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Theorem 4atexlemc 30551
Description: Lemma for 4atexlem7 30557. (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 30539 . . . 4  |-  ( ph  ->  K  e.  Lat )
4 4thatlem0.j . . . . 5  |-  .\/  =  ( join `  K )
5 4thatlem0.a . . . . 5  |-  A  =  ( Atoms `  K )
62, 4, 54atexlemqtb 30543 . . . 4  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
72, 4, 54atexlempsb 30542 . . . 4  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
8 eqid 2404 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
9 4thatlem0.m . . . . 5  |-  ./\  =  ( meet `  K )
108, 9latmcom 14459 . . . 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 1184 . . 3  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  =  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) ) )
121, 11syl5eq 2448 . 2  |-  ( ph  ->  C  =  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) ) )
1324atexlemk 30529 . . 3  |-  ( ph  ->  K  e.  HL )
1424atexlemp 30532 . . 3  |-  ( ph  ->  P  e.  A )
1524atexlems 30534 . . 3  |-  ( ph  ->  S  e.  A )
1624atexlemq 30533 . . 3  |-  ( ph  ->  Q  e.  A )
1724atexlemt 30535 . . 3  |-  ( ph  ->  T  e.  A )
18 4thatlem0.l . . . 4  |-  .<_  =  ( le `  K )
192, 18, 4, 54atexlempns 30544 . . 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 30550 . . . 4  |-  ( ph  ->  -.  T  .<_  ( P 
.\/  Q ) )
2418, 4, 5atnlej2 29862 . . . . 5  |-  ( ( K  e.  HL  /\  ( T  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  T  .<_  ( P  .\/  Q
) )  ->  T  =/=  Q )
2524necomd 2650 . . . 4  |-  ( ( K  e.  HL  /\  ( T  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  T  .<_  ( P  .\/  Q
) )  ->  Q  =/=  T )
2613, 17, 14, 16, 23, 25syl131anc 1197 . . 3  |-  ( ph  ->  Q  =/=  T )
2724atexlempnq 30537 . . . 4  |-  ( ph  ->  P  =/=  Q )
2824atexlemnslpq 30538 . . . 4  |-  ( ph  ->  -.  S  .<_  ( P 
.\/  Q ) )
2918, 4, 54atlem0ae 30076 . . . 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 1202 . . 3  |-  ( ph  ->  -.  Q  .<_  ( P 
.\/  S ) )
318, 5atbase 29772 . . . . 5  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
3217, 31syl 16 . . . 4  |-  ( ph  ->  T  e.  ( Base `  K ) )
332, 18, 4, 9, 5, 20, 214atexlemu 30546 . . . . 5  |-  ( ph  ->  U  e.  A )
342, 18, 4, 9, 5, 20, 21, 224atexlemv 30547 . . . . 5  |-  ( ph  ->  V  e.  A )
358, 4, 5hlatjcl 29849 . . . . 5  |-  ( ( K  e.  HL  /\  U  e.  A  /\  V  e.  A )  ->  ( U  .\/  V
)  e.  ( Base `  K ) )
3613, 33, 34, 35syl3anc 1184 . . . 4  |-  ( ph  ->  ( U  .\/  V
)  e.  ( Base `  K ) )
378, 5atbase 29772 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
3816, 37syl 16 . . . . 5  |-  ( ph  ->  Q  e.  ( Base `  K ) )
398, 4latjcl 14434 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  Q  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  .\/  Q )  e.  ( Base `  K ) )
403, 7, 38, 39syl3anc 1184 . . . 4  |-  ( ph  ->  ( ( P  .\/  S )  .\/  Q )  e.  ( Base `  K
) )
4124atexlemkc 30540 . . . . 5  |-  ( ph  ->  K  e.  CvLat )
422, 18, 4, 9, 5, 20, 21, 224atexlemunv 30548 . . . . 5  |-  ( ph  ->  U  =/=  V )
4324atexlemutvt 30536 . . . . 5  |-  ( ph  ->  ( U  .\/  T
)  =  ( V 
.\/  T ) )
445, 18, 4cvlsupr4 29828 . . . . 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 1202 . . . 4  |-  ( ph  ->  T  .<_  ( U  .\/  V ) )
