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Theorem dalem13 29790
Description: Lemma for dalem14 29791. (Contributed by NM, 21-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 )
dalem13.o  |-  O  =  ( LPlanes `  K )
dalem13.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem13.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem13.w  |-  W  =  ( Y  .\/  C
)
Assertion
Ref Expression
dalem13  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( Y  .\/  Z )  =  W )

Proof of Theorem dalem13
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 ) ) ) ) )
21dalemkehl 29737 . . 3  |-  ( ph  ->  K  e.  HL )
32adantr 452 . 2  |-  ( (
ph  /\  Y  =/=  Z )  ->  K  e.  HL )
41dalemyeo 29746 . . 3  |-  ( ph  ->  Y  e.  O )
54adantr 452 . 2  |-  ( (
ph  /\  Y  =/=  Z )  ->  Y  e.  O )
61dalemzeo 29747 . . 3  |-  ( ph  ->  Z  e.  O )
76adantr 452 . 2  |-  ( (
ph  /\  Y  =/=  Z )  ->  Z  e.  O )
8 dalemc.l . . 3  |-  .<_  =  ( le `  K )
9 dalemc.j . . 3  |-  .\/  =  ( join `  K )
10 dalemc.a . . 3  |-  A  =  ( Atoms `  K )
11 dalem13.o . . 3  |-  O  =  ( LPlanes `  K )
12 eqid 2387 . . 3  |-  ( LVols `  K )  =  (
LVols `  K )
13 dalem13.y . . 3  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
14 dalem13.z . . 3  |-  Z  =  ( ( S  .\/  T )  .\/  U )
15 dalem13.w . . 3  |-  W  =  ( Y  .\/  C
)
161, 8, 9, 10, 11, 12, 13, 14, 15dalem9 29786 . 2  |-  ( (
ph  /\  Y  =/=  Z )  ->  W  e.  ( LVols `  K )
)
171dalemkelat 29738 . . . . 5  |-  ( ph  ->  K  e.  Lat )
181, 11dalemyeb 29763 . . . . 5  |-  ( ph  ->  Y  e.  ( Base `  K ) )
191, 10dalemceb 29752 . . . . 5  |-  ( ph  ->  C  e.  ( Base `  K ) )
20 eqid 2387 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
2120, 8, 9latlej1 14416 . . . . 5  |-  ( ( K  e.  Lat  /\  Y  e.  ( Base `  K )  /\  C  e.  ( Base `  K
) )  ->  Y  .<_  ( Y  .\/  C
) )
2217, 18, 19, 21syl3anc 1184 . . . 4  |-  ( ph  ->  Y  .<_  ( Y  .\/  C ) )
2322, 15syl6breqr 4193 . . 3  |-  ( ph  ->  Y  .<_  W )
2423adantr 452 . 2  |-  ( (
ph  /\  Y  =/=  Z )  ->  Y  .<_  W )
251, 8, 9, 10, 11, 13, 14, 15dalem8 29784 . . 3  |-  ( ph  ->  Z  .<_  W )
2625adantr 452 . 2  |-  ( (
ph  /\  Y  =/=  Z )  ->  Z  .<_  W )
27 simpr 448 . 2  |-  ( (
ph  /\  Y  =/=  Z )  ->  Y  =/=  Z )
288, 9, 11, 122lplnj 29734 . 2  |-  ( ( K  e.  HL  /\  ( Y  e.  O  /\  Z  e.  O  /\  W  e.  ( LVols `  K ) )  /\  ( Y  .<_  W  /\  Z  .<_  W  /\  Y  =/=  Z ) )  ->  ( Y  .\/  Z )  =  W )
293, 5, 7, 16, 24, 26, 27, 28syl133anc 1207 1  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( Y  .\/  Z )  =  W )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   Basecbs 13396   lecple 13463   joincjn 14328   Latclat 14401   Atomscatm 29378   HLchlt 29465   LPlanesclpl 29606   LVolsclvol 29607
This theorem is referenced by:  dalem14  29791
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-llines 29612  df-lplanes 29613  df-lvols 29614
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