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Theorem dalem11 30156
Description: Lemma for dath 30218. Analog of dalem10 30155 for  E. (Contributed by NM, 23-Jul-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 )
dalem11.m  |-  ./\  =  ( meet `  K )
dalem11.o  |-  O  =  ( LPlanes `  K )
dalem11.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem11.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem11.x  |-  X  =  ( Y  ./\  Z
)
dalem11.e  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
Assertion
Ref Expression
dalem11  |-  ( ph  ->  E  .<_  X )

Proof of Theorem dalem11
StepHypRef Expression
1 dalema.ph . . . 4  |-  ( 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 ) ) ) ) )
2 dalemc.l . . . 4  |-  .<_  =  ( le `  K )
3 dalemc.j . . . 4  |-  .\/  =  ( join `  K )
4 dalemc.a . . . 4  |-  A  =  ( Atoms `  K )
5 dalem11.y . . . 4  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
6 dalem11.z . . . 4  |-  Z  =  ( ( S  .\/  T )  .\/  U )
71, 2, 3, 4, 5, 6dalemrot 30139 . . 3  |-  ( ph  ->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( Q  e.  A  /\  R  e.  A  /\  P  e.  A )  /\  ( T  e.  A  /\  U  e.  A  /\  S  e.  A
) )  /\  (
( ( Q  .\/  R )  .\/  P )  e.  O  /\  (
( T  .\/  U
)  .\/  S )  e.  O )  /\  (
( -.  C  .<_  ( Q  .\/  R )  /\  -.  C  .<_  ( R  .\/  P )  /\  -.  C  .<_  ( P  .\/  Q ) )  /\  ( -.  C  .<_  ( T  .\/  U )  /\  -.  C  .<_  ( U  .\/  S )  /\  -.  C  .<_  ( S  .\/  T
) )  /\  ( C  .<_  ( Q  .\/  T )  /\  C  .<_  ( R  .\/  U )  /\  C  .<_  ( P 
.\/  S ) ) ) ) )
8 biid 228 . . . 4  |-  ( ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( Q  e.  A  /\  R  e.  A  /\  P  e.  A )  /\  ( T  e.  A  /\  U  e.  A  /\  S  e.  A
) )  /\  (
( ( Q  .\/  R )  .\/  P )  e.  O  /\  (
( T  .\/  U
)  .\/  S )  e.  O )  /\  (
( -.  C  .<_  ( Q  .\/  R )  /\  -.  C  .<_  ( R  .\/  P )  /\  -.  C  .<_  ( P  .\/  Q ) )  /\  ( -.  C  .<_  ( T  .\/  U )  /\  -.  C  .<_  ( U  .\/  S )  /\  -.  C  .<_  ( S  .\/  T
) )  /\  ( C  .<_  ( Q  .\/  T )  /\  C  .<_  ( R  .\/  U )  /\  C  .<_  ( P 
.\/  S ) ) ) )  <->  ( (
( K  e.  HL  /\  C  e.  ( Base `  K ) )  /\  ( Q  e.  A  /\  R  e.  A  /\  P  e.  A
)  /\  ( T  e.  A  /\  U  e.  A  /\  S  e.  A ) )  /\  ( ( ( Q 
.\/  R )  .\/  P )  e.  O  /\  ( ( T  .\/  U )  .\/  S )  e.  O )  /\  ( ( -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
)  /\  -.  C  .<_  ( P  .\/  Q
) )  /\  ( -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S )  /\  -.  C  .<_  ( S 
.\/  T ) )  /\  ( C  .<_  ( Q  .\/  T )  /\  C  .<_  ( R 
.\/  U )  /\  C  .<_  ( P  .\/  S ) ) ) ) )
9 dalem11.m . . . 4  |-  ./\  =  ( meet `  K )
