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Theorem cdlema2N 29249
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 28861 . . . . 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 2482 . . 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 225 . 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 28818 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
1614, 15syl 17 . . 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 28775 . . 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 203 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 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685    =/= wne 2448   class class class wbr 4025   ` cfv 5222  (class class class)co 5820   Basecbs 13143   lecple 13210   joincjn 14073   meetcmee 14074   0.cp0 14138   Atomscatm 28721   AtLatcal 28722   HLchlt 28808
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-poset 14075  df-plt 14087  df-lub 14103  df-glb 14104  df-join 14105  df-meet 14106  df-p0 14140  df-lat 14147  df-covers 28724  df-ats 28725  df-atl 28756  df-cvlat 28780  df-hlat 28809
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