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Theorem dalawlem2 30683
Description: Lemma for dalaw 30697. 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 955 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  ->  K  e.  HL )
2 hllat 30175 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 15 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  ->  K  e.  Lat )
4 simp2l 981 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  ->  P  e.  A )
5 simp2r 982 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  ->  Q  e.  A )
6 eqid 2296 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
7 dalawlem.j . . . . . . 7  |-  .\/  =  ( join `  K )
8 dalawlem.a . . . . . . 7  |-  A  =  ( Atoms `  K )
96, 7, 8hlatjcl 30178 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
101, 4, 5, 9syl3anc 1182 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
11 simp3r 984 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  ->  T  e.  A )
126, 8atbase 30101 . . . . . 6  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
1311, 12syl 15 . . . . 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 14182 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
)  ->  ( P  .\/  Q )  .<_  ( ( P  .\/  Q ) 
.\/  T ) )
163, 10, 13, 15syl3anc 1182 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( P  .\/  Q
)  .<_  ( ( P 
.\/  Q )  .\/  T ) )
17 simp3l 983 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  ->  S  e.  A )
186, 8atbase 30101 . . . . . 6  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
1917, 18syl 15 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  ->  S  e.  ( Base `  K ) )
206, 14, 7latlej1 14182 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  ( P  .\/  Q )  .<_  ( ( P  .\/  Q ) 
.\/  S ) )
213, 10, 19, 20syl3anc 1182 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( P  .\/  Q
)  .<_  ( ( P 
.\/  Q )  .\/  S ) )
226, 7latjcl 14172 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  T )  e.  ( Base `  K ) )
233, 10, 13, 22syl3anc 1182 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( ( P  .\/  Q )  .\/  T )  e.  ( Base `  K
) )
246, 7latjcl 14172 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  S )  e.  ( Base `  K ) )
253, 10, 19, 24syl3anc 1182 . . . . 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 14200 . . . . 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 1184 . . . 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 887 . . 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 14173 . . . . 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 1182 . . . 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 30178 . . . . 5  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
331, 17, 11, 32syl3anc 1182 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( S  .\/  T
)  e.  ( Base `  K ) )
346, 14, 26latmlem1 14203 . . . 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 1184 . . 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 14 . 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 14183 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  S  .<_  ( ( P  .\/  Q
)  .\/  S )
)
383, 10, 19, 37syl3anc 1182 . . . . 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 30675 . . . . 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 1195 . . . 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 5890 . . 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 14173 . . . . 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 1182 . . . 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 14215 . . . . 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 1184 . . . 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 30673 . . . 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 1195 . . 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 30173 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
491, 48syl 15 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  ->  K  e.  OL )
506, 26latmassOLD 30041 . . . 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 1184 . . 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 2339 . 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 4063 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Latclat 14167   OLcol 29986   Atomscatm 30075   HLchlt 30162
This theorem is referenced by:  dalawlem5  30686  dalawlem8  30689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-psubsp 30314  df-pmap 30315  df-padd 30607
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