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Theorem dalemply 33617
Description: Lemma for dath 33699. 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 33587 . . . 4  |-  ( ph  ->  K  e.  Lat )
3 dalemc.a . . . . 5  |-  A  =  ( Atoms `  K )
41, 3dalempeb 33602 . . . 4  |-  ( ph  ->  P  e.  ( Base `  K ) )
51dalemkehl 33586 . . . . 5  |-  ( ph  ->  K  e.  HL )
61dalemqea 33590 . . . . 5  |-  ( ph  ->  Q  e.  A )
71dalemrea 33591 . . . . 5  |-  ( ph  ->  R  e.  A )
8 eqid 2452 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
9 dalemc.j . . . . . 6  |-  .\/  =  ( join `  K )
108, 9, 3hlatjcl 33330 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
115, 6, 7, 10syl3anc 1219 . . . 4  |-  ( ph  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
12 dalemc.l . . . . 5  |-  .<_  =  ( le `  K )
138, 12, 9latlej1 15344 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
) )  ->  P  .<_  ( P  .\/  ( Q  .\/  R ) ) )
142, 4, 11, 13syl3anc 1219 . . 3  |-  ( ph  ->  P  .<_  ( P  .\/  ( Q  .\/  R
) ) )
151dalempea 33589 . . . 4  |-  ( ph  ->  P  e.  A )
169, 3hlatjass 33333 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( P  .\/  ( Q  .\/  R ) ) )
175, 15, 6, 7, 16syl13anc 1221 . . 3  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  R )  =  ( P  .\/  ( Q  .\/  R ) ) )
1814, 17breqtrrd 4421 . 2  |-  ( ph  ->  P  .<_  ( ( P  .\/  Q )  .\/  R ) )
19 dalempnes.y . 2  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
2018, 19syl6breqr 4435 1  |-  ( ph  ->  P  .<_  Y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   Basecbs 14287   lecple 14359   joincjn 15228   Latclat 15329   Atomscatm 33227   HLchlt 33314   LPlanesclpl 33455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-poset 15230  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-lat 15330  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315
This theorem is referenced by:  dalem21  33657  dalem23  33659  dalem24  33660  dalem27  33662
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