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Theorem 4atexlemc 30880
Description: Lemma for 4atexlem7 30886. (Contributed by NM, 24-Nov-2012.)
Hypotheses
Ref Expression
4thatlem.ph  |-  ( ph  <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) ) )
4thatlem0.l  |-  .<_  =  ( le `  K )
4thatlem0.j  |-  .\/  =  ( join `  K )
4thatlem0.m  |-  ./\  =  ( meet `  K )
4thatlem0.a  |-  A  =  ( Atoms `  K )
4thatlem0.h  |-  H  =  ( LHyp `  K
)
4thatlem0.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
4thatlem0.v  |-  V  =  ( ( P  .\/  S )  ./\  W )
4thatlem0.c  |-  C  =  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )
Assertion
Ref Expression
4atexlemc  |-  ( ph  ->  C  e.  A )

Proof of Theorem 4atexlemc
StepHypRef Expression
1 4thatlem0.c . . 3  |-  C  =  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )
2 4thatlem.ph . . . . 5  |-  ( ph  <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) ) )
324atexlemkl 30868 . . . 4  |-  ( ph  ->  K  e.  Lat )
4 4thatlem0.j . . . . 5  |-  .\/  =  ( join `  K )
5 4thatlem0.a . . . . 5  |-  A  =  ( Atoms `  K )
62, 4, 54atexlemqtb 30872 . . . 4  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
72, 4, 54atexlempsb 30871 . . . 4  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
8 eqid 2296 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
9 4thatlem0.m . . . . 5  |-  ./\  =  ( meet `  K )
108, 9latmcom 14197 . . . 4  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  =  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) ) )
113, 6, 7, 10syl3anc 1182 . . 3  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  =  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) ) )
121, 11syl5eq 2340 . 2  |-  ( ph  ->  C  =  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) ) )
1324atexlemk 30858 . . 3  |-  ( ph  ->  K  e.  HL )
1424atexlemp 30861 . . 3  |-  ( ph  ->  P  e.  A )
1524atexlems 30863 . . 3  |-  ( ph  ->  S  e.  A )
1624atexlemq 30862 . . 3  |-  ( ph  ->  Q  e.  A )
1724atexlemt 30864 . . 3  |-  ( ph  ->  T  e.  A )
18 4thatlem0.l . . . 4  |-  .<_  =  ( le `  K )
192, 18, 4, 54atexlempns 30873 . . 3  |-  ( ph  ->  P  =/=  S )
20 4thatlem0.h . . . . 5  |-  H  =  ( LHyp `  K
)
21 4thatlem0.u . . . . 5  |-  U  =  ( ( P  .\/  Q )  ./\  W )
22 4thatlem0.v . . . . 5  |-  V  =  ( ( P  .\/  S )  ./\  W )
232, 18, 4, 9, 5, 20, 21, 224atexlemntlpq 30879 . . . 4  |-  ( ph  ->  -.  T  .<_  ( P 
.\/  Q ) )
2418, 4, 5atnlej2 30191 . . . . 5  |-  ( ( K  e.  HL  /\  ( T  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  T  .<_  ( P  .\/  Q
) )  ->  T  =/=  Q )
2524necomd 2542 . . . 4  |-  ( ( K  e.  HL  /\  ( T  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  T  .<_  ( P  .\/  Q
) )  ->  Q  =/=  T )
2613, 17, 14, 16, 23, 25syl131anc 1195 . . 3  |-  ( ph  ->  Q  =/=  T )
2724atexlempnq 30866 . . . 4  |-  ( ph  ->  P  =/=  Q )
2824atexlemnslpq 30867 . . . 4  |-  ( ph  ->  -.  S  .<_  ( P 
.\/  Q ) )
2918, 4, 54atlem0ae 30405 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  -.  Q  .<_  ( P 
.\/  S ) )
3013, 14, 16, 15, 27, 28, 29syl132anc 1200 . . 3  |-  ( ph  ->  -.  Q  .<_  ( P 
.\/  S ) )
318, 5atbase 30101 . . . . 5  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
3217, 31syl 15 . . . 4  |-  ( ph  ->  T  e.  ( Base `  K ) )
332, 18, 4, 9, 5, 20, 214atexlemu 30875 . . . . 5  |-  ( ph  ->  U  e.  A )
342, 18, 4, 9, 5, 20, 21, 224atexlemv 30876 . . . . 5  |-  ( ph  ->  V  e.  A )
358, 4, 5hlatjcl 30178 . . . . 5  |-  ( ( K  e.  HL  /\  U  e.  A  /\  V  e.  A )  ->  ( U  .\/  V
)  e.  ( Base `  K ) )
3613, 33, 34, 35syl3anc 1182 . . . 4  |-  ( ph  ->  ( U  .\/  V
)  e.  ( Base `  K ) )
378, 5atbase 30101 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
3816, 37syl 15 . . . . 5  |-  ( ph  ->  Q  e.  ( Base `  K ) )
398, 4latjcl 14172 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  Q  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  .\/  Q )  e.  ( Base `  K ) )
403, 7, 38, 39syl3anc 1182 . . . 4  |-  ( ph  ->  ( ( P  .\/  S )  .\/  Q )  e.  ( Base `  K
) )
4124atexlemkc 30869 . . . . 5  |-  ( ph  ->  K  e.  CvLat )
422, 18, 4, 9, 5, 20, 21, 224atexlemunv 30877 . . . . 5  |-  ( ph  ->  U  =/=  V )
4324atexlemutvt 30865 . . . . 5  |-  ( ph  ->  ( U  .\/  T
)  =  ( V 
.\/  T ) )
