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Theorem dalem62 30432
Description: Lemma for dath 30434. Eliminate the condition  ps containing dummy variables  c and  d. (Contributed by NM, 11-Aug-2012.)
Hypotheses
Ref Expression
dalem62.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 ) ) ) ) )
dalem62.l  |-  .<_  =  ( le `  K )
dalem62.j  |-  .\/  =  ( join `  K )
dalem62.a  |-  A  =  ( Atoms `  K )
dalem62.m  |-  ./\  =  ( meet `  K )
dalem62.o  |-  O  =  ( LPlanes `  K )
dalem62.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem62.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem62.d  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
dalem62.e  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
dalem62.f  |-  F  =  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )
Assertion
Ref Expression
dalem62  |-  ( (
ph  /\  Y  =  Z )  ->  F  .<_  ( D  .\/  E
) )

Proof of Theorem dalem62
Dummy variables  c 
d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dalem62.ph . . 3  |-  ( 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 dalem62.l . . 3  |-  .<_  =  ( le `  K )
3 dalem62.j . . 3  |-  .\/  =  ( join `  K )
4 dalem62.a . . 3  |-  A  =  ( Atoms `  K )
5 biid 228 . . 3  |-  ( ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) )  <->  ( (
c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) ) )
6 dalem62.o . . 3  |-  O  =  ( LPlanes `  K )
7 dalem62.y . . 3  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
8 dalem62.z . . 3  |-  Z  =  ( ( S  .\/  T )  .\/  U )
91, 2, 3, 4, 5, 6, 7, 8dalem20 30391 . 2  |-  ( (
ph  /\  Y  =  Z )  ->  E. c E. d ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) ) )
10 dalem62.m . . . . 5  |-  ./\  =  ( meet `  K )
11 dalem62.d . . . . 5  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
12 dalem62.e . . . . 5  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
13 dalem62.f . . . . 5  |-  F  =  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )
141, 2, 3, 4, 5, 10, 6, 7, 8, 11, 12, 13dalem61 30431 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) ) )  ->  F  .<_  ( D  .\/  E ) )
15143expia 1155 . . 3  |-  ( (
ph  /\  Y  =  Z )  ->  (
( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  Y  /\  (
d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) )  ->  F  .<_  ( D 
.\/  E ) ) )
1615exlimdvv 1647 . 2  |-  ( (
ph  /\  Y  =  Z )  ->  ( E. c E. d ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) )  ->  F  .<_  ( D  .\/  E ) ) )
179, 16mpd 15 1  |-  ( (
ph  /\  Y  =  Z )  ->  F  .<_  ( D  .\/  E
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13459   lecple 13526   joincjn 14391   meetcmee 14392   Atomscatm 29962   HLchlt 30049   LPlanesclpl 30190
This theorem is referenced by:  dalem63  30433
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-3or 937  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 14393  df-plt 14405  df-lub 14421  df-glb 14422  df-join 14423  df-meet 14424  df-p0 14458  df-lat 14465  df-clat 14527  df-oposet 29875  df-ol 29877  df-oml 29878  df-covers 29965  df-ats 29966  df-atl 29997  df-cvlat 30021  df-hlat 30050  df-llines 30196  df-lplanes 30197  df-lvols 30198
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