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Theorem dochvalr 36029
Description: Orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.)
Hypotheses
Ref Expression
dochvalr.o  |-  ._|_  =  ( oc `  K )
dochvalr.h  |-  H  =  ( LHyp `  K
)
dochvalr.i  |-  I  =  ( ( DIsoH `  K
) `  W )
dochvalr.n  |-  N  =  ( ( ocH `  K
) `  W )
Assertion
Ref Expression
dochvalr  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  ( N `  X )  =  ( I `  (  ._|_  `  ( `' I `  X )
) ) )

Proof of Theorem dochvalr
Dummy variables  z 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dochvalr.h . . . 4  |-  H  =  ( LHyp `  K
)
2 eqid 2460 . . . 4  |-  ( (
DVecH `  K ) `  W )  =  ( ( DVecH `  K ) `  W )
3 dochvalr.i . . . 4  |-  I  =  ( ( DIsoH `  K
) `  W )
4 eqid 2460 . . . 4  |-  ( Base `  ( ( DVecH `  K
) `  W )
)  =  ( Base `  ( ( DVecH `  K
) `  W )
)
51, 2, 3, 4dihrnss 35950 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  X  C_  ( Base `  (
( DVecH `  K ) `  W ) ) )
6 eqid 2460 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
7 eqid 2460 . . . 4  |-  ( glb `  K )  =  ( glb `  K )
8 dochvalr.o . . . 4  |-  ._|_  =  ( oc `  K )
9 dochvalr.n . . . 4  |-  N  =  ( ( ocH `  K
) `  W )
106, 7, 8, 1, 3, 2, 4, 9dochval 36023 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  ( Base `  ( ( DVecH `  K ) `  W
) ) )  -> 
( N `  X
)  =  ( I `
 (  ._|_  `  (
( glb `  K
) `  { y  e.  ( Base `  K
)  |  X  C_  ( I `  y
) } ) ) ) )
115, 10syldan 470 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  ( N `  X )  =  ( I `  (  ._|_  `  ( ( glb `  K ) `  { y  e.  (
Base `  K )  |  X  C_  ( I `
 y ) } ) ) ) )
12 eqid 2460 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
13 hllat 34035 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
1413ad2antrr 725 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  K  e.  Lat )
15 hlclat 34030 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  CLat )
1615ad2antrr 725 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  K  e.  CLat )
17 ssrab2 3578 . . . . . 6  |-  { y  e.  ( Base `  K
)  |  X  C_  ( I `  y
) }  C_  ( Base `  K )
186, 7clatglbcl 15590 . . . . . 6  |-  ( ( K  e.  CLat  /\  {
y  e.  ( Base `  K )  |  X  C_  ( I `  y
) }  C_  ( Base `  K ) )  ->  ( ( glb `  K ) `  {
y  e.  ( Base `  K )  |  X  C_  ( I `  y
) } )  e.  ( Base `  K
) )
1916, 17, 18sylancl 662 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  (
( glb `  K
) `  { y  e.  ( Base `  K
)  |  X  C_  ( I `  y
) } )  e.  ( Base `  K
) )
206, 1, 3dihcnvcl 35943 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  ( `' I `  X )  e.  ( Base `  K
) )
2117a1i 11 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  { y  e.  ( Base `  K
)  |  X  C_  ( I `  y
) }  C_  ( Base `  K ) )
22 ssid 3516 . . . . . . . 8  |-  X  C_  X
231, 3dihcnvid2 35945 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  (
I `  ( `' I `  X )
)  =  X )
2422, 23syl5sseqr 3546 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  X  C_  ( I `  ( `' I `  X ) ) )
25 fveq2 5857 . . . . . . . . 9  |-  ( y  =  ( `' I `  X )  ->  (
I `  y )  =  ( I `  ( `' I `  X ) ) )
