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Theorem dalem63 30221
Description: Lemma for dath 30222. Combine the cases where  Y and  Z are different planes with the case where  Y and 
Z are the same plane. (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
dalem63  |-  ( ph  ->  F  .<_  ( D  .\/  E ) )

Proof of Theorem dalem63
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 dalem62.m . . 3  |-  ./\  =  ( meet `  K )
6 dalem62.o . . 3  |-  O  =  ( LPlanes `  K )
7 dalem62.y . . 3  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
8 dalem62.z . . 3  |-  Z  =  ( ( S  .\/  T )  .\/  U )
9 dalem62.d . . 3  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
10 dalem62.e . . 3  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
11 dalem62.f . . 3  |-  F  =  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11dalem62 30220 . 2  |-  ( (
ph  /\  Y  =  Z )  ->  F  .<_  ( D  .\/  E
) )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11dalem16 30165 . 2  |-  ( (
ph  /\  Y  =/=  Z )  ->  F  .<_  ( D  .\/  E ) )
1412, 13pm2.61dane 2649 1  |-  ( ph  ->  F  .<_  ( D  .\/  E ) )
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 4176   ` cfv 5417  (class class class)co 6044   Basecbs 13428   lecple 13495   joincjn 14360   meetcmee 14361   Atomscatm 29750   HLchlt 29837   LPlanesclpl 29978
This theorem is referenced by:  dath  30222
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-lat 14434  df-clat 14496  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-llines 29984  df-lplanes 29985  df-lvols 29986
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