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Theorem 2llnma1b 33152
Description: Generalization of 2llnma1 33153. (Contributed by NM, 26-Apr-2013.)
Hypotheses
Ref Expression
2llnma1b.b  |-  B  =  ( Base `  K
)
2llnma1b.l  |-  .<_  =  ( le `  K )
2llnma1b.j  |-  .\/  =  ( join `  K )
2llnma1b.m  |-  ./\  =  ( meet `  K )
2llnma1b.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
2llnma1b  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  (
( P  .\/  X
)  ./\  ( P  .\/  Q ) )  =  P )

Proof of Theorem 2llnma1b
StepHypRef Expression
1 hllat 32730 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 1004 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  K  e.  Lat )
3 simp22 1017 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  P  e.  A )
4 2llnma1b.b . . . . . . 7  |-  B  =  ( Base `  K
)
5 2llnma1b.a . . . . . . 7  |-  A  =  ( Atoms `  K )
64, 5atbase 32656 . . . . . 6  |-  ( P  e.  A  ->  P  e.  B )
73, 6syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  P  e.  B )
8 simp21 1016 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  X  e.  B )
9 2llnma1b.l . . . . . 6  |-  .<_  =  ( le `  K )
10 2llnma1b.j . . . . . 6  |-  .\/  =  ( join `  K )
114, 9, 10latlej1 15226 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  P  .<_  ( P  .\/  X ) )
122, 7, 8, 11syl3anc 1213 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  P  .<_  ( P  .\/  X
) )
13 simp23 1018 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  Q  e.  A )
144, 5atbase 32656 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  B )
1513, 14syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  Q  e.  B )
164, 9, 10latlej1 15226 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  P  .<_  ( P  .\/  Q ) )
172, 7, 15, 16syl3anc 1213 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  P  .<_  ( P  .\/  Q
) )
184, 10latjcl 15217 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  ( P  .\/  X
)  e.  B )
192, 7, 8, 18syl3anc 1213 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  ( P  .\/  X )  e.  B )
20 simp1 983 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  K  e.  HL )
214, 10, 5hlatjcl 32733 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  B )
2220, 3, 13, 21syl3anc 1213 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  ( P  .\/  Q )  e.  B )
23 2llnma1b.m . . . . . 6  |-  ./\  =  ( meet `  K )
244, 9, 23latlem12 15244 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  e.  B  /\  ( P  .\/  X
)  e.  B  /\  ( P  .\/  Q )  e.  B ) )  ->  ( ( P 
.<_  ( P  .\/  X
)  /\  P  .<_  ( P  .\/  Q ) )  <->  P  .<_  ( ( P  .\/  X ) 
./\  ( P  .\/  Q ) ) ) )
252, 7, 19, 22, 24syl13anc 1215 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  (
( P  .<_  ( P 
.\/  X )  /\  P  .<_  ( P  .\/  Q ) )  <->  P  .<_  ( ( P  .\/  X
)  ./\  ( P  .\/  Q ) ) ) )
2612, 17, 25mpbi2and 907 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  P  .<_  ( ( P  .\/  X )  ./\  ( P  .\/  Q ) ) )
27 hlatl 32727 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
28273ad2ant1 1004 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  K  e.  AtLat )
29 simp3 985 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  -.  Q  .<_  ( P  .\/  X ) )
30 nbrne2 4307 . . . . . 6  |-  ( ( P  .<_  ( P  .\/  X )  /\  -.  Q  .<_  ( P  .\/  X ) )  ->  P  =/=  Q )
3112, 29, 30syl2anc 656 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  P  =/=  Q )
324, 10latjcl 15217 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  X )  e.  B  /\  Q  e.  B )  ->  (
( P  .\/  X
)  .\/  Q )  e.  B )
332, 19, 15, 32syl3anc 1213 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  (
( P  .\/  X
)  .\/  Q )  e.  B )
344, 9, 10latlej1 15226 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  X )  e.  B  /\  Q  e.  B )  ->  ( P  .\/  X )  .<_  ( ( P  .\/  X )  .\/  Q ) )
352, 19, 15, 34syl3anc 1213 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  ( P  .\/  X )  .<_  ( ( P  .\/  X )  .\/  Q ) )
364, 9, 2, 7, 19, 33, 12, 35lattrd 15224 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  P  .<_  ( ( P  .\/  X )  .\/  Q ) )
374, 9, 10, 23, 5cvrat3 32808 . . . . . 6  |-  ( ( K  e.  HL  /\  ( ( P  .\/  X )  e.  B  /\  P  e.  A  /\  Q  e.  A )
)  ->  ( ( P  =/=  Q  /\  -.  Q  .<_  ( P  .\/  X )  /\  P  .<_  ( ( P  .\/  X
)  .\/  Q )
)  ->  ( ( P  .\/  X )  ./\  ( P  .\/  Q ) )  e.  A ) )
38373impia 1179 . . . . 5  |-  ( ( K  e.  HL  /\  ( ( P  .\/  X )  e.  B  /\  P  e.  A  /\  Q  e.  A )  /\  ( P  =/=  Q  /\  -.  Q  .<_  ( P 
.\/  X )  /\  P  .<_  ( ( P 
.\/  X )  .\/  Q ) ) )  -> 
( ( P  .\/  X )  ./\  ( P  .\/  Q ) )  e.  A )
3920, 19, 3, 13, 31, 29, 36, 38syl133anc 1236 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  (
( P  .\/  X
)  ./\  ( P  .\/  Q ) )  e.  A )
409, 5atcmp 32678 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  (
( P  .\/  X
)  ./\  ( P  .\/  Q ) )  e.  A )  ->  ( P  .<_  ( ( P 
.\/  X )  ./\  ( P  .\/  Q ) )  <->  P  =  (
( P  .\/  X
)  ./\  ( P  .\/  Q ) ) ) )
4128, 3, 39, 40syl3anc 1213 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  ( P  .<_  ( ( P 
.\/  X )  ./\  ( P  .\/  Q ) )  <->  P  =  (
( P  .\/  X
)  ./\  ( P  .\/  Q ) ) ) )
4226, 41mpbid 210 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  P  =  ( ( P 
.\/  X )  ./\  ( P  .\/  Q ) ) )
4342eqcomd 2446 1  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  (
( P  .\/  X
)  ./\  ( P  .\/  Q ) )  =  P )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   Basecbs 14170   lecple 14241   joincjn 15110   meetcmee 15111   Latclat 15211   Atomscatm 32630   AtLatcal 32631   HLchlt 32717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-poset 15112  df-plt 15124  df-lub 15140  df-glb 15141  df-join 15142  df-meet 15143  df-p0 15205  df-lat 15212  df-clat 15274  df-oposet 32543  df-ol 32545  df-oml 32546  df-covers 32633  df-ats 32634  df-atl 32665  df-cvlat 32689  df-hlat 32718
This theorem is referenced by:  2llnma1  33153  cdlemg4  33983  cdlemkfid1N  34287
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