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Theorem lhplt 30724
Description: An atom under a co-atom is strictly less than it. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
Hypotheses
Ref Expression
lhplt.l  |-  .<_  =  ( le `  K )
lhplt.s  |-  .<  =  ( lt `  K )
lhplt.a  |-  A  =  ( Atoms `  K )
lhplt.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
lhplt  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  P  .<_  W ) )  ->  P  .<  W )

Proof of Theorem lhplt
StepHypRef Expression
1 simpll 731 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  P  .<_  W ) )  ->  K  e.  HL )
2 simprl 733 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  P  .<_  W ) )  ->  P  e.  A )
3 eqid 2435 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
4 lhplt.h . . . 4  |-  H  =  ( LHyp `  K
)
53, 4lhpbase 30722 . . 3  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
65ad2antlr 708 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  P  .<_  W ) )  ->  W  e.  ( Base `  K
) )
7 eqid 2435 . . . 4  |-  ( 1.
`  K )  =  ( 1. `  K
)
8 eqid 2435 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
97, 8, 4lhp1cvr 30723 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W (  <o  `  K
) ( 1. `  K ) )
109adantr 452 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  P  .<_  W ) )  ->  W
(  <o  `  K )
( 1. `  K
) )
11 simprr 734 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  P  .<_  W ) )  ->  P  .<_  W )
12 lhplt.l . . 3  |-  .<_  =  ( le `  K )
13 lhplt.s . . 3  |-  .<  =  ( lt `  K )
14 lhplt.a . . 3  |-  A  =  ( Atoms `  K )
153, 12, 13, 7, 8, 141cvratlt 30198 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  W  e.  ( Base `  K ) )  /\  ( W (  <o  `  K
) ( 1. `  K )  /\  P  .<_  W ) )  ->  P  .<  W )
161, 2, 6, 10, 11, 15syl32anc 1192 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  P  .<_  W ) )  ->  P  .<  W )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446   Basecbs 13461   lecple 13528   ltcplt 14390   1.cp1 14459    <o ccvr 29987   Atomscatm 29988   HLchlt 30075   LHypclh 30708
This theorem is referenced by:  lhpexle1  30732
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
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  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-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  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-undef 6535  df-riota 6541  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-oposet 29901  df-ol 29903  df-oml 29904  df-covers 29991  df-ats 29992  df-atl 30023  df-cvlat 30047  df-hlat 30076  df-lhyp 30712
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