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Theorem dalawlem2 34885
Description: Lemma for dalaw 34899. Utility lemma that breaks  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) ) into a join of two pieces. (Contributed by NM, 6-Oct-2012.)
Hypotheses
Ref Expression
dalawlem.l  |-  .<_  =  ( le `  K )
dalawlem.j  |-  .\/  =  ( join `  K )
dalawlem.m  |-  ./\  =  ( meet `  K )
dalawlem.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
dalawlem2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( ( ( P  .\/  Q ) 
.\/  T )  ./\  S )  .\/  ( ( ( P  .\/  Q
)  .\/  S )  ./\  T ) ) )

Proof of Theorem dalawlem2
StepHypRef Expression
1 simp1 996 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  ->  K  e.  HL )
2 hllat 34377 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  ->  K  e.  Lat )
4 simp2l 1022 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  ->  P  e.  A )
5 simp2r 1023 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  ->  Q  e.  A )
6 eqid 2467 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
7 dalawlem.j . . . . . . 7  |-  .\/  =  ( join `  K )
8 dalawlem.a . . . . . . 7  |-  A  =  ( Atoms `  K )
96, 7, 8hlatjcl 34380 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
101, 4, 5, 9syl3anc 1228 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
11 simp3r 1025 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  ->  T  e.  A )
126, 8atbase 34303 . . . . . 6  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
1311, 12syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  ->  T  e.  ( Base `  K ) )
14 dalawlem.l . . . . . 6  |-  .<_  =  ( le `  K )
156, 14, 7latlej1 15550 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
)  ->  ( P  .\/  Q )  .<_  ( ( P  .\/  Q ) 
.\/  T ) )
163, 10, 13, 15syl3anc 1228 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( P  .\/  Q
)  .<_  ( ( P 
.\/  Q )  .\/  T ) )
17 simp3l 1024 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  ->  S  e.  A )
186, 8atbase 34303 . . . . . 6  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
1917, 18syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  ->  S  e.  ( Base `  K ) )
206, 14, 7latlej1 15550 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  ( P  .\/  Q )  .<_  ( ( P  .\/  Q ) 
.\/  S ) )
213, 10, 19, 20syl3anc 1228 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( P  .\/  Q
)  .<_  ( ( P 
.\/  Q )  .\/  S ) )
226, 7latjcl 15541 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  T )  e.  ( Base `  K ) )
233, 10, 13, 22syl3anc 1228 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( ( P  .\/  Q )  .\/  T )  e.  ( Base `  K
) )
246, 7latjcl 15541 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  S )  e.  ( Base `  K ) )
253, 10, 19, 24syl3anc 1228 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( ( P  .\/  Q )  .\/  S )  e.  ( Base `  K
) )
26 dalawlem.m . . . . . 6  |-  ./\  =  ( meet `  K )
276, 14, 26latlem12 15568 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  e.  ( Base `  K )  /\  (
( P  .\/  Q
)  .\/  T )  e.  ( Base `  K
)  /\  ( ( P  .\/  Q )  .\/  S )  e.  ( Base `  K ) ) )  ->  ( ( ( P  .\/  Q ) 
