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Theorem dochss 31848
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 29841 . . . . . 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 3388 . . . . . 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 3318 . . . . . . 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 3384 . . . . 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 2404 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
12 eqid 2404 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
13 eqid 2404 . . . . . 6  |-  ( glb `  K )  =  ( glb `  K )
1411, 12, 13clatglbss 14509 . . . . 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 29845 . . . . . 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 14496 . . . . . 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 3388 . . . . . 6  |-  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) }  C_  ( Base `  K
)
2111, 13clatglbcl 14496 . . . . . 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 2404 . . . . . 6  |-  ( oc
`  K )  =  ( oc `  K
)
2411, 12, 23oplecon3b 29683 . . . . 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 29677 . . . . 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 29677 . . . . 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 2404 . . . . 5  |-  ( (
DIsoH `  K ) `  W )  =  ( ( DIsoH `  K ) `  W )
3411, 12, 32, 33dihord 31747 . . . 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 31834 . . 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 3318 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  X  C_  V
)
4511, 13, 23, 32, 33, 37, 38, 39dochval 31834 . . 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 3351 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 1649    e. wcel 1721   {crab 2670    C_ wss 3280   class class class wbr 4172   ` cfv 5413   Basecbs 13424   lecple 13491   occoc 13492   glbcglb 14355   CLatccla 14491   OPcops 29655   HLchlt 29833   LHypclh 30466   DVecHcdvh 31561   DIsoHcdih 31711   ocHcoch 31830
This theorem is referenced by:  dochsscl  31851  dochord  31853  dihoml4  31860  dochocsp  31862  dochdmj1  31873  dochpolN  31973  lclkrlem2p  32005  lclkrslem1  32020  lclkrslem2  32021  lcfrvalsnN  32024  mapdsn  32124
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-undef 6502  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-0g 13682  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-cntz 15071  df-lsm 15225  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-drng 15792  df-lmod 15907  df-lss 15964  df-lsp 16003  df-lvec 16130  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641  df-tendo 31237  df-edring 31239  df-disoa 31512  df-dvech 31562  df-dib 31622  df-dic 31656  df-dih 31712  df-doch 31831
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