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Theorem 2atmat0 33010
Description: The meet of two unequal lines (expressed as joins of atoms) is an atom or zero. (Contributed by NM, 2-Dec-2012.)
Hypotheses
Ref Expression
2atmatz.j  |-  .\/  =  ( join `  K )
2atmatz.m  |-  ./\  =  ( meet `  K )
2atmatz.z  |-  .0.  =  ( 0. `  K )
2atmatz.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
2atmat0  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  ( (
( P  .\/  Q
)  ./\  ( R  .\/  S ) )  e.  A  \/  ( ( P  .\/  Q ) 
./\  ( R  .\/  S ) )  =  .0.  ) )

Proof of Theorem 2atmat0
StepHypRef Expression
1 simpl 457 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A ) )
2 simpr1 994 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  R  e.  A )
3 simpr2 995 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  S  e.  A )
43orcd 392 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  ( S  e.  A  \/  S  =  .0.  ) )
5 simpr3 996 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  ( P  .\/  Q )  =/=  ( R  .\/  S ) )
6 2atmatz.j . . 3  |-  .\/  =  ( join `  K )
7 2atmatz.m . . 3  |-  ./\  =  ( meet `  K )
8 2atmatz.z . . 3  |-  .0.  =  ( 0. `  K )
9 2atmatz.a . . 3  |-  A  =  ( Atoms `  K )
106, 7, 8, 92at0mat0 33009 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  ( S  e.  A  \/  S  =  .0.  )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( (
( P  .\/  Q
)  ./\  ( R  .\/  S ) )  e.  A  \/  ( ( P  .\/  Q ) 
./\  ( R  .\/  S ) )  =  .0.  ) )
111, 2, 4, 5, 10syl13anc 1220 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  ( (
( P  .\/  Q
)  ./\  ( R  .\/  S ) )  e.  A  \/  ( ( P  .\/  Q ) 
./\  ( R  .\/  S ) )  =  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   ` cfv 5413  (class class class)co 6086   joincjn 15106   meetcmee 15107   0.cp0 15199   Atomscatm 32748   HLchlt 32835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-lat 15208  df-clat 15270  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836  df-llines 32982
This theorem is referenced by:  2atm  33011  trlval3  33671  cdleme22b  33825  cdlemg31b0N  34178  cdlemh  34301
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