.<_  ( ( P  .\/  Q )  .\/  T )  /\  ( P  .\/  Q )  .<_  ( ( P  .\/  Q )  .\/  S ) )  <->  ( P  .\/  Q )  .<_  ( ( ( P  .\/  Q
)  .\/  T )  ./\  ( ( P  .\/  Q )  .\/  S ) ) ) )
283, 10, 23, 25, 27syl13anc 1230 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( ( ( P 
.\/  Q )  .<_  ( ( P  .\/  Q )  .\/  T )  /\  ( P  .\/  Q )  .<_  ( ( P  .\/  Q )  .\/  S ) )  <->  ( P  .\/  Q )  .<_  ( ( ( P  .\/  Q
)  .\/  T )  ./\  ( ( P  .\/  Q )  .\/  S ) ) ) )
2916, 21, 28mpbi2and 919 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( P  .\/  Q
)  .<_  ( ( ( P  .\/  Q ) 
.\/  T )  ./\  ( ( P  .\/  Q )  .\/  S ) ) )
306, 26latmcl 15542 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  .\/  T )  e.  ( Base `  K
)  /\  ( ( P  .\/  Q )  .\/  S )  e.  ( Base `  K ) )  -> 
( ( ( P 
.\/  Q )  .\/  T )  ./\  ( ( P  .\/  Q )  .\/  S ) )  e.  (
Base `  K )
)
313, 23, 25, 30syl3anc 1228 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( ( ( P 
.\/  Q )  .\/  T )  ./\  ( ( P  .\/  Q )  .\/  S ) )  e.  (
Base `  K )
)
326, 7, 8hlatjcl 34380 . . . . 5  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
331, 17, 11, 32syl3anc 1228 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( S  .\/  T
)  e.  ( Base `  K ) )
346, 14, 26latmlem1 15571 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  e.  ( Base `  K )  /\  (
( ( P  .\/  Q )  .\/  T ) 
./\  ( ( P 
.\/  Q )  .\/  S ) )  e.  (
Base `  K )  /\  ( S  .\/  T
)  e.  ( Base `  K ) ) )  ->  ( ( P 
.\/  Q )  .<_  ( ( ( P 
.\/  Q )  .\/  T )  ./\  ( ( P  .\/  Q )  .\/  S ) )  ->  (
( P  .\/  Q
)  ./\  ( S  .\/  T ) )  .<_  ( ( ( ( P  .\/  Q ) 
.\/  T )  ./\  ( ( P  .\/  Q )  .\/  S ) )  ./\  ( S  .\/  T ) ) ) )
353, 10, 31, 33, 34syl13anc 1230 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( ( P  .\/  Q )  .<_  ( (
( P  .\/  Q
)  .\/  T )  ./\  ( ( P  .\/  Q )  .\/  S ) )  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( (
( ( P  .\/  Q )  .\/  T ) 
./\  ( ( P 
.\/  Q )  .\/  S ) )  ./\  ( S  .\/  T ) ) ) )
3629, 35mpd 15 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( ( ( P  .\/  Q ) 
.\/  T )  ./\  ( ( P  .\/  Q )  .\/  S ) )  ./\  ( S  .\/  T ) ) )
376, 14, 7latlej2 15551 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  S  .<_  ( ( P  .\/  Q
)  .\/  S )
)
383, 10, 19, 37syl3anc 1228 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  ->  S  .<_  ( ( P 
.\/  Q )  .\/  S ) )
396, 14, 7, 26, 8atmod3i1 34877 . . . . 5  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  ( ( P  .\/  Q )  .\/  S )  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  S )
)  ->  ( S  .\/  ( ( ( P 
.\/  Q )  .\/  S )  ./\  T )
)  =  ( ( ( P  .\/  Q
)  .\/  S )  ./\  ( S  .\/  T
) ) )
401, 17, 25, 13, 38, 39syl131anc 1241 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( S  .\/  (
( ( P  .\/  Q )  .\/  S ) 
./\  T ) )  =  ( ( ( P  .\/  Q ) 
.\/  S )  ./\  ( S  .\/  T ) ) )
4140oveq2d 6301 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( ( ( P 
.\/  Q )  .\/  T )  ./\  ( S  .\/  ( ( ( P 
.\/  Q )  .\/  S )  ./\  T )
) )  =  ( ( ( P  .\/  Q )  .\/  T ) 
./\  ( ( ( P  .\/  Q ) 
.\/  S )  ./\  ( S  .\/  T ) ) ) )
426, 26latmcl 15542 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  .\/  S )  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
