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Theorem 2llnma1b 34459
Description: Generalization of 2llnma1 34460. (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 34037 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 1012 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  K  e.  Lat )
3 simp22 1025 . . . . . 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 33963 . . . . . 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 1024 . . . . 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 15538 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  P  .<_  ( P  .\/  X ) )
122, 7, 8, 11syl3anc 1223 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  P  .<_  ( P  .\/  X
) )
13 simp23 1026 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  Q  e.  A )
144, 5atbase 33963 . . . . . 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 15538 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  P  .<_  ( P  .\/  Q ) )
172, 7, 15, 16syl3anc 1223 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  P  .<_  ( P  .\/  Q
) )
184, 10latjcl 15529 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  ( P  .\/  X
)  e.  B )
192, 7, 8, 18syl3anc 1223 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  ( P  .\/  X )  e.  B )
20 simp1 991 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  K  e.  HL )
214, 10, 5hlatjcl 34040 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  B )
2220, 3, 13, 21syl3anc 1223 . . . . 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 15556 . . . . 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 1225 . . . 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 914 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  P  .<_  ( ( P  .\/  X )  ./\  ( P  .\/  Q ) ) )
27 hlatl 34034 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
28273ad2ant1 1012 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  K  e.  AtLat )
29 simp3 993 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  -.  Q  .<_  ( P  .\/  X ) )
30 nbrne2 4460 . . . . . 6  |-  ( ( P  .<_  ( P  .\/  X )  /\  -.  Q  .<_  ( P  .\/  X ) )  ->  P  =/=  Q )
3112, 29, 30syl2anc 661 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  P  =/=  Q )
324, 10latjcl 15529 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  X )  e.  B  /\  Q  e.  B )  ->  (
( P  .\/  X
)  .\/  Q )  e.  B )
332, 19, 15, 32syl3anc 1223 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  (
( P  .\/  X
)  .\/  Q )  e.  B )
344, 9, 10latlej1 15538 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  X )  e.  B  /\  Q  e.  B )  ->  ( P  .\/  X )  .<_  ( ( P  .\/  X )  .\/  Q ) )
352, 19, 15, 34syl3anc 1223 . . . . . 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 15536 . . . . 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 34115 . . . . . 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 1188 . . . . 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 1246 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  (
( P  .\/  X
)  ./\  ( P  .\/  Q ) )  e.  A )
409, 5atcmp 33985 . . . 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 1223 . . 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 2470 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 968    = wceq 1374    e. wcel 1762    =/= wne 2657   class class class wbr 4442   ` cfv 5581  (class class class)co 6277   Basecbs 14481   lecple 14553   joincjn 15422   meetcmee 15423   Latclat 15523   Atomscatm 33937   AtLatcal 33938   HLchlt 34024
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-poset 15424  df-plt 15436  df-lub 15452  df-glb 15453  df-join 15454  df-meet 15455  df-p0 15517  df-lat 15524  df-clat 15586  df-oposet 33850  df-ol 33852  df-oml 33853  df-covers 33940  df-ats 33941  df-atl 33972  df-cvlat 33996  df-hlat 34025
This theorem is referenced by:  2llnma1  34460  cdlemg4  35290  cdlemkfid1N  35594
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