Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  4atexlemc Structured version   Unicode version

Theorem 4atexlemc 33597
Description: Lemma for 4atexlem7 33603. (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 33585 . . . 4  |-  ( ph  ->  K  e.  Lat )
4 4thatlem0.j . . . . 5  |-  .\/  =  ( join `  K )
5 4thatlem0.a . . . . 5  |-  A  =  ( Atoms `  K )
62, 4, 54atexlemqtb 33589 . . . 4  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
72, 4, 54atexlempsb 33588 . . . 4  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
8 eqid 2423 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
9 4thatlem0.m . . . . 5  |-  ./\  =  ( meet `  K )
108, 9latmcom 16314 . . . 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 1265 . . 3  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  =  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) ) )
121, 11syl5eq 2476 . 2  |-  ( ph  ->  C  =  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) ) )
1324atexlemk 33575 . . 3  |-  ( ph  ->  K  e.  HL )
1424atexlemp 33578 . . 3  |-  ( ph  ->  P  e.  A )
1524atexlems 33580 . . 3  |-  ( ph  ->  S  e.  A )
1624atexlemq 33579 . . 3  |-  ( ph  ->  Q  e.  A )
1724atexlemt 33581 . . 3  |-  ( ph  ->  T  e.  A )
18 4thatlem0.l . . . 4  |-  .<_  =  ( le `  K )
192, 18, 4, 54atexlempns 33590 . . 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 33596 . . . 4  |-  ( ph  ->  -.  T  .<_  ( P 
.\/  Q ) )
2418, 4, 5atnlej2 32908 . . . . 5  |-  ( ( K  e.  HL  /\  ( T  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  T  .<_  ( P  .\/  Q
) )  ->  T  =/=  Q )
2524necomd 2696 . . . 4  |-  ( ( K  e.  HL  /\  ( T  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  T  .<_  ( P  .\/  Q
) )  ->  Q  =/=  T )
2613, 17, 14, 16, 23, 25syl131anc 1278 . . 3  |-  ( ph  ->  Q  =/=  T )
2724atexlempnq 33583 . . . 4  |-  ( ph  ->  P  =/=  Q )
2824atexlemnslpq 33584 . . . 4  |-  ( ph  ->  -.  S  .<_  ( P 
.\/  Q ) )
2918, 4, 54atlem0ae 33122 . . . 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 1283 . . 3  |-  ( ph  ->  -.  Q  .<_  ( P 
.\/  S ) )
318, 5atbase 32818 . . . . 5  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
3217, 31syl 17 . . . 4  |-  ( ph  ->  T  e.  ( Base `  K ) )
332, 18, 4, 9, 5, 20, 214atexlemu 33592 . . . . 5  |-  ( ph  ->  U  e.  A )
342, 18, 4, 9, 5, 20, 21, 224atexlemv 33593 . . . . 5  |-  ( ph  ->  V  e.  A )
358, 4, 5hlatjcl 32895 . . . . 5  |-  ( ( K  e.  HL  /\  U  e.  A  /\  V  e.  A )  ->  ( U  .\/  V
)  e.  ( Base `  K ) )
3613, 33, 34, 35syl3anc 1265 . . . 4  |-  ( ph  ->  ( U  .\/  V
)  e.  ( Base `  K ) )
378, 5atbase 32818 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
3816, 37syl 17 . . . . 5  |-  ( ph  ->  Q  e.  ( Base `  K ) )
398, 4latjcl 16290 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  Q  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  .\/  Q )  e.  ( Base `  K ) )
403, 7, 38, 39syl3anc 1265 . . . 4  |-  ( ph  ->  ( ( P  .\/  S )  .\/  Q )  e.  ( Base `  K
) )
4124atexlemkc 33586 . . . . 5  |-  ( ph  ->  K  e.  CvLat )
422, 18, 4, 9, 5, 20, 21, 224atexlemunv 33594 . . . . 5  |-  ( ph  ->  U  =/=  V )
4324atexlemutvt 33582 . . . . 5  |-  ( ph  ->  ( U  .\/  T
)  =  ( V 
.\/  T ) )
445, 18, 4cvlsupr4 32874 . . . . 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 1283 . . . 4  |-  ( ph  ->  T  .<_  ( U  .\/  V ) )
468, 4, 5hlatjcl 32895 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
4713, 14, 16, 46syl3anc 1265 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
