  ->  Fun  I )
119, 10syl 15 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Fun  I )
1211adantr 451 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  Fun  I )
13 hllat 29553 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
1413ad2antrr 706 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  K  e.  Lat )
15 hlop 29552 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OP )
1615ad2antrr 706 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  K  e.  OP )
17 simpl 443 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  ( K  e.  HL  /\  W  e.  H ) )
18 ssrab2 3258 . . . . . . . . . . 11  |-  { z  e.  ran  I  |  X  C_  z }  C_ 
ran  I
1918a1i 10 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  { z  e.  ran  I  |  X  C_  z }  C_  ran  I )
204, 5, 6dia1elN 31244 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  T  e.  ran  I
)
2120anim1i 551 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  ( T  e.  ran  I  /\  X  C_  T ) )
22 sseq2 3200 . . . . . . . . . . . . 13  |-  ( z  =  T  ->  ( X  C_  z  <->  X  C_  T
) )
2322elrab 2923 . . . . . . . . . . . 12  |-  ( T  e.  { z  e. 
ran  I  |  X  C_  z }  <->  ( T  e.  ran  I  /\  X  C_  T ) )
2421, 23sylibr 203 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  T  e.  { z  e.  ran  I  |  X  C_  z } )
25 ne0i 3461 . . . . . . . . . . 11  |-  ( T  e.  { z  e. 
ran  I  |  X  C_  z }  ->  { z  e.  ran  I  |  X  C_  z }  =/=  (/) )
2624, 25syl 15 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  { z  e.  ran  I  |  X  C_  z }  =/=  (/) )
274, 6diaintclN 31248 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( { z  e.  ran  I  |  X  C_  z }  C_ 
ran  I  /\  {
z  e.  ran  I  |  X  C_  z }  =/=  (/) ) )  ->  |^| { z  e.  ran  I  |  X  C_  z }  e.  ran  I )
2817, 19, 26, 27syl12anc 1180 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  |^| { z  e.  ran  I  |  X  C_  z }  e.  ran  I )
294, 6diacnvclN 31241 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  |^| { z  e. 
ran  I  |  X  C_  z }  e.  ran  I )  ->  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
)  e.  dom  I
)
3028, 29syldan 456 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
)  e.  dom  I
)
31 eqid 2283 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
3231, 4, 6diadmclN 31227 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( `' I `  |^| { z  e. 
ran  I  |  X  C_  z } )  e. 
dom  I )  -> 
( `' I `  |^| { z  e.  ran  I  |  X  C_  z } )  e.  (
Base `  K )
)
3330, 32syldan 456 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
)  e.  ( Base `  K ) )
3431, 3opoccl 29384 . . . . . . 7  |-  ( ( K  e.  OP  /\  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
)  e.  ( Base `  K ) )  -> 
( ( oc `  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) )  e.  (
Base `  K )
)
3516, 33, 34syl2anc 642 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  ( ( oc `  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) )  e.  (
Base `  K )
)
3631, 4lhpbase 30187 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
3736ad2antlr 707 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  W  e.  ( Base `  K )
)
3831, 3opoccl 29384 . . . . . . 7  |-  ( ( K  e.  OP  /\  W  e.  ( Base `  K ) )  -> 
( ( oc `  K ) `  W
)  e.  ( Base `  K ) )
3916, 37, 38syl2anc 642 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  ( ( oc `  K ) `  W )  e.  (
Base `  K )
)
4031, 1latjcl 14156 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) )  e.  (
Base `  K )  /\  ( ( oc `  K ) `  W
)  e.  ( Base `  K ) )  -> 
( ( ( oc
`  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ( join `  K ) ( ( oc `  K ) `
 W ) )  e.  ( Base `  K
) )
4114, 35, 39, 40syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  ( (
( oc `  K
) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ( join `  K ) ( ( oc `  K ) `
 W ) )  e.  ( Base `  K
) )
4231, 2latmcl 14157 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( ( oc
`  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ( join `  K ) ( ( oc `  K ) `
 W ) )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( (
( ( oc `  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W )  e.  ( Base `  K
) )
4314, 41, 37, 42syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  ( (
( ( oc `  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W )  e.  ( Base `  K
) )
44 eqid 2283 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
4531, 44, 2latmle2 14183 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( ( oc
`  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ( join `  K ) ( ( oc `  K ) `
 W ) )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( (
( ( oc `  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W ) ( le `  K ) W )
4614, 41, 37, 45syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  ( (
( ( oc `  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W ) ( le `  K ) W )
4731, 44, 4, 6diaeldm 31226 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( ( ( oc `  K
) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W )  e. 
dom  I  <->  ( (
( ( ( oc
`  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W )  e.  ( Base `  K
)  /\  ( (
( ( oc `  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W ) ( le `  K ) W ) ) )
4847adantr 451 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  ( (
( ( ( oc
`  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W )  e. 
dom  I  <->  ( (
( ( ( oc
`  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W )  e.  ( Base `  K
)  /\  ( (
( ( oc `  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W ) ( le `  K ) W ) ) )
4943, 46, 48mpbir2and 888 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  ( (
( ( oc `  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W )  e. 
dom  I )
50 fvelrn 5661 . . 3  |-  ( ( Fun  I  /\  (
( ( ( oc
`  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W )  e. 
dom  I )  -> 
( I `  (
( ( ( oc
`  K ) `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
) ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W ) )  e.  ran  I )
5112, 49, 50syl2anc 642 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  ( I `  ( ( ( ( oc `  K ) `
 ( `' I `  |^| { z  e. 
ran  I  |  X  C_  z } ) ) ( join `  K
) ( ( oc
`  K ) `  W ) ) (
meet `  K ) W ) )  e. 
ran  I )
528, 51eqeltrd 2357 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  (  ._|_  `  X )  e.  ran  I )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547    C_ wss 3152   (/)c0 3455   |^|cint 3862   class class class wbr 4023   `'ccnv 4688   dom cdm 4689   ran crn 4690   Fun wfun 5249   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   occoc 13216   joincjn 14078   meetcmee 14079   Latclat 14151   OPcops 29362   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   DIsoAcdia 31218   ocAcocaN 31309
This theorem is referenced by:  dvadiaN  31318  djaclN  31326
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-disoa 31219  df-docaN 31310
  Copyright terms: Public domain W3C validator