2625sseq2d 3525 . . . . . . . 8  |-  ( y  =  ( `' I `  X )  ->  ( X  C_  ( I `  y )  <->  X  C_  (
I `  ( `' I `  X )
) ) )
2726elrab 3254 . . . . . . 7  |-  ( ( `' I `  X )  e.  { y  e.  ( Base `  K
)  |  X  C_  ( I `  y
) }  <->  ( ( `' I `  X )  e.  ( Base `  K
)  /\  X  C_  (
I `  ( `' I `  X )
) ) )
2820, 24, 27sylanbrc 664 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  ( `' I `  X )  e.  { y  e.  ( Base `  K
)  |  X  C_  ( I `  y
) } )
296, 12, 7clatglble 15601 . . . . . 6  |-  ( ( K  e.  CLat  /\  {
y  e.  ( Base `  K )  |  X  C_  ( I `  y
) }  C_  ( Base `  K )  /\  ( `' I `  X )  e.  { y  e.  ( Base `  K
)  |  X  C_  ( I `  y
) } )  -> 
( ( glb `  K
) `  { y  e.  ( Base `  K
)  |  X  C_  ( I `  y
) } ) ( le `  K ) ( `' I `  X ) )
3016, 21, 28, 29syl3anc 1223 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  (
( glb `  K
) `  { y  e.  ( Base `  K
)  |  X  C_  ( I `  y
) } ) ( le `  K ) ( `' I `  X ) )
31 fveq2 5857 . . . . . . . . . 10  |-  ( y  =  z  ->  (
I `  y )  =  ( I `  z ) )
3231sseq2d 3525 . . . . . . . . 9  |-  ( y  =  z  ->  ( X  C_  ( I `  y )  <->  X  C_  (
I `  z )
) )
3332elrab 3254 . . . . . . . 8  |-  ( z  e.  { y  e.  ( Base `  K
)  |  X  C_  ( I `  y
) }  <->  ( z  e.  ( Base `  K
)  /\  X  C_  (
I `  z )
) )
3423adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  /\  z  e.  ( Base `  K ) )  -> 
( I `  ( `' I `  X ) )  =  X )
3534sseq1d 3524 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  /\  z  e.  ( Base `  K ) )  -> 
( ( I `  ( `' I `  X ) )  C_  ( I `  z )  <->  X  C_  (
I `  z )
) )
36 simpll 753 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  /\  z  e.  ( Base `  K ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
3720adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  /\  z  e.  ( Base `  K ) )  -> 
( `' I `  X )  e.  (
Base `  K )
)
38 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  /\  z  e.  ( Base `  K ) )  -> 
z  e.  ( Base `  K ) )
396, 12, 1, 3dihord 35936 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( `' I `  X )  e.  (
Base `  K )  /\  z  e.  ( Base `  K ) )  ->  ( ( I `
 ( `' I `  X ) )  C_  ( I `  z
)  <->  ( `' I `  X ) ( le
`  K ) z ) )
4036, 37, 38, 39syl3anc 1223 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  /\  z  e.  ( Base `  K ) )  -> 
( ( I `  ( `' I `  X ) )  C_  ( I `  z )  <->  ( `' I `  X )
( le `  K
) z ) )
4135, 40bitr3d 255 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  /\  z  e.  ( Base `  K ) )  -> 
( X  C_  (
I `  z )  <->  ( `' I `  X ) ( le `  K
) z ) )
4241biimpd 207 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  /\  z  e.  ( Base `  K ) )  -> 
( X  C_  (
I `  z )  ->  ( `' I `  X ) ( le
`  K ) z ) )
4342expimpd 603 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  (
( z  e.  (
Base `  K )  /\  X  C_  ( I `
 z ) )  ->  ( `' I `  X ) ( le
`  K ) z ) )
4433, 43syl5bi 217 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  (
z  e.  { y  e.  ( Base `  K
)  |  X  C_  ( I `  y
) }  ->  ( `' I `  X ) ( le `  K
) z ) )
4544ralrimiv 2869 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  A. z  e.  { y  e.  (
Base `  K )  |  X  C_  ( I `