482, 204atexlemwb 33587 . . . . . . . 8  |-  ( ph  ->  W  e.  ( Base `  K ) )
498, 18, 9latmle1 16315 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q ) )
503, 47, 48, 49syl3anc 1265 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q
) )
5121, 50syl5eqbr 4455 . . . . . 6  |-  ( ph  ->  U  .<_  ( P  .\/  Q ) )
528, 18, 9latmle1 16315 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S ) )
533, 7, 48, 52syl3anc 1265 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S
) )
5422, 53syl5eqbr 4455 . . . . . 6  |-  ( ph  ->  V  .<_  ( P  .\/  S ) )
558, 5atbase 32818 . . . . . . . 8  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
5633, 55syl 17 . . . . . . 7  |-  ( ph  ->  U  e.  ( Base `  K ) )
578, 5atbase 32818 . . . . . . . 8  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
5834, 57syl 17 . . . . . . 7  |-  ( ph  ->  V  e.  ( Base `  K ) )
598, 18, 4latjlej12 16306 . . . . . . 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 1274 . . . . . 6  |-  ( ph  ->  ( ( U  .<_  ( P  .\/  Q )  /\  V  .<_  ( P 
.\/  S ) )  ->  ( U  .\/  V )  .<_  ( ( P  .\/  Q )  .\/  ( P  .\/  S ) ) ) )
6151, 54, 60mp2and 684 . . . . 5  |-  ( ph  ->  ( U  .\/  V
)  .<_  ( ( P 
.\/  Q )  .\/  ( P  .\/  S ) ) )
624, 5hlatjass 32898 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  S  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  S )  =  ( P  .\/  ( Q  .\/  S ) ) )
6313, 14, 16, 15, 62syl13anc 1267 . . . . . 6  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  S )  =  ( P  .\/  ( Q  .\/  S ) ) )
648, 5atbase 32818 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
6514, 64syl 17 . . . . . . 7  |-  ( ph  ->  P  e.  ( Base `  K ) )
668, 5atbase 32818 . . . . . . . 8  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
6715, 66syl 17 . . . . . . 7  |-  ( ph  ->  S  e.  ( Base `  K ) )
688, 4latj32 16336 . . . . . . 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 1267 . . . . . 6  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  S )  =  ( ( P 
.\/  S )  .\/  Q ) )
708, 4latjjdi 16342 . . . . . . 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 1267 . . . . . 6  |-  ( ph  ->  ( P  .\/  ( Q  .\/  S ) )  =  ( ( P 
.\/  Q )  .\/  ( P  .\/  S ) ) )
7263, 69, 713eqtr3rd 2473 . . . . 5  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  ( P 
.\/  S ) )  =  ( ( P 
.\/  S )  .\/  Q ) )
7361, 72breqtrd 4446 . . . 4  |-  ( ph  ->  ( U  .\/  V
)  .<_  ( ( P 
.\/  S )  .\/  Q ) )
748, 18, 3, 32, 36, 40, 45, 73lattrd 16297 . . 3  |-  ( ph  ->  T  .<_  ( ( P  .\/  S )  .\/  Q ) )
7518, 4, 9, 52atmat 33089 . . 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 1297 . 2  |-  ( ph  ->  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  e.  A )
7712, 76eqeltrd 2511 1  |-  ( ph  ->  C  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869    =/= wne 2619   class class class wbr 4421   ` cfv 5599  (class class class)co 6303   Basecbs 15114   lecple 15190   joincjn 16182   meetcmee 16183   Latclat 16284   Atomscatm 32792   CvLatclc 32794   HLchlt 32879   LHypclh 33512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-preset 16166  df-poset 16184  df-plt 16197  df-lub 16213  df-glb 16214  df-join 16215  df-meet 16216  df-p0 16278  df-p1 16279  df-lat 16285  df-clat 16347  df-oposet 32705  df-ol 32707  df-oml 32708  df-covers 32795  df-ats 32796  df-atl 32827  df-cvlat 32851  df-hlat 32880  df-llines 33026  df-lplanes 33027  df-lhyp 33516
This theorem is referenced by:  4atexlemnclw  33598  4atexlemex2  33599  4atexlemcnd  33600
  Copyright terms: Public domain W3C validator