445, 18, 4cvlsupr4 30157 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( U  e.  A  /\  V  e.  A  /\  T  e.  A )  /\  ( U  =/=  V  /\  ( U  .\/  T
)  =  ( V 
.\/  T ) ) )  ->  T  .<_  ( U  .\/  V ) )
4541, 33, 34, 17, 42, 43, 44syl132anc 1200 . . . 4  |-  ( ph  ->  T  .<_  ( U  .\/  V ) )
468, 4, 5hlatjcl 30178 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
4713, 14, 16, 46syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
482, 204atexlemwb 30870 . . . . . . . 8  |-  ( ph  ->  W  e.  ( Base `  K ) )
498, 18, 9latmle1 14198 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q ) )
503, 47, 48, 49syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q
) )
5121, 50syl5eqbr 4072 . . . . . 6  |-  ( ph  ->  U  .<_  ( P  .\/  Q ) )
528, 18, 9latmle1 14198 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S ) )
533, 7, 48, 52syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S
) )
5422, 53syl5eqbr 4072 . . . . . 6  |-  ( ph  ->  V  .<_  ( P  .\/  S ) )
558, 5atbase 30101 . . . . . . . 8  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
5633, 55syl 15 . . . . . . 7  |-  ( ph  ->  U  e.  ( Base `  K ) )
578, 5atbase 30101 . . . . . . . 8  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
5834, 57syl 15 . . . . . . 7  |-  ( ph  ->  V  e.  ( Base `  K ) )
598, 18, 4latjlej12 14189 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( U  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  /\  ( V  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
) ) )  -> 
( ( U  .<_  ( P  .\/  Q )  /\  V  .<_  ( P 
.\/  S ) )  ->  ( U  .\/  V )  .<_  ( ( P  .\/  Q )  .\/  ( P  .\/  S ) ) ) )
603, 56, 47, 58, 7, 59syl122anc 1191 . . . . . 6  |-  ( ph  ->  ( ( U  .<_  ( P  .\/  Q )  /\  V  .<_  ( P 
.\/  S ) )  ->  ( U  .\/  V )  .<_  ( ( P  .\/  Q )  .\/  ( P  .\/  S ) ) ) )
6151, 54, 60mp2and 660 . . . . 5  |-  ( ph  ->  ( U  .\/  V
)  .<_  ( ( P 
.\/  Q )  .\/  ( P  .\/  S ) ) )
624, 5hlatjass 30181 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  S  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  S )  =  ( P  .\/  ( Q  .\/  S ) ) )
6313, 14, 16, 15, 62syl13anc 1184 . . . . . 6  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  S )  =  ( P  .\/  ( Q  .\/  S ) ) )
648, 5atbase 30101 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
6514, 64syl 15 . . . . . . 7  |-  ( ph  ->  P  e.  ( Base `  K ) )
668, 5atbase 30101 . . . . . . . 8  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
6715, 66syl 15 . . . . . . 7  |-  ( ph  ->  S  e.  ( Base `  K ) )
688, 4latj32 14219 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) ) )  -> 
( ( P  .\/  Q )  .\/  S )  =  ( ( P 
.\/  S )  .\/  Q ) )
693, 65, 38, 67, 68syl13anc 1184 . . . . . 6  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  S )  =  ( ( P 
.\/  S )  .\/  Q ) )
708, 4latjjdi 14225 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) ) )  -> 
( P  .\/  ( Q  .\/  S ) )  =  ( ( P 
.\/  Q )  .\/  ( P  .\/  S ) ) )
713, 65, 38, 67, 70syl13anc 1184 . . . . . 6  |-  ( ph  ->  ( P  .\/  ( Q  .\/  S ) )  =  ( ( P 
.\/  Q )  .\/  ( P  .\/  S ) ) )
7263, 69, 713eqtr3rd 2337 . . . . 5  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  ( P 
.\/  S ) )  =  ( ( P 
.\/  S )  .\/  Q ) )
7361, 72breqtrd 4063 . . . 4  |-  ( ph  ->  ( U  .\/  V
)  .<_  ( ( P 
.\/  S )  .\/  Q ) )
748, 18, 3, 32, 36, 40, 45, 73lattrd 14180 . . 3  |-  ( ph  ->  T  .<_  ( ( P  .\/  S )  .\/  Q ) )
7518, 4, 9, 52atmat 30372 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  /\  ( Q  e.  A  /\  T  e.  A  /\  P  =/=  S
)  /\  ( Q  =/=  T  /\  -.  Q  .<_  ( P  .\/  S
)  /\  T  .<_  ( ( P  .\/  S
)  .\/  Q )
) )  ->  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  e.  A )
7613, 14, 15, 16, 17, 19, 26, 30, 74, 75syl333anc 1214 . 2  |-  ( ph  ->  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  e.  A )
7712, 76eqeltrd 2370 1  |-  ( ph  ->  C  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Latclat 14167   Atomscatm 30075   CvLatclc 30077   HLchlt 30162   LHypclh 30795
This theorem is referenced by:  4atexlemnclw  30881  4atexlemex2  30882  4atexlemcnd  30883
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-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-p1 14162  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-llines 30309  df-lplanes 30310  df-lhyp 30799
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