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Theorem 2llnma1b 32816
Description: Generalization of 2llnma1 32817. (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 32394 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 1020 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  K  e.  Lat )
3 simp22 1033 . . . . . 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 32320 . . . . . 6  |-  ( P  e.  A  ->  P  e.  B )
73, 6syl 17 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  P  e.  B )
8 simp21 1032 . . . . 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 16016 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  P  .<_  ( P  .\/  X ) )
122, 7, 8, 11syl3anc 1232 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  P  .<_  ( P  .\/  X
) )
13 simp23 1034 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  Q  e.  A )
144, 5atbase 32320 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  B )
1513, 14syl 17 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  Q  e.  B )
164, 9, 10latlej1 16016 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  P  .<_  ( P  .\/  Q ) )
172, 7, 15, 16syl3anc 1232 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  P  .<_  ( P  .\/  Q
) )
184, 10latjcl 16007 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  ( P  .\/  X
)  e.  B )
192, 7, 8, 18syl3anc 1232 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  ( P  .\/  X )  e.  B )
20 simp1 999 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  K  e.  HL )
214, 10, 5hlatjcl 32397 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  B )
2220, 3, 13, 21syl3anc 1232 . . . . 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 16034 . . . . 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 1234 . . . 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 924 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  P  .<_  ( ( P  .\/  X )  ./\  ( P  .\/  Q ) ) )
27 hlatl 32391 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
28273ad2ant1 1020 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  K  e.  AtLat )
29 simp3 1001 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  -.  Q  .<_  ( P  .\/  X ) )
30 nbrne2 4415 . . . . . 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 16007 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  X )  e.  B  /\  Q  e.  B )  ->  (
( P  .\/  X
)  .\/  Q )  e.  B )
332, 19, 15, 32syl3anc 1232 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  (
( P  .\/  X
)  .\/  Q )  e.  B )
344, 9, 10latlej1 16016 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  X )  e.  B  /\  Q  e.  B )  ->  ( P  .\/  X )  .<_  ( ( P  .\/  X )  .\/  Q ) )
352, 19, 15, 34syl3anc 1232 . . . . . 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 16014 . . . . 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 32472 . . . . . 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 1196 . . . . 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 1255 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  (
( P  .\/  X
)  ./\  ( P  .\/  Q ) )  e.  A )
409, 5atcmp 32342 . . . 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 1232 . . 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 212 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  P  =  ( ( P 
.\/  X )  ./\  ( P  .\/  Q ) ) )
4342eqcomd 2412 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 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844    =/= wne 2600   class class class wbr 4397   ` cfv 5571  (class class class)co 6280   Basecbs 14843   lecple 14918   joincjn 15899   meetcmee 15900   Latclat 16001   Atomscatm 32294   AtLatcal 32295   HLchlt 32381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-preset 15883  df-poset 15901  df-plt 15914  df-lub 15930  df-glb 15931  df-join 15932  df-meet 15933  df-p0 15995  df-lat 16002  df-clat 16064  df-oposet 32207  df-ol 32209  df-oml 32210  df-covers 32297  df-ats 32298  df-atl 32329  df-cvlat 32353  df-hlat 32382
This theorem is referenced by:  2llnma1  32817  cdlemg4  33649  cdlemkfid1N  33953
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