 y ) }  ( `' I `  X ) ( le
`  K ) z )
466, 12, 7clatleglb 15602 . . . . . . 7  |-  ( ( K  e.  CLat  /\  ( `' I `  X )  e.  ( Base `  K
)  /\  { y  e.  ( Base `  K
)  |  X  C_  ( I `  y
) }  C_  ( Base `  K ) )  ->  ( ( `' I `  X ) ( le `  K
) ( ( glb `  K ) `  {
y  e.  ( Base `  K )  |  X  C_  ( I `  y
) } )  <->  A. z  e.  { y  e.  (
Base `  K )  |  X  C_  ( I `
 y ) }  ( `' I `  X ) ( le
`  K ) z ) )
4716, 20, 21, 46syl3anc 1223 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  (
( `' I `  X ) ( le
`  K ) ( ( glb `  K
) `  { y  e.  ( Base `  K
)  |  X  C_  ( I `  y
) } )  <->  A. z  e.  { y  e.  (
Base `  K )  |  X  C_  ( I `
 y ) }  ( `' I `  X ) ( le
`  K ) z ) )
4845, 47mpbird 232 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  ( `' I `  X ) ( le `  K
) ( ( glb `  K ) `  {
y  e.  ( Base `  K )  |  X  C_  ( I `  y
) } ) )
496, 12, 14, 19, 20, 30, 48latasymd 15533 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  (
( glb `  K
) `  { y  e.  ( Base `  K
)  |  X  C_  ( I `  y
) } )  =  ( `' I `  X ) )
5049fveq2d 5861 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  (  ._|_  `  ( ( glb `  K ) `  {
y  e.  ( Base `  K )  |  X  C_  ( I `  y
) } ) )  =  (  ._|_  `  ( `' I `  X ) ) )
5150fveq2d 5861 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  (
I `  (  ._|_  `  ( ( glb `  K
) `  { y  e.  ( Base `  K
)  |  X  C_  ( I `  y
) } ) ) )  =  ( I `
 (  ._|_  `  ( `' I `  X ) ) ) )
5211, 51eqtrd 2501 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  ( N `  X )  =  ( I `  (  ._|_  `  ( `' I `  X )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807   {crab 2811    C_ wss 3469   class class class wbr 4440   `'ccnv 4991   ran crn 4993   ` cfv 5579   Basecbs 14479   lecple 14551   occoc 14552   glbcglb 15419   Latclat 15521   CLatccla 15583   HLchlt 34022   LHypclh 34655   DVecHcdvh 35750   DIsoHcdih 35900   ocHcoch 36019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-riotaBAD 33631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-tpos 6945  df-undef 6992  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-sca 14560  df-vsca 14561  df-0g 14686  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-p1 15516  df-lat 15522  df-clat 15584  df-mnd 15721  df-submnd 15771  df-grp 15851  df-minusg 15852  df-sbg 15853  df-subg 15986  df-cntz 16143  df-lsm 16445  df-cmn 16589  df-abl 16590  df-mgp 16925  df-ur 16937  df-rng 16981  df-oppr 17049  df-dvdsr 17067  df-unit 17068  df-invr 17098  df-dvr 17109  df-drng 17174  df-lmod 17290  df-lss 17355  df-lsp 17394  df-lvec 17525  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-llines 34169  df-lplanes 34170  df-lvols 34171  df-lines 34172  df-psubsp 34174  df-pmap 34175  df-padd 34467  df-lhyp 34659  df-laut 34660  df-ldil 34775  df-ltrn 34776  df-trl 34830  df-tendo 35426  df-edring 35428  df-disoa 35701  df-dvech 35751  df-dib 35811  df-dic 35845  df-dih 35901  df-doch 36020
This theorem is referenced by:  doch0  36030  doch1  36031  dochvalr2  36034  dochvalr3  36035  dochocss  36038  dochoc  36039  dochnoncon  36063
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