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Theorem 2atjm 33397
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 33316 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 1009 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  K  e.  Lat )
3 simp21 1021 . . . . . 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 33242 . . . . . 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 1022 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  Q  e.  A )
94, 5atbase 33242 . . . . . 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 15334 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  P  .<_  ( P  .\/  Q ) )
142, 7, 10, 13syl3anc 1219 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  .<_  ( P  .\/  Q ) )
15 simp3l 1016 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  .<_  X )
16 simp1 988 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  K  e.  HL )
174, 12, 5hlatjcl 33319 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  B )
1816, 3, 8, 17syl3anc 1219 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( P  .\/  Q
)  e.  B )
19 simp23 1023 . . . . 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 15352 . . . . 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 1221 . . . 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 912 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  .<_  ( ( P 
.\/  Q )  ./\  X ) )
24 hlatl 33313 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
25243ad2ant1 1009 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  K  e.  AtLat )
264, 20latmcom 15349 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  X  e.  B )  ->  (
( P  .\/  Q
)  ./\  X )  =  ( X  ./\  ( P  .\/  Q ) ) )
272, 18, 19, 26syl3anc 1219 . . . . 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 1168 . . . . . 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 4410 . . . . . . 7  |-  ( ( P  .<_  X  /\  -.  Q  .<_  X )  ->  P  =/=  Q
)
30293ad2ant3 1011 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  =/=  Q )
31 simp3r 1017 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  -.  Q  .<_  X )
324, 12latjcl 15325 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Q  e.  B )  ->  ( X  .\/  Q
)  e.  B )
332, 19, 10, 32syl3anc 1219 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( X  .\/  Q
)  e.  B )
344, 11, 12latlej1 15334 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Q  e.  B )  ->  X  .<_  ( X  .\/  Q ) )
352, 19, 10, 34syl3anc 1219 . . . . . . 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 15332 . . . . . 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 33394 . . . . . . 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 429 . . . . . 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 1226 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( X  ./\  ( P  .\/  Q ) )  e.  A )
4027, 39eqeltrd 2539 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( ( P  .\/  Q )  ./\  X )  e.  A )
4111, 5atcmp 33264 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  (
( P  .\/  Q
)  ./\  X )  e.  A )  ->  ( P  .<_  ( ( P 
.\/  Q )  ./\  X )  <->  P  =  (
( P  .\/  Q
)  ./\  X )
) )
4225, 3, 40, 41syl3anc 1219 . . 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 210 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  =  ( ( P  .\/  Q )  ./\  X ) )
4443eqcomd 2459 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   Basecbs 14278   lecple 14349   joincjn 15218   meetcmee 15219   Latclat 15319   Atomscatm 33216   AtLatcal 33217   HLchlt 33303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-poset 15220  df-plt 15232  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-p0 15313  df-lat 15320  df-clat 15382  df-oposet 33129  df-ol 33131  df-oml 33132  df-covers 33219  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304
This theorem is referenced by:  atbtwn  33398  dalem24  33649  dalem25  33650
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