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Theorem 2llnma1b 29975
Description: Generalization of 2llnma1 29976. (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 29553 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 976 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  K  e.  Lat )
3 simp22 989 . . . . . 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 29479 . . . . . 6  |-  ( P  e.  A  ->  P  e.  B )
73, 6syl 15 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  P  e.  B )
8 simp21 988 . . . . 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 14166 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  P  .<_  ( P  .\/  X ) )
122, 7, 8, 11syl3anc 1182 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  P  .<_  ( P  .\/  X
) )
13 simp23 990 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  Q  e.  A )
144, 5atbase 29479 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  B )
1513, 14syl 15 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  Q  e.  B )
164, 9, 10latlej1 14166 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  P  .<_  ( P  .\/  Q ) )
172, 7, 15, 16syl3anc 1182 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  P  .<_  ( P  .\/  Q
) )
184, 10latjcl 14156 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  ( P  .\/  X
)  e.  B )
192, 7, 8, 18syl3anc 1182 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  ( P  .\/  X )  e.  B )
20 simp1 955 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  K  e.  HL )
214, 10, 5hlatjcl 29556 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  B )
2220, 3, 13, 21syl3anc 1182 . . . . 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 14184 . . . . 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 1184 . . . 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 887 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  P  .<_  ( ( P  .\/  X )  ./\  ( P  .\/  Q ) ) )
27 hlatl 29550 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
28273ad2ant1 976 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  K  e.  AtLat )
29 simp3 957 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  -.  Q  .<_  ( P  .\/  X ) )
30 nbrne2 4041 . . . . . 6  |-  ( ( P  .<_  ( P  .\/  X )  /\  -.  Q  .<_  ( P  .\/  X ) )  ->  P  =/=  Q )
3112, 29, 30syl2anc 642 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  P  =/=  Q )
324, 10latjcl 14156 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  X )  e.  B  /\  Q  e.  B )  ->  (
( P  .\/  X
)  .\/  Q )  e.  B )
332, 19, 15, 32syl3anc 1182 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  (
( P  .\/  X
)  .\/  Q )  e.  B )
344, 9, 10latlej1 14166 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  X )  e.  B  /\  Q  e.  B )  ->  ( P  .\/  X )  .<_  ( ( P  .\/  X )  .\/  Q ) )
352, 19, 15, 34syl3anc 1182 . . . . . 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 14164 . . . . 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 29631 . . . . . 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 1148 . . . . 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 1205 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  (
( P  .\/  X
)  ./\  ( P  .\/  Q ) )  e.  A )
409, 5atcmp 29501 . . . 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 1182 . . 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 201 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  Q  .<_  ( P  .\/  X
) )  ->  P  =  ( ( P 
.\/  X )  ./\  ( P  .\/  Q ) ) )
4342eqcomd 2288 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 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 29453   AtLatcal 29454   HLchlt 29540
This theorem is referenced by:  2llnma1  29976  cdlemg4  30806  cdlemkfid1N  31110
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 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541
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