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Theorem dalem24 33680
Description: Lemma for dath 33719. 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 6211 . . . 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 33606 . . . . . . 7  |-  ( ph  ->  K  e.  HL )
5 hlol 33345 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OL )
64, 5syl 16 . . . . . 6  |-  ( ph  ->  K  e.  OL )
763ad2ant1 1009 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  OL )
843ad2ant1 1009 . . . . . 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 33666 . . . . . . 7  |-  ( ps 
->  c  e.  A
)
11103ad2ant3 1011 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
123dalempea 33609 . . . . . . 7  |-  ( ph  ->  P  e.  A )
13123ad2ant1 1009 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  e.  A )
14 eqid 2454 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
15 dalem.j . . . . . . 7  |-  .\/  =  ( join `  K )
16 dalem.a . . . . . . 7  |-  A  =  ( Atoms `  K )
1714, 15, 16hlatjcl 33350 . . . . . 6  |-  ( ( K  e.  HL  /\  c  e.  A  /\  P  e.  A )  ->  ( c  .\/  P
)  e.  ( Base `  K ) )
188, 11, 13, 17syl3anc 1219 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  P
)  e.  ( Base `  K ) )
199dalemddea 33667 . . . . . . 7  |-  ( ps 
->  d  e.  A
)
20193ad2ant3 1011 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  e.  A )
213dalemsea 33612 . . . . . . 7  |-  ( ph  ->  S  e.  A )
22213ad2ant1 1009 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  e.  A )
2314, 15, 16hlatjcl 33350 . . . . . 6  |-  ( ( K  e.  HL  /\  d  e.  A  /\  S  e.  A )  ->  ( d  .\/  S
)  e.  ( Base `  K ) )
248, 20, 22, 23syl3anc 1219 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .\/  S
)  e.  ( Base `  K ) )
25 dalem23.o . . . . . . 7  |-  O  =  ( LPlanes `  K )
263, 25dalemyeb 33632 . . . . . 6  |-  ( ph  ->  Y  e.  ( Base `  K ) )
27263ad2ant1 1009 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  e.  ( Base `  K ) )
28 dalem23.m . . . . . 6  |-  ./\  =  ( meet `  K )
2914, 28latmmdir 33219 . . . . 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 1221 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( c 
.\/  P )  ./\  ( d  .\/  S
) )  ./\  Y
)  =  ( ( ( c  .\/  P
)  ./\  Y )  ./\  ( ( d  .\/  S )  ./\  Y )
) )
312, 30syl5eq 2507 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  ./\  Y
)  =  ( ( ( c  .\/  P
)  ./\  Y )  ./\  ( ( d  .\/  S )  ./\  Y )
) )
3215, 16hlatjcom 33351 . . . . . . 7  |-  ( ( K  e.  HL  /\  c  e.  A  /\  P  e.  A )  ->  ( c  .\/  P
)  =  ( P 
.\/  c ) )
338, 11, 13, 32syl3anc 1219 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  P
)  =  ( P 
.\/  c ) )
3433oveq1d 6216 . . . . 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 33637 . . . . . . 7  |-  ( ph  ->  P  .<_  Y )
38373ad2ant1 1009 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  .<_  Y )
399dalem-ccly 33668 . . . . . . 7  |-  ( ps 
->  -.  c  .<_  Y )
40393ad2ant3 1011 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  Y )
4114, 35, 15, 28, 162atjm 33428 . . . . . 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 1237 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( P  .\/  c )  ./\  Y
)  =  P )
4334, 42eqtrd 2495 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  P )  ./\  Y )  =  P )
4415, 16hlatjcom 33351 . . . . . . 7  |-  ( ( K  e.  HL  /\  d  e.  A  /\  S  e.  A )  ->  ( d  .\/  S
)  =  ( S 
.\/  d ) )
458, 20, 22, 44syl3anc 1219 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .\/  S
)  =  ( S 
.\/  d ) )
4645oveq1d 6216 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( d  .\/  S )  ./\  Y )  =  ( ( S 
.\/  d )  ./\  Y ) )
47 dalem23.z . . . . . . . 8  |-  Z  =  ( ( S  .\/  T )  .\/  U )
483, 35, 15, 16, 47dalemsly 33638 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z )  ->  S  .<_  Y )
49483adant3 1008 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  .<_  Y )
509dalem-ddly 33669 . . . . . . 7  |-  ( ps 
->  -.  d  .<_  Y )
51503ad2ant3 1011 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  d  .<_  Y )
5214, 35, 15, 28, 162atjm 33428 . . . . . 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 1237 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( S  .\/  d )  ./\  Y
)  =  S )
5446, 53eqtrd 2495 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( d  .\/  S )  ./\  Y )  =  S )
5543, 54oveq12d 6219 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( c 
.\/  P )  ./\  Y )  ./\  ( (
d  .\/  S )  ./\  Y ) )  =  ( P  ./\  S
) )
563, 35, 15, 16, 25, 36dalempnes 33634 . . . . 5  |-  ( ph  ->  P  =/=  S )
57 hlatl 33344 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  AtLat )
584, 57syl 16 . . . . . 6  |-  ( ph  ->  K  e.  AtLat )
59 eqid 2454 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
6028, 59, 16atnem0 33302 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  S  e.  A )  ->  ( P  =/=  S  <->  ( P  ./\ 
S )  =  ( 0. `  K ) ) )
6158, 12, 21, 60syl3anc 1219 . . . . 5  |-  ( ph  ->  ( P  =/=  S  <->  ( P  ./\  S )  =  ( 0. `  K ) ) )
6256, 61mpbid 210 . . . 4  |-  ( ph  ->  ( P  ./\  S
)  =  ( 0.
`  K ) )
63623ad2ant1 1009 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  ./\  S
)  =  ( 0.
`  K ) )
6431, 55, 633eqtrd 2499 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  ./\  Y
)  =  ( 0.
`  K ) )
65583ad2ant1 1009 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  AtLat )
663, 35, 15, 16, 9, 28, 25, 36, 47, 1dalem23 33679 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
6714, 35, 28, 59, 16atnle 33301 . . 3  |-  ( ( K  e.  AtLat  /\  G  e.  A  /\  Y  e.  ( Base `  K
) )  ->  ( -.  G  .<_  Y  <->  ( G  ./\ 
Y )  =  ( 0. `  K ) ) )
6865, 66, 27, 67syl3anc 1219 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( -.  G  .<_  Y  <-> 
( G  ./\  Y
)  =  ( 0.
`  K ) ) )
6964, 68mpbird 232 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  G  .<_  Y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   Basecbs 14293   lecple 14365   joincjn 15234   meetcmee 15235   0.cp0 15327   OLcol 33158   Atomscatm 33247   AtLatcal 33248   HLchlt 33334   LPlanesclpl 33475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-lat 15336  df-clat 15398  df-oposet 33160  df-ol 33162  df-oml 33163  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335  df-llines 33481  df-lplanes 33482
This theorem is referenced by:  dalem27  33682  dalem30  33685  dalem54  33709
  Copyright terms: Public domain W3C validator