Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lhpjat1 Structured version   Unicode version

Theorem lhpjat1 34691
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 753 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  HL )
2 eqid 2460 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
3 lhpjat.h . . . 4  |-  H  =  ( LHyp `  K
)
42, 3lhpbase 34669 . . 3  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
54ad2antlr 726 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  ( Base `  K ) )
6 simprl 755 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A )
7 lhpjat.u . . . 4  |-  .1.  =  ( 1. `  K )
8 eqid 2460 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
97, 8, 3lhp1cvr 34670 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W (  <o  `  K
)  .1.  )
109adantr 465 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W (  <o  `  K
)  .1.  )
11 simprr 756 . 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 34146 . 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 1231 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 369    = wceq 1374    e. wcel 1762   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   lecple 14551   joincjn 15420   1.cp1 15514    <o ccvr 33934   Atomscatm 33935   HLchlt 34022   LHypclh 34655
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
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  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-ral 2812  df-rex 2813  df-reu 2814  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-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  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-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-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-lhyp 34659
This theorem is referenced by:  lhpjat2  34692  lhpj1  34693  trljat1  34837  trljat2  34838  cdlemc1  34862  cdlemc6  34867  cdleme20c  34982  cdleme20j  34989  trlcolem  35397
  Copyright terms: Public domain W3C validator