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Theorem dalem22 30566
Description: Lemma for dath 30607. Show that lines  c d and  P S determine a plane. (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
) ) ) )
dalem22.o  |-  O  =  ( LPlanes `  K )
dalem22.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem22.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
Assertion
Ref Expression
dalem22  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  d )  .\/  ( P  .\/  S ) )  e.  O )

Proof of Theorem dalem22
StepHypRef Expression
1 dalem.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 dalem.l . . 3  |-  .<_  =  ( le `  K )
3 dalem.j . . 3  |-  .\/  =  ( join `  K )
4 dalem.a . . 3  |-  A  =  ( Atoms `  K )
5 dalem.ps . . 3  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
6 eqid 2438 . . 3  |-  ( meet `  K )  =  (
meet `  K )
7 dalem22.o . . 3  |-  O  =  ( LPlanes `  K )
8 dalem22.y . . 3  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
9 dalem22.z . . 3  |-  Z  =  ( ( S  .\/  T )  .\/  U )
101, 2, 3, 4, 5, 6, 7, 8, 9dalem21 30565 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  d ) ( meet `  K ) ( P 
.\/  S ) )  e.  A )
111dalemkehl 30494 . . . . 5  |-  ( ph  ->  K  e.  HL )
1211adantr 453 . . . 4  |-  ( (
ph  /\  ps )  ->  K  e.  HL )
131, 2, 3, 4, 5dalemcjden 30563 . . . 4  |-  ( (
ph  /\  ps )  ->  ( c  .\/  d
)  e.  ( LLines `  K ) )
141, 2, 3, 4, 7, 8dalempjsen 30524 . . . . 5  |-  ( ph  ->  ( P  .\/  S
)  e.  ( LLines `  K ) )
1514adantr 453 . . . 4  |-  ( (
ph  /\  ps )  ->  ( P  .\/  S
)  e.  ( LLines `  K ) )
16 eqid 2438 . . . . 5  |-  ( LLines `  K )  =  (
LLines `  K )
173, 6, 4, 16, 72llnmj 30431 . . . 4  |-  ( ( K  e.  HL  /\  ( c  .\/  d
)  e.  ( LLines `  K )  /\  ( P  .\/  S )  e.  ( LLines `  K )
)  ->  ( (
( c  .\/  d
) ( meet `  K
) ( P  .\/  S ) )  e.  A  <->  ( ( c  .\/  d
)  .\/  ( P  .\/  S ) )  e.  O ) )
1812, 13, 15, 17syl3anc 1185 . . 3  |-  ( (
ph  /\  ps )  ->  ( ( ( c 
.\/  d ) (
meet `  K )
( P  .\/  S
) )  e.  A  <->  ( ( c  .\/  d
)  .\/  ( P  .\/  S ) )  e.  O ) )
19183adant2 977 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( c 
.\/  d ) (
meet `  K )
( P  .\/  S
) )  e.  A  <->  ( ( c  .\/  d
)  .\/  ( P  .\/  S ) )  e.  O ) )
2010, 19mpbid 203 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  d )  .\/  ( P  .\/  S ) )  e.  O )
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   meetcmee 14407   Atomscatm 30135   HLchlt 30222   LLinesclln 30362   LPlanesclpl 30363
This theorem is referenced by:  dalem23  30567
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-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
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