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Theorem dalemply 34806
Description: Lemma for dath 34888. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
Hypotheses
Ref Expression
dalema.ph  |-  ( ph  <->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) ) )
dalemc.l  |-  .<_  =  ( le `  K )
dalemc.j  |-  .\/  =  ( join `  K )
dalemc.a  |-  A  =  ( Atoms `  K )
dalempnes.o  |-  O  =  ( LPlanes `  K )
dalempnes.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
Assertion
Ref Expression
dalemply  |-  ( ph  ->  P  .<_  Y )

Proof of Theorem dalemply
StepHypRef Expression
1 dalema.ph . . . . 5  |-  ( ph  <->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) ) )
21dalemkelat 34776 . . . 4  |-  ( ph  ->  K  e.  Lat )
3 dalemc.a . . . . 5  |-  A  =  ( Atoms `  K )
41, 3dalempeb 34791 . . . 4  |-  ( ph  ->  P  e.  ( Base `  K ) )
51dalemkehl 34775 . . . . 5  |-  ( ph  ->  K  e.  HL )
61dalemqea 34779 . . . . 5  |-  ( ph  ->  Q  e.  A )
71dalemrea 34780 . . . . 5  |-  ( ph  ->  R  e.  A )
8 eqid 2467 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
9 dalemc.j . . . . . 6  |-  .\/  =  ( join `  K )
108, 9, 3hlatjcl 34519 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
115, 6, 7, 10syl3anc 1228 . . . 4  |-  ( ph  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
12 dalemc.l . . . . 5  |-  .<_  =  ( le `  K )
138, 12, 9latlej1 15564 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
) )  ->  P  .<_  ( P  .\/  ( Q  .\/  R ) ) )
142, 4, 11, 13syl3anc 1228 . . 3  |-  ( ph  ->  P  .<_  ( P  .\/  ( Q  .\/  R
) ) )
151dalempea 34778 . . . 4  |-  ( ph  ->  P  e.  A )
169, 3hlatjass 34522 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( P  .\/  ( Q  .\/  R ) ) )
175, 15, 6, 7, 16syl13anc 1230 . . 3  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  R )  =  ( P  .\/  ( Q  .\/  R ) ) )
1814, 17breqtrrd 4479 . 2  |-  ( ph  ->  P  .<_  ( ( P  .\/  Q )  .\/  R ) )
19 dalempnes.y . 2  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
2018, 19syl6breqr 4493 1  |-  ( ph  ->  P  .<_  Y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   lecple 14579   joincjn 15448   Latclat 15549   Atomscatm 34416   HLchlt 34503   LPlanesclpl 34644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-poset 15450  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-lat 15550  df-ats 34420  df-atl 34451  df-cvlat 34475  df-hlat 34504
This theorem is referenced by:  dalem21  34846  dalem23  34848  dalem24  34849  dalem27  34851
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