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Theorem 2atjm 30243
Description: The meet of a line (expressed with 2 atoms) and a lattice element. (Contributed by NM, 30-Jul-2012.)
Hypotheses
Ref Expression
2atjm.b  |-  B  =  ( Base `  K
)
2atjm.l  |-  .<_  =  ( le `  K )
2atjm.j  |-  .\/  =  ( join `  K )
2atjm.m  |-  ./\  =  ( meet `  K )
2atjm.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
2atjm  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( ( P  .\/  Q )  ./\  X )  =  P )

Proof of Theorem 2atjm
StepHypRef Expression
1 hllat 30162 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 979 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  K  e.  Lat )
3 simp21 991 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  e.  A )
4 2atjm.b . . . . . . 7  |-  B  =  ( Base `  K
)
5 2atjm.a . . . . . . 7  |-  A  =  ( Atoms `  K )
64, 5atbase 30088 . . . . . 6  |-  ( P  e.  A  ->  P  e.  B )
73, 6syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  e.  B )
8 simp22 992 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  Q  e.  A )
94, 5atbase 30088 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  B )
108, 9syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  Q  e.  B )
11 2atjm.l . . . . . 6  |-  .<_  =  ( le `  K )
12 2atjm.j . . . . . 6  |-  .\/  =  ( join `  K )
134, 11, 12latlej1 14490 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  P  .<_  ( P  .\/  Q ) )
142, 7, 10, 13syl3anc 1185 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  .<_  ( P  .\/  Q ) )
15 simp3l 986 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  .<_  X )
16 simp1 958 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  K  e.  HL )
174, 12, 5hlatjcl 30165 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  B )
1816, 3, 8, 17syl3anc 1185 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( P  .\/  Q
)  e.  B )
19 simp23 993 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  X  e.  B )
20 2atjm.m . . . . . 6  |-  ./\  =  ( meet `  K )
214, 11, 20latlem12 14508 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  e.  B  /\  ( P  .\/  Q
)  e.  B  /\  X  e.  B )
)  ->  ( ( P  .<_  ( P  .\/  Q )  /\  P  .<_  X )  <->  P  .<_  ( ( P  .\/  Q ) 
./\  X ) ) )
222, 7, 18, 19, 21syl13anc 1187 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( ( P  .<_  ( P  .\/  Q )  /\  P  .<_  X )  <-> 
P  .<_  ( ( P 
.\/  Q )  ./\  X ) ) )
2314, 15, 22mpbi2and 889 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  .<_  ( ( P 
.\/  Q )  ./\  X ) )
24 hlatl 30159 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
25243ad2ant1 979 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  K  e.  AtLat )
264, 20latmcom 14505 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  X  e.  B )  ->  (
( P  .\/  Q
)  ./\  X )  =  ( X  ./\  ( P  .\/  Q ) ) )
272, 18, 19, 26syl3anc 1185 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( ( P  .\/  Q )  ./\  X )  =  ( X  ./\  ( P  .\/  Q ) ) )
2819, 3, 83jca 1135 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )
29 nbrne2 4231 . . . . . . 7  |-  ( ( P  .<_  X  /\  -.  Q  .<_  X )  ->  P  =/=  Q
)
30293ad2ant3 981 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  =/=  Q )
31 simp3r 987 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  -.  Q  .<_  X )
324, 12latjcl 14480 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Q  e.  B )  ->  ( X  .\/  Q
)  e.  B )
332, 19, 10, 32syl3anc 1185 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( X  .\/  Q
)  e.  B )
344, 11, 12latlej1 14490 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Q  e.  B )  ->  X  .<_  ( X  .\/  Q ) )
352, 19, 10, 34syl3anc 1185 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  X  .<_  ( X  .\/  Q ) )
364, 11, 2, 7, 19, 33, 15, 35lattrd 14488 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  .<_  ( X  .\/  Q ) )
374, 11, 12, 20, 5cvrat3 30240 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( P  =/=  Q  /\  -.  Q  .<_  X  /\  P  .<_  ( X  .\/  Q ) )  ->  ( X  ./\  ( P  .\/  Q ) )  e.  A
) )
3837imp 420 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  ( P  =/=  Q  /\  -.  Q  .<_  X  /\  P  .<_  ( X  .\/  Q
) ) )  -> 
( X  ./\  ( P  .\/  Q ) )  e.  A )
3916, 28, 30, 31, 36, 38syl23anc 1192 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( X  ./\  ( P  .\/  Q ) )  e.  A )
4027, 39eqeltrd 2511 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( ( P  .\/  Q )  ./\  X )  e.  A )
4111, 5atcmp 30110 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  (
( P  .\/  Q
)  ./\  X )  e.  A )  ->  ( P  .<_  ( ( P 
.\/  Q )  ./\  X )  <->  P  =  (
( P  .\/  Q
)  ./\  X )
) )
4225, 3, 40, 41syl3anc 1185 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( P  .<_  ( ( P  .\/  Q ) 
./\  X )  <->  P  =  ( ( P  .\/  Q )  ./\  X )
) )
4323, 42mpbid 203 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  =  ( ( P  .\/  Q )  ./\  X ) )
4443eqcomd 2442 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( ( P  .\/  Q )  ./\  X )  =  P )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   Basecbs 13470   lecple 13537   joincjn 14402   meetcmee 14403   Latclat 14475   Atomscatm 30062   AtLatcal 30063   HLchlt 30149
This theorem is referenced by:  atbtwn  30244  dalem24  30495  dalem25  30496
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-undef 6544  df-riota 6550  df-poset 14404  df-plt 14416  df-lub 14432  df-glb 14433  df-join 14434  df-meet 14435  df-p0 14469  df-lat 14476  df-clat 14538  df-oposet 29975  df-ol 29977  df-oml 29978  df-covers 30065  df-ats 30066  df-atl 30097  df-cvlat 30121  df-hlat 30150
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