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Theorem dochss 32100
Description: Subset law for orthocomplement. (Contributed by NM, 16-Apr-2014.)
Hypotheses
Ref Expression
dochss.h  |-  H  =  ( LHyp `  K
)
dochss.u  |-  U  =  ( ( DVecH `  K
) `  W )
dochss.v  |-  V  =  ( Base `  U
)
dochss.o  |-  ._|_  =  ( ( ocH `  K
) `  W )
Assertion
Ref Expression
dochss  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  (  ._|_  `  Y
)  C_  (  ._|_  `  X ) )

Proof of Theorem dochss
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 simp1l 981 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  K  e.  HL )
2 hlclat 30093 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CLat )
31, 2syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  K  e.  CLat )
4 ssrab2 3420 . . . . . 6  |-  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) }  C_  ( Base `  K
)
54a1i 11 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) }  C_  ( Base `  K
) )
6 simpll3 998 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  /\  z  e.  ( Base `  K
) )  /\  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z )
)  ->  X  C_  Y
)
7 simpr 448 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  /\  z  e.  ( Base `  K
) )  /\  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z )
)  ->  Y  C_  (
( ( DIsoH `  K
) `  W ) `  z ) )
86, 7sstrd 3350 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  /\  z  e.  ( Base `  K
) )  /\  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z )
)  ->  X  C_  (
( ( DIsoH `  K
) `  W ) `  z ) )
98ex 424 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  /\  z  e.  ( Base `  K
) )  ->  ( Y  C_  ( ( (
DIsoH `  K ) `  W ) `  z
)  ->  X  C_  (
( ( DIsoH `  K
) `  W ) `  z ) ) )
109ss2rabdv 3416 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) }  C_  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } )
11 eqid 2435 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
12 eqid 2435 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
13 eqid 2435 . . . . . 6  |-  ( glb `  K )  =  ( glb `  K )
1411, 12, 13clatglbss 14546 . . . . 5  |-  ( ( K  e.  CLat  /\  {
z  e.  ( Base `  K )  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) }  C_  ( Base `  K
)  /\  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) }  C_  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } )  ->  (
( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ( le
`  K ) ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) )
153, 5, 10, 14syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  ( ( glb `  K ) `  {
z  e.  ( Base `  K )  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ( le
`  K ) ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) )
16 hlop 30097 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
171, 16syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  K  e.  OP )
1811, 13clatglbcl 14533 . . . . . 6  |-  ( ( K  e.  CLat  /\  {
z  e.  ( Base `  K )  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) }  C_  ( Base `  K
) )  ->  (
( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } )  e.  (
Base `  K )
)
193, 4, 18sylancl 644 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  ( ( glb `  K ) `  {
z  e.  ( Base `  K )  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } )  e.  (
Base `  K )
)
20 ssrab2 3420 . . . . . 6  |-  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) }  C_  ( Base `  K
)
2111, 13clatglbcl 14533 . . . . . 6  |-  ( ( K  e.  CLat  /\  {
z  e.  ( Base `  K )  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) }  C_  ( Base `  K
) )  ->  (
( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } )  e.  (
Base `  K )
)
223, 20, 21sylancl 644 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  ( ( glb `  K ) `  {
z  e.  ( Base `  K )  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } )  e.  (
Base `  K )
)
23 eqid 2435 . . . . . 6  |-  ( oc
`  K )  =  ( oc `  K
)
2411, 12, 23oplecon3b 29935 . . . . 5  |-  ( ( K  e.  OP  /\  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } )  e.  (
Base `  K )  /\  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } )  e.  (
Base `  K )
)  ->  ( (
( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ( le
`  K ) ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } )  <->  ( ( oc `  K ) `  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) ( le `  K ) ( ( oc `  K ) `  (
( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) ) )
2517, 19, 22, 24syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  ( ( ( glb `  K ) `
 { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ( le
`  K ) ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } )  <->  ( ( oc `  K ) `  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) ( le `  K ) ( ( oc `  K ) `  (
( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) ) )
2615, 25mpbid 202 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  ( ( oc
`  K ) `  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) ( le `  K ) ( ( oc `  K ) `  (
( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) )
27 simp1 957 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  ( K  e.  HL  /\  W  e.  H ) )