)  ->  ( (
( P  .\/  Q
)  .\/  S )  ./\  T )  e.  (
Base `  K )
)
433, 25, 13, 42syl3anc 1228 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( ( ( P 
.\/  Q )  .\/  S )  ./\  T )  e.  ( Base `  K
) )
446, 14, 7, 26latmlej22 15583 . . . . 5  |-  ( ( K  e.  Lat  /\  ( T  e.  ( Base `  K )  /\  ( ( P  .\/  Q )  .\/  S )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
) )  ->  (
( ( P  .\/  Q )  .\/  S ) 
./\  T )  .<_  ( ( P  .\/  Q )  .\/  T ) )
453, 13, 25, 10, 44syl13anc 1230 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( ( ( P 
.\/  Q )  .\/  S )  ./\  T )  .<_  ( ( P  .\/  Q )  .\/  T ) )
466, 14, 7, 26, 8atmod2i2 34875 . . . 4  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  ( ( P  .\/  Q )  .\/  T )  e.  ( Base `  K
)  /\  ( (
( P  .\/  Q
)  .\/  S )  ./\  T )  e.  (
Base `  K )
)  /\  ( (
( P  .\/  Q
)  .\/  S )  ./\  T )  .<_  ( ( P  .\/  Q ) 
.\/  T ) )  ->  ( ( ( ( P  .\/  Q
)  .\/  T )  ./\  S )  .\/  (
( ( P  .\/  Q )  .\/  S ) 
./\  T ) )  =  ( ( ( P  .\/  Q ) 
.\/  T )  ./\  ( S  .\/  ( ( ( P  .\/  Q
)  .\/  S )  ./\  T ) ) ) )
471, 17, 23, 43, 45, 46syl131anc 1241 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( ( ( ( P  .\/  Q ) 
.\/  T )  ./\  S )  .\/  ( ( ( P  .\/  Q
)  .\/  S )  ./\  T ) )  =  ( ( ( P 
.\/  Q )  .\/  T )  ./\  ( S  .\/  ( ( ( P 
.\/  Q )  .\/  S )  ./\  T )
) ) )
48 hlol 34375 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
491, 48syl 16 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  ->  K  e.  OL )
506, 26latmassOLD 34243 . . . 4  |-  ( ( K  e.  OL  /\  ( ( ( P 
.\/  Q )  .\/  T )  e.  ( Base `  K )  /\  (
( P  .\/  Q
)  .\/  S )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
) )  ->  (
( ( ( P 
.\/  Q )  .\/  T )  ./\  ( ( P  .\/  Q )  .\/  S ) )  ./\  ( S  .\/  T ) )  =  ( ( ( P  .\/  Q ) 
.\/  T )  ./\  ( ( ( P 
.\/  Q )  .\/  S )  ./\  ( S  .\/  T ) ) ) )
5149, 23, 25, 33, 50syl13anc 1230 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( ( ( ( P  .\/  Q ) 
.\/  T )  ./\  ( ( P  .\/  Q )  .\/  S ) )  ./\  ( S  .\/  T ) )  =  ( ( ( P 
.\/  Q )  .\/  T )  ./\  ( (
( P  .\/  Q
)  .\/  S )  ./\  ( S  .\/  T
) ) ) )
5241, 47, 513eqtr4rd 2519 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( ( ( ( P  .\/  Q ) 
.\/  T )  ./\  ( ( P  .\/  Q )  .\/  S ) )  ./\  ( S  .\/  T ) )  =  ( ( ( ( P  .\/  Q ) 
.\/  T )  ./\  S )  .\/  ( ( ( P  .\/  Q
)  .\/  S )  ./\  T ) ) )
5336, 52breqtrd 4471 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( ( ( P  .\/  Q ) 
.\/  T )  ./\  S )  .\/  ( ( ( P  .\/  Q
)  .\/  S )  ./\  T ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   Basecbs 14493   lecple 14565   joincjn 15434   meetcmee 15435   Latclat 15535   OLcol 34188   Atomscatm 34277   HLchlt 34364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-poset 15436  df-plt 15448  df-lub 15464  df-glb 15465  df-join 15466  df-meet 15467  df-p0 15529  df-lat 15536  df-clat 15598  df-oposet 34190  df-ol 34192  df-oml 34193  df-covers 34280  df-ats 34281  df-atl 34312  df-cvlat 34336  df-hlat 34365  df-psubsp 34516  df-pmap 34517  df-padd 34809
This theorem is referenced by:  dalawlem5  34888  dalawlem8  34891
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