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Theorem 2atjm 29007
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 28926 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 976 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  K  e.  Lat )
3 simp21 988 . . . . . 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 28852 . . . . . 6  |-  ( P  e.  A  ->  P  e.  B )
73, 6syl 15 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  e.  B )
8 simp22 989 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  Q  e.  A )
94, 5atbase 28852 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  B )
108, 9syl 15 . . . . 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 14166 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  P  .<_  ( P  .\/  Q ) )
142, 7, 10, 13syl3anc 1182 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  .<_  ( P  .\/  Q ) )
15 simp3l 983 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  .<_  X )
16 simp1 955 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  K  e.  HL )
174, 12, 5hlatjcl 28929 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  B )
1816, 3, 8, 17syl3anc 1182 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( P  .\/  Q
)  e.  B )
19 simp23 990 . . . . 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 14184 . . . . 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 1184 . . . 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 887 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  .<_  ( ( P 
.\/  Q )  ./\  X ) )
24 hlatl 28923 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
25243ad2ant1 976 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  K  e.  AtLat )
264, 20latmcom 14181 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  X  e.  B )  ->  (
( P  .\/  Q
)  ./\  X )  =  ( X  ./\  ( P  .\/  Q ) ) )
272, 18, 19, 26syl3anc 1182 . . . . 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 1132 . . . . . 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 4041 . . . . . . 7  |-  ( ( P  .<_  X  /\  -.  Q  .<_  X )  ->  P  =/=  Q
)
30293ad2ant3 978 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  =/=  Q )
31 simp3r 984 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  -.  Q  .<_  X )
324, 12latjcl 14156 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Q  e.  B )  ->  ( X  .\/  Q
)  e.  B )
332, 19, 10, 32syl3anc 1182 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( X  .\/  Q
)  e.  B )
344, 11, 12latlej1 14166 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Q  e.  B )  ->  X  .<_  ( X  .\/  Q ) )
352, 19, 10, 34syl3anc 1182 . . . . . . 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 14164 . . . . . 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 29004 . . . . . . 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 418 . . . . . 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 1189 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( X  ./\  ( P  .\/  Q ) )  e.  A )
4027, 39eqeltrd 2357 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( ( P  .\/  Q )  ./\  X )  e.  A )
4111, 5atcmp 28874 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  (
( P  .\/  Q
)  ./\  X )  e.  A )  ->  ( P  .<_  ( ( P 
.\/  Q )  ./\  X )  <->  P  =  (
( P  .\/  Q
)  ./\  X )
) )
4225, 3, 40, 41syl3anc 1182 . . 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 201 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  =  ( ( P  .\/  Q )  ./\  X ) )
4443eqcomd 2288 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Latclat 14151   Atomscatm 28826   AtLatcal 28827   HLchlt 28913
This theorem is referenced by:  atbtwn  29008  dalem24  29259  dalem25  29260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 28739  df-ol 28741  df-oml 28742  df-covers 28829  df-ats 28830  df-atl 28861  df-cvlat 28885  df-hlat 28914
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