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Theorem cdlema2N 29908
Description: A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 9-May-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlema2.b  |-  B  =  ( Base `  K
)
cdlema2.l  |-  .<_  =  ( le `  K )
cdlema2.j  |-  .\/  =  ( join `  K )
cdlema2.m  |-  ./\  =  ( meet `  K )
cdlema2.z  |-  .0.  =  ( 0. `  K )
cdlema2.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cdlema2N  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  -> 
( R  ./\  X
)  =  .0.  )

Proof of Theorem cdlema2N
StepHypRef Expression
1 simp3ll 1028 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  R  =/=  P )
2 simp3rl 1030 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  P  .<_  X )
3 simp3rr 1031 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  -.  Q  .<_  X )
4 simp3lr 1029 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  R  .<_  ( P  .\/  Q ) )
52, 3, 43jca 1134 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  -> 
( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )
6 cdlema2.b . . . . . 6  |-  B  =  ( Base `  K
)
7 cdlema2.l . . . . . 6  |-  .<_  =  ( le `  K )
8 cdlema2.j . . . . . 6  |-  .\/  =  ( join `  K )
9 cdlema2.a . . . . . 6  |-  A  =  ( Atoms `  K )
106, 7, 8, 9exatleN 29520 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q
) ) )  -> 
( R  .<_  X  <->  R  =  P ) )
115, 10syld3an3 1229 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  -> 
( R  .<_  X  <->  R  =  P ) )
1211necon3bbid 2586 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  -> 
( -.  R  .<_  X  <-> 
R  =/=  P ) )
131, 12mpbird 224 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  -.  R  .<_  X )
14 simp1l 981 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  K  e.  HL )
15 hlatl 29477 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
1614, 15syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  K  e.  AtLat )
17 simp23 992 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  R  e.  A )
18 simp1r 982 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  ->  X  e.  B )
19 cdlema2.m . . . 4  |-  ./\  =  ( meet `  K )
20 cdlema2.z . . . 4  |-  .0.  =  ( 0. `  K )
216, 7, 19, 20, 9atnle 29434 . . 3  |-  ( ( K  e.  AtLat  /\  R  e.  A  /\  X  e.  B )  ->  ( -.  R  .<_  X  <->  ( R  ./\ 
X )  =  .0.  ) )
2216, 17, 18, 21syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  -> 
( -.  R  .<_  X  <-> 
( R  ./\  X
)  =  .0.  )
)
2313, 22mpbid 202 1  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  (
( R  =/=  P  /\  R  .<_  ( P 
.\/  Q ) )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) )  -> 
( R  ./\  X
)  =  .0.  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2552   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   Basecbs 13398   lecple 13465   joincjn 14330   meetcmee 14331   0.cp0 14395   Atomscatm 29380   AtLatcal 29381   HLchlt 29467
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-poset 14332  df-plt 14344  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-p0 14397  df-lat 14404  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468
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