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Theorem dalem13 30547
Description: Lemma for dalem14 30548. (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 30494 . . 3  |-  ( ph  ->  K  e.  HL )
32adantr 453 . 2  |-  ( (
ph  /\  Y  =/=  Z )  ->  K  e.  HL )
41dalemyeo 30503 . . 3  |-  ( ph  ->  Y  e.  O )
54adantr 453 . 2  |-  ( (
ph  /\  Y  =/=  Z )  ->  Y  e.  O )
61dalemzeo 30504 . . 3  |-  ( ph  ->  Z  e.  O )
76adantr 453 . 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 2438 . . 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 30543 . 2  |-  ( (
ph  /\  Y  =/=  Z )  ->  W  e.  ( LVols `  K )
)
171dalemkelat 30495 . . . . 5  |-  ( ph  ->  K  e.  Lat )
181, 11dalemyeb 30520 . . . . 5  |-  ( ph  ->  Y  e.  ( Base `  K ) )
191, 10dalemceb 30509 . . . . 5  |-  ( ph  ->  C  e.  ( Base `  K ) )
20 eqid 2438 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
2120, 8, 9latlej1 14494 . . . . 5  |-  ( ( K  e.  Lat  /\  Y  e.  ( Base `  K )  /\  C  e.  ( Base `  K
) )  ->  Y  .<_  ( Y  .\/  C
) )
2217, 18, 19, 21syl3anc 1185 . . . 4  |-  ( ph  ->  Y  .<_  ( Y  .\/  C ) )
2322, 15syl6breqr 4255 . . 3  |-  ( ph  ->  Y  .<_  W )
2423adantr 453 . 2  |-  ( (
ph  /\  Y  =/=  Z )  ->  Y  .<_  W )
251, 8, 9, 10, 11, 13, 14, 15dalem8 30541 . . 3  |-  ( ph  ->  Z  .<_  W )
2625adantr 453 . 2  |-  ( (
ph  /\  Y  =/=  Z )  ->  Z  .<_  W )
27 simpr 449 . 2  |-  ( (
ph  /\  Y  =/=  Z )  ->  Y  =/=  Z )
288, 9, 11, 122lplnj 30491 . 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 1208 1  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( Y  .\/  Z )  =  W )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   joincjn 14406   Latclat 14479   Atomscatm 30135   HLchlt 30222   LPlanesclpl 30363   LVolsclvol 30364
This theorem is referenced by:  dalem14  30548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-llines 30369  df-lplanes 30370  df-lvols 30371
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