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Theorem dalawlem2 30669
Description: Lemma for dalaw 30683. 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 957 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  ->  K  e.  HL )
2 hllat 30161 . . . . . 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 983 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  ->  P  e.  A )
5 simp2r 984 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  ->  Q  e.  A )
6 eqid 2436 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
7 dalawlem.j . . . . . . 7  |-  .\/  =  ( join `  K )
8 dalawlem.a . . . . . . 7  |-  A  =  ( Atoms `  K )
96, 7, 8hlatjcl 30164 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
101, 4, 5, 9syl3anc 1184 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
11 simp3r 986 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  ->  T  e.  A )
126, 8atbase 30087 . . . . . 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 14489 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
)  ->  ( P  .\/  Q )  .<_  ( ( P  .\/  Q ) 
.\/  T ) )
163, 10, 13, 15syl3anc 1184 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( P  .\/  Q
)  .<_  ( ( P 
.\/  Q )  .\/  T ) )
17 simp3l 985 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  ->  S  e.  A )
186, 8atbase 30087 . . . . . 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 14489 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  ( P  .\/  Q )  .<_  ( ( P  .\/  Q ) 
.\/  S ) )
213, 10, 19, 20syl3anc 1184 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( P  .\/  Q
)  .<_  ( ( P 
.\/  Q )  .\/  S ) )
226, 7latjcl 14479 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  T )  e.  ( Base `  K ) )
233, 10, 13, 22syl3anc 1184 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( ( P  .\/  Q )  .\/  T )  e.  ( Base `  K
) )
246, 7latjcl 14479 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  S )  e.  ( Base `  K ) )
253, 10, 19, 24syl3anc 1184 . . . . 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 14507 . . . . 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 1186 . . . 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 888 . . 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 14480 . . . . 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 1184 . . . 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 30164 . . . . 5  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
331, 17, 11, 32syl3anc 1184 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( S  .\/  T
)  e.  ( Base `  K ) )
346, 14, 26latmlem1 14510 . . . 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 1186 . . 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 14490 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  S  .<_  ( ( P  .\/  Q
)  .\/  S )
)
383, 10, 19, 37syl3anc 1184 . . . . 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 30661 . . . . 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 1197 . . . 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 6097 . . 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 14480 . . . . 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 1184 . . . 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 14522 . . . . 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 1186 . . . 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 30659 . . . 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 1197 . . 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 30159 . . . . 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 30027 . . . 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 1186 . . 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 2479 . 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 4236 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   joincjn 14401   meetcmee 14402   Latclat 14474   OLcol 29972   Atomscatm 30061   HLchlt 30148
This theorem is referenced by:  dalawlem5  30672  dalawlem8  30675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-psubsp 30300  df-pmap 30301  df-padd 30593
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