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Theorem dalem24 30556
Description: Lemma for dath 30595. Show that auxiliary atom  G is outside of plane  Y. (Contributed by NM, 2-Aug-2012.)
Hypotheses
Ref Expression
dalem.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 ) ) ) ) )
dalem.l  |-  .<_  =  ( le `  K )
dalem.j  |-  .\/  =  ( join `  K )
dalem.a  |-  A  =  ( Atoms `  K )
dalem.ps  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
dalem23.m  |-  ./\  =  ( meet `  K )
dalem23.o  |-  O  =  ( LPlanes `  K )
dalem23.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem23.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem23.g  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
Assertion
Ref Expression
dalem24  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  G  .<_  Y )

Proof of Theorem dalem24
StepHypRef Expression
1 dalem23.g . . . . 5  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
21oveq1i 6093 . . . 4  |-  ( G 
./\  Y )  =  ( ( ( c 
.\/  P )  ./\  ( d  .\/  S
) )  ./\  Y
)
3 dalem.ph . . . . . . . 8  |-  ( 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 ) ) ) ) )
43dalemkehl 30482 . . . . . . 7  |-  ( ph  ->  K  e.  HL )
5 hlol 30221 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OL )
64, 5syl 16 . . . . . 6  |-  ( ph  ->  K  e.  OL )
763ad2ant1 979 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  OL )
843ad2ant1 979 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
9 dalem.ps . . . . . . . 8  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
109dalemccea 30542 . . . . . . 7  |-  ( ps 
->  c  e.  A
)
11103ad2ant3 981 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
123dalempea 30485 . . . . . . 7  |-  ( ph  ->  P  e.  A )
13123ad2ant1 979 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  e.  A )
14 eqid 2438 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
15 dalem.j . . . . . . 7  |-  .\/  =  ( join `  K )
16 dalem.a . . . . . . 7  |-  A  =  ( Atoms `  K )
1714, 15, 16hlatjcl 30226 . . . . . 6  |-  ( ( K  e.  HL  /\  c  e.  A  /\  P  e.  A )  ->  ( c  .\/  P
)  e.  ( Base `  K ) )
188, 11, 13, 17syl3anc 1185 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  P
)  e.  ( Base `  K ) )
199dalemddea 30543 . . . . . . 7  |-  ( ps 
->  d  e.  A
)
20193ad2ant3 981 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  e.  A )
213dalemsea 30488 . . . . . . 7  |-  ( ph  ->  S  e.  A )
22213ad2ant1 979 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  e.  A )
2314, 15, 16hlatjcl 30226 . . . . . 6  |-  ( ( K  e.  HL  /\  d  e.  A  /\  S  e.  A )  ->  ( d  .\/  S
)  e.  ( Base `  K ) )
248, 20, 22, 23syl3anc 1185 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .\/  S
)  e.  ( Base `  K ) )
25 dalem23.o . . . . . . 7  |-  O  =  ( LPlanes `  K )
263, 25dalemyeb 30508 . . . . . 6  |-  ( ph  ->  Y  e.  ( Base `  K ) )
27263ad2ant1 979 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  e.  ( Base `  K ) )
28 dalem23.m . . . . . 6  |-  ./\  =  ( meet `  K )
2914, 28latmmdir 30095 . . . . 5  |-  ( ( K  e.  OL  /\  ( ( c  .\/  P )  e.  ( Base `  K )  /\  (
d  .\/  S )  e.  ( Base `  K
)  /\  Y  e.  ( Base `  K )
) )  ->  (
( ( c  .\/  P )  ./\  ( d  .\/  S ) )  ./\  Y )  =  ( ( ( c  .\/  P
)  ./\  Y )  ./\  ( ( d  .\/  S )  ./\  Y )
) )
307, 18, 24, 27, 29syl13anc 1187 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( c 
.\/  P )  ./\  ( d  .\/  S
) )  ./\  Y
)  =  ( ( ( c  .\/  P
)  ./\  Y )  ./\  ( ( d  .\/  S )  ./\  Y )
) )
312, 30syl5eq 2482 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  ./\  Y
)  =  ( ( ( c  .\/  P
)  ./\  Y )  ./\  ( ( d  .\/  S )  ./\  Y )
) )
3215, 16hlatjcom 30227 . . . . . . 7  |-  ( ( K  e.  HL  /\  c  e.  A  /\  P  e.  A )  ->  ( c  .\/  P
)  =  ( P 
.\/  c ) )
338, 11, 13, 32syl3anc 1185 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  P
)  =  ( P 
.\/  c ) )
3433oveq1d 6098 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  P )  ./\  Y )  =  ( ( P 
.\/  c )  ./\  Y ) )
35 dalem.l . . . . . . . 8  |-  .<_  =  ( le `  K )