468, 4, 5hlatjcl 29849 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
4713, 14, 16, 46syl3anc 1184 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
482, 204atexlemwb 30541 . . . . . . . 8  |-  ( ph  ->  W  e.  ( Base `  K ) )
498, 18, 9latmle1 14460 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q ) )
503, 47, 48, 49syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q
) )
5121, 50syl5eqbr 4205 . . . . . 6  |-  ( ph  ->  U  .<_  ( P  .\/  Q ) )
528, 18, 9latmle1 14460 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S ) )
533, 7, 48, 52syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S
) )
5422, 53syl5eqbr 4205 . . . . . 6  |-  ( ph  ->  V  .<_  ( P  .\/  S ) )
558, 5atbase 29772 . . . . . . . 8  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
5633, 55syl 16 . . . . . . 7  |-  ( ph  ->  U  e.  ( Base `  K ) )
578, 5atbase 29772 . . . . . . . 8  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
5834, 57syl 16 . . . . . . 7  |-  ( ph  ->  V  e.  ( Base `  K ) )
598, 18, 4latjlej12 14451 . . . . . . 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 1193 . . . . . 6  |-  ( ph  ->  ( ( U  .<_  ( P  .\/  Q )  /\  V  .<_  ( P 
.\/  S ) )  ->  ( U  .\/  V )  .<_  ( ( P  .\/  Q )  .\/  ( P  .\/  S ) ) ) )
6151, 54, 60mp2and 661 . . . . 5  |-  ( ph  ->  ( U  .\/  V
)  .<_  ( ( P 
.\/  Q )  .\/  ( P  .\/  S ) ) )
624, 5hlatjass 29852 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  S  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  S )  =  ( P  .\/  ( Q  .\/  S ) ) )
6313, 14, 16, 15, 62syl13anc 1186 . . . . . 6  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  S )  =  ( P  .\/  ( Q  .\/  S ) ) )
648, 5atbase 29772 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
6514, 64syl 16 . . . . . . 7  |-  ( ph  ->  P  e.  ( Base `  K ) )
668, 5atbase 29772 . . . . . . . 8  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
6715, 66syl 16 . . . . . . 7  |-  ( ph  ->  S  e.  ( Base `  K ) )
688, 4latj32 14481 . . . . . . 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 1186 . . . . . 6  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  S )  =  ( ( P 
.\/  S )  .\/  Q ) )
708, 4latjjdi 14487 . . . . . . 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 1186 . . . . . 6  |-  ( ph  ->  ( P  .\/  ( Q  .\/  S ) )  =  ( ( P 
.\/  Q )  .\/  ( P  .\/  S ) ) )
7263, 69, 713eqtr3rd 2445 . . . . 5  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  ( P 
.\/  S ) )  =  ( ( P 
.\/  S )  .\/  Q ) )
7361, 72breqtrd 4196 . . . 4  |-  ( ph  ->  ( U  .\/  V
)  .<_  ( ( P 
.\/  S )  .\/  Q ) )
748, 18, 3, 32, 36, 40, 45, 73lattrd 14442 . . 3  |-  ( ph  ->  T  .<_  ( ( P  .\/  S )  .\/  Q ) )
7518, 4, 9, 52atmat 30043 . . 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 1216 . 2  |-  ( ph  ->  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  e.  A )
7712, 76eqeltrd 2478 1  |-  ( ph  ->  C  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Latclat 14429   Atomscatm 29746   CvLatclc 29748   HLchlt 29833   LHypclh 30466
This theorem is referenced by:  4atexlemnclw  30552  4atexlemex2  30553  4atexlemcnd  30554
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lhyp 30470
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