2811, 23opoccl 29929 . . . . 5  |-  ( ( K  e.  OP  /\  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } )  e.  (
Base `  K )
)  ->  ( ( oc `  K ) `  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) )  e.  ( Base `  K
) )
2917, 22, 28syl2anc 643 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  ( ( oc
`  K ) `  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) )  e.  ( Base `  K
) )
3011, 23opoccl 29929 . . . . 5  |-  ( ( K  e.  OP  /\  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } )  e.  (
Base `  K )
)  ->  ( ( oc `  K ) `  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) )  e.  ( Base `  K
) )
3117, 19, 30syl2anc 643 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  ( ( oc
`  K ) `  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) )  e.  ( Base `  K
) )
32 dochss.h . . . . 5  |-  H  =  ( LHyp `  K
)
33 eqid 2435 . . . . 5  |-  ( (
DIsoH `  K ) `  W )  =  ( ( DIsoH `  K ) `  W )
3411, 12, 32, 33dihord 31999 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( oc
`  K ) `  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) )  e.  ( Base `  K
)  /\  ( ( oc `  K ) `  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) )  e.  ( Base `  K
) )  ->  (
( ( ( DIsoH `  K ) `  W
) `  ( ( oc `  K ) `  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) ) 
C_  ( ( (
DIsoH `  K ) `  W ) `  (
( oc `  K
) `  ( ( glb `  K ) `  { z  e.  (
Base `  K )  |  X  C_  ( ( ( DIsoH `  K ) `  W ) `  z
) } ) ) )  <->  ( ( oc
`  K ) `  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) ( le `  K ) ( ( oc `  K ) `  (
( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) ) )
3527, 29, 31, 34syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  ( ( ( ( DIsoH `  K ) `  W ) `  (
( oc `  K
) `  ( ( glb `  K ) `  { z  e.  (
Base `  K )  |  Y  C_  ( ( ( DIsoH `  K ) `  W ) `  z
) } ) ) )  C_  ( (
( DIsoH `  K ) `  W ) `  (
( oc `  K
) `  ( ( glb `  K ) `  { z  e.  (
Base `  K )  |  X  C_  ( ( ( DIsoH `  K ) `  W ) `  z
) } ) ) )  <->  ( ( oc
`  K ) `  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) ( le `  K ) ( ( oc `  K ) `  (
( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) ) )
3626, 35mpbird 224 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  ( ( (
DIsoH `  K ) `  W ) `  (
( oc `  K
) `  ( ( glb `  K ) `  { z  e.  (
Base `  K )  |  Y  C_  ( ( ( DIsoH `  K ) `  W ) `  z
) } ) ) )  C_  ( (
( DIsoH `  K ) `  W ) `  (
( oc `  K
) `  ( ( glb `  K ) `  { z  e.  (
Base `  K )  |  X  C_  ( ( ( DIsoH `  K ) `  W ) `  z
) } ) ) ) )
37 dochss.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
38 dochss.v . . . 4  |-  V  =  ( Base `  U
)
39 dochss.o . . . 4  |-  ._|_  =  ( ( ocH `  K
) `  W )
4011, 13, 23, 32, 33, 37, 38, 39dochval 32086 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V
)  ->  (  ._|_  `  Y )  =  ( ( ( DIsoH `  K
) `  W ) `  ( ( oc `  K ) `  (
( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) ) )
41403adant3 977 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  (  ._|_  `  Y
)  =  ( ( ( DIsoH `  K ) `  W ) `  (
( oc `  K
) `  ( ( glb `  K ) `  { z  e.  (
Base `  K )  |  Y  C_  ( ( ( DIsoH `  K ) `  W ) `  z
) } ) ) ) )
42 simp3 959 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  X  C_  Y
)
43 simp2 958 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  Y  C_  V
)
4442, 43sstrd 3350 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  X  C_  V
)
4511, 13, 23, 32, 33, 37, 38, 39dochval 32086 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  (  ._|_  `  X )  =  ( ( ( DIsoH `  K
) `  W ) `  ( ( oc `  K ) `  (
( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) ) )
4627, 44, 45syl2anc 643 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  (  ._|_  `  X
)  =  ( ( ( DIsoH `  K ) `  W ) `  (
( oc `  K
) `  ( ( glb `  K ) `  { z  e.  (
Base `  K )  |  X  C_  ( ( ( DIsoH `  K ) `  W ) `  z
) } ) ) ) )
4736, 41, 463sstr4d 3383 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  (  ._|_  `  Y
)  C_  (  ._|_  `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {crab 2701    C_ wss 3312   class class class wbr 4204   ` cfv 5446   Basecbs 13461   lecple 13528   occoc 13529   glbcglb 14392   CLatccla 14528   OPcops 29907   HLchlt 30085   LHypclh 30718   DVecHcdvh 31813   DIsoHcdih 31963   ocHcoch 32082
This theorem is referenced by:  dochsscl  32103  dochord  32105  dihoml4  32112  dochocsp  32114  dochdmj1  32125  dochpolN  32225  lclkrlem2p  32257  lclkrslem1  32272  lclkrslem2  32273  lcfrvalsnN  32276  mapdsn  32376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-undef 6535  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-0g 13719  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-cntz 15108  df-lsm 15262  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-dvr 15780  df-drng 15829  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lvec 16167  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lvols 30234  df-lines 30235  df-psubsp 30237  df-pmap 30238  df-padd 30530  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839  df-trl 30893  df-tendo 31489  df-edring 31491  df-disoa 31764  df-dvech 31814  df-dib 31874  df-dic 31908  df-dih 31964  df-doch 32083
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