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Theorem lhpjat1 33018
Description: The join of a co-atom (hyperplane) and an atom not under it is the lattice unit. (Contributed by NM, 18-May-2012.)
Hypotheses
Ref Expression
lhpjat.l  |-  .<_  =  ( le `  K )
lhpjat.j  |-  .\/  =  ( join `  K )
lhpjat.u  |-  .1.  =  ( 1. `  K )
lhpjat.a  |-  A  =  ( Atoms `  K )
lhpjat.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
lhpjat1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( W  .\/  P
)  =  .1.  )

Proof of Theorem lhpjat1
StepHypRef Expression
1 simpll 752 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  HL )
2 eqid 2402 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
3 lhpjat.h . . . 4  |-  H  =  ( LHyp `  K
)
42, 3lhpbase 32996 . . 3  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
54ad2antlr 725 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  ( Base `  K ) )
6 simprl 756 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A )
7 lhpjat.u . . . 4  |-  .1.  =  ( 1. `  K )
8 eqid 2402 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
97, 8, 3lhp1cvr 32997 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W (  <o  `  K
)  .1.  )
109adantr 463 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W (  <o  `  K
)  .1.  )
11 simprr 758 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  W )
12 lhpjat.l . . 3  |-  .<_  =  ( le `  K )
13 lhpjat.j . . 3  |-  .\/  =  ( join `  K )
14 lhpjat.a . . 3  |-  A  =  ( Atoms `  K )
152, 12, 13, 7, 8, 141cvrjat 32473 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  ( Base `  K )  /\  P  e.  A )  /\  ( W (  <o  `  K
)  .1.  /\  -.  P  .<_  W ) )  ->  ( W  .\/  P )  =  .1.  )
161, 5, 6, 10, 11, 15syl32anc 1238 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( W  .\/  P
)  =  .1.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   class class class wbr 4394   ` cfv 5525  (class class class)co 6234   Basecbs 14733   lecple 14808   joincjn 15789   1.cp1 15884    <o ccvr 32261   Atomscatm 32262   HLchlt 32349   LHypclh 32982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-preset 15773  df-poset 15791  df-plt 15804  df-lub 15820  df-glb 15821  df-join 15822  df-meet 15823  df-p0 15885  df-p1 15886  df-lat 15892  df-clat 15954  df-oposet 32175  df-ol 32177  df-oml 32178  df-covers 32265  df-ats 32266  df-atl 32297  df-cvlat 32321  df-hlat 32350  df-lhyp 32986
This theorem is referenced by:  lhpjat2  33019  lhpj1  33020  trljat1  33165  trljat2  33166  cdlemc1  33190  cdlemc6  33195  cdleme20c  33311  cdleme20j  33318  trlcolem  33726
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