36 dalem23.y . . . . . . . 8  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
373, 35, 15, 16, 25, 36dalemply 30513 . . . . . . 7  |-  ( ph  ->  P  .<_  Y )
38373ad2ant1 979 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  .<_  Y )
399dalem-ccly 30544 . . . . . . 7  |-  ( ps 
->  -.  c  .<_  Y )
40393ad2ant3 981 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  Y )
4114, 35, 15, 28, 162atjm 30304 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  c  e.  A  /\  Y  e.  ( Base `  K ) )  /\  ( P  .<_  Y  /\  -.  c  .<_  Y ) )  -> 
( ( P  .\/  c )  ./\  Y
)  =  P )
428, 13, 11, 27, 38, 40, 41syl132anc 1203 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( P  .\/  c )  ./\  Y
)  =  P )
4334, 42eqtrd 2470 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  P )  ./\  Y )  =  P )
4415, 16hlatjcom 30227 . . . . . . 7  |-  ( ( K  e.  HL  /\  d  e.  A  /\  S  e.  A )  ->  ( d  .\/  S
)  =  ( S 
.\/  d ) )
458, 20, 22, 44syl3anc 1185 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .\/  S
)  =  ( S 
.\/  d ) )
4645oveq1d 6098 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( d  .\/  S )  ./\  Y )  =  ( ( S 
.\/  d )  ./\  Y ) )
47 dalem23.z . . . . . . . 8  |-  Z  =  ( ( S  .\/  T )  .\/  U )
483, 35, 15, 16, 47dalemsly 30514 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z )  ->  S  .<_  Y )
49483adant3 978 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  .<_  Y )
509dalem-ddly 30545 . . . . . . 7  |-  ( ps 
->  -.  d  .<_  Y )
51503ad2ant3 981 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  d  .<_  Y )
5214, 35, 15, 28, 162atjm 30304 . . . . . 6  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  d  e.  A  /\  Y  e.  ( Base `  K ) )  /\  ( S  .<_  Y  /\  -.  d  .<_  Y ) )  -> 
( ( S  .\/  d )  ./\  Y
)  =  S )
538, 22, 20, 27, 49, 51, 52syl132anc 1203 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( S  .\/  d )  ./\  Y
)  =  S )
5446, 53eqtrd 2470 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( d  .\/  S )  ./\  Y )  =  S )
5543, 54oveq12d 6101 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( c 
.\/  P )  ./\  Y )  ./\  ( (
d  .\/  S )  ./\  Y ) )  =  ( P  ./\  S
) )
563, 35, 15, 16, 25, 36dalempnes 30510 . . . . 5  |-  ( ph  ->  P  =/=  S )
57 hlatl 30220 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  AtLat )
584, 57syl 16 . . . . . 6  |-  ( ph  ->  K  e.  AtLat )
59 eqid 2438 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
6028, 59, 16atnem0 30178 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  S  e.  A )  ->  ( P  =/=  S  <->  ( P  ./\ 
S )  =  ( 0. `  K ) ) )
6158, 12, 21, 60syl3anc 1185 . . . . 5  |-  ( ph  ->  ( P  =/=  S  <->  ( P  ./\  S )  =  ( 0. `  K ) ) )
6256, 61mpbid 203 . . . 4  |-  ( ph  ->  ( P  ./\  S
)  =  ( 0.
`  K ) )
63623ad2ant1 979 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  ./\  S
)  =  ( 0.
`  K ) )
6431, 55, 633eqtrd 2474 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  ./\  Y
)  =  ( 0.
`  K ) )
65583ad2ant1 979 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  AtLat )
663, 35, 15, 16, 9, 28, 25, 36, 47, 1dalem23 30555 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
6714, 35, 28, 59, 16atnle 30177 . . 3  |-  ( ( K  e.  AtLat  /\  G  e.  A  /\  Y  e.  ( Base `  K
) )  ->  ( -.  G  .<_  Y  <->  ( G  ./\ 
Y )  =  ( 0. `  K ) ) )
6865, 66, 27, 67syl3anc 1185 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( -.  G  .<_  Y  <-> 
( G  ./\  Y
)  =  ( 0.
`  K ) ) )
6964, 68mpbird 225 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  G  .<_  Y )
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 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   meetcmee 14404   0.cp0 14468   OLcol 30034   Atomscatm 30123   AtLatcal 30124   HLchlt 30210   LPlanesclpl 30351
This theorem is referenced by:  dalem27  30558  dalem30  30561  dalem54  30585
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  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 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-llines 30357  df-lplanes 30358
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