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Theorem docaclN 31984
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 2438 . . 3  |-  ( join `  K )  =  (
join `  K )
2 eqid 2438 . . 3  |-  ( meet `  K )  =  (
meet `  K )
3 eqid 2438 . . 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 31983 . 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 31909 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I : dom  I -1-1-onto-> ran  I )
10 f1ofun 5678 . . . . 5  |-  ( I : dom  I -1-1-onto-> ran  I  ->  Fun  I )
119, 10syl 16 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Fun  I )
1211adantr 453 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  Fun  I )
13 hllat 30223 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
1413ad2antrr 708 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  K  e.  Lat )
15 hlop 30222 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OP )
1615ad2antrr 708 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  K  e.  OP )
17 simpl 445 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  ( K  e.  HL  /\  W  e.  H ) )
18 ssrab2 3430 . . . . . . . . . . 11  |-  { z  e.  ran  I  |  X  C_  z }  C_ 
ran  I
1918a1i 11 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  { z  e.  ran  I  |  X  C_  z }  C_  ran  I )
204, 5, 6dia1elN 31914 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  T  e.  ran  I
)
2120anim1i 553 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  ( T  e.  ran  I  /\  X  C_  T ) )
22 sseq2 3372 . . . . . . . . . . . . 13  |-  ( z  =  T  ->  ( X  C_  z  <->  X  C_  T
) )
2322elrab 3094 . . . . . . . . . . . 12  |-  ( T  e.  { z  e. 
ran  I  |  X  C_  z }  <->  ( T  e.  ran  I  /\  X  C_  T ) )
2421, 23sylibr 205 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  T  e.  { z  e.  ran  I  |  X  C_  z } )
25 ne0i 3636 . . . . . . . . . . 11  |-  ( T  e.  { z  e. 
ran  I  |  X  C_  z }  ->  { z  e.  ran  I  |  X  C_  z }  =/=  (/) )
2624, 25syl 16 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  { z  e.  ran  I  |  X  C_  z }  =/=  (/) )
274, 6diaintclN 31918 . . . . . . . . . 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 1183 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  |^| { z  e.  ran  I  |  X  C_  z }  e.  ran  I )
294, 6diacnvclN 31911 . . . . . . . . 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 458 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
)  e.  dom  I
)
31 eqid 2438 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
3231, 4, 6diadmclN 31897 . . . . . . . 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 458 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z }
)  e.  ( Base `  K ) )
3431, 3opoccl 30054 . . . . . . 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 644 . . . . . 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 30857 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
3736ad2antlr 709 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  W  e.  ( Base `  K )
)
3831, 3opoccl 30054 . . . . . . 7  |-  ( ( K  e.  OP  /\  W  e.  ( Base `  K ) )  -> 
( ( oc `  K ) `  W
)  e.  ( Base `  K ) )
3916, 37, 38syl2anc 644 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T
)  ->  ( ( oc `  K ) `  W )  e.  (
Base `  K )
)
4031, 1latjcl 14481 . . . . . 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 1185 . . . . 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 14482 . . . . 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 1185 . . . 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 2438 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
4531, 44, 2latmle2 14508 . . . . 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 1185 . . . 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 31896 . . . . 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 453 . . . 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 890 . . 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 5868 . . 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 644 . 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 2512 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 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   {crab 2711    C_ wss 3322   (/)c0 3630   |^|cint 4052   class class class wbr 4214   `'ccnv 4879   dom cdm 4880   ran crn 4881   Fun wfun 5450   -1-1-onto->wf1o 5455   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   occoc 13539   joincjn 14403   meetcmee 14404   Latclat 14476   OPcops 30032   HLchlt 30210   LHypclh 30843   LTrncltrn 30960   DIsoAcdia 31888   ocAcocaN 31979
This theorem is referenced by:  dvadiaN  31988  djaclN  31996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-map 7022  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-llines 30357  df-lplanes 30358  df-lvols 30359  df-lines 30360  df-psubsp 30362  df-pmap 30363  df-padd 30655  df-lhyp 30847  df-laut 30848  df-ldil 30963  df-ltrn 30964  df-trl 31018  df-disoa 31889  df-docaN 31980
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