10 dalem11.o . . . 4  |-  O  =  ( LPlanes `  K )
11 eqid 2404 . . . 4  |-  ( ( Q  .\/  R ) 
.\/  P )  =  ( ( Q  .\/  R )  .\/  P )
12 eqid 2404 . . . 4  |-  ( ( T  .\/  U ) 
.\/  S )  =  ( ( T  .\/  U )  .\/  S )
13 eqid 2404 . . . 4  |-  ( ( ( Q  .\/  R
)  .\/  P )  ./\  ( ( T  .\/  U )  .\/  S ) )  =  ( ( ( Q  .\/  R
)  .\/  P )  ./\  ( ( T  .\/  U )  .\/  S ) )
14 dalem11.e . . . 4  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
158, 2, 3, 4, 9, 10, 11, 12, 13, 14dalem10 30155 . . 3  |-  ( ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( Q  e.  A  /\  R  e.  A  /\  P  e.  A )  /\  ( T  e.  A  /\  U  e.  A  /\  S  e.  A
) )  /\  (
( ( Q  .\/  R )  .\/  P )  e.  O  /\  (
( T  .\/  U
)  .\/  S )  e.  O )  /\  (
( -.  C  .<_  ( Q  .\/  R )  /\  -.  C  .<_  ( R  .\/  P )  /\  -.  C  .<_  ( P  .\/  Q ) )  /\  ( -.  C  .<_  ( T  .\/  U )  /\  -.  C  .<_  ( U  .\/  S )  /\  -.  C  .<_  ( S  .\/  T
) )  /\  ( C  .<_  ( Q  .\/  T )  /\  C  .<_  ( R  .\/  U )  /\  C  .<_  ( P 
.\/  S ) ) ) )  ->  E  .<_  ( ( ( Q 
.\/  R )  .\/  P )  ./\  ( ( T  .\/  U )  .\/  S ) ) )
167, 15syl 16 . 2  |-  ( ph  ->  E  .<_  ( (
( Q  .\/  R
)  .\/  P )  ./\  ( ( T  .\/  U )  .\/  S ) ) )
17 dalem11.x . . 3  |-  X  =  ( Y  ./\  Z
)
181, 3, 4dalemqrprot 30130 . . . . 5  |-  ( ph  ->  ( ( Q  .\/  R )  .\/  P )  =  ( ( P 
.\/  Q )  .\/  R ) )
1918, 5syl6reqr 2455 . . . 4  |-  ( ph  ->  Y  =  ( ( Q  .\/  R ) 
.\/  P ) )
201dalemkehl 30105 . . . . . 6  |-  ( ph  ->  K  e.  HL )
211dalemtea 30112 . . . . . 6  |-  ( ph  ->  T  e.  A )
221dalemuea 30113 . . . . . 6  |-  ( ph  ->  U  e.  A )
231dalemsea 30111 . . . . . 6  |-  ( ph  ->  S  e.  A )
243, 4hlatjrot 29855 . . . . . 6  |-  ( ( K  e.  HL  /\  ( T  e.  A  /\  U  e.  A  /\  S  e.  A
) )  ->  (
( T  .\/  U
)  .\/  S )  =  ( ( S 
.\/  T )  .\/  U ) )
2520, 21, 22, 23, 24syl13anc 1186 . . . . 5  |-  ( ph  ->  ( ( T  .\/  U )  .\/  S )  =  ( ( S 
.\/  T )  .\/  U ) )
2625, 6syl6reqr 2455 . . . 4  |-  ( ph  ->  Z  =  ( ( T  .\/  U ) 
.\/  S ) )
2719, 26oveq12d 6058 . . 3  |-  ( ph  ->  ( Y  ./\  Z
)  =  ( ( ( Q  .\/  R
)  .\/  P )  ./\  ( ( T  .\/  U )  .\/  S ) ) )
2817, 27syl5eq 2448 . 2  |-  ( ph  ->  X  =  ( ( ( Q  .\/  R
)  .\/  P )  ./\  ( ( T  .\/  U )  .\/  S ) ) )
2916, 28breqtrrd 4198 1  |-  ( ph  ->  E  .<_  X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   HLchlt 29833   LPlanesclpl 29974
This theorem is referenced by:  dalem12  30157  dalem16  30161
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-lat 14430  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-lplanes 29981
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