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

Theorem 4atexlemnclw 33068
Description: Lemma for 4atexlem7 33073. (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
4atexlemnclw  |-  ( ph  ->  -.  C  .<_  W )

Proof of Theorem 4atexlemnclw
StepHypRef Expression
1 4thatlem0.c . . . 4  |-  C  =  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )
2 4thatlem.ph . . . . . 6  |-  ( 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 33055 . . . . 5  |-  ( ph  ->  K  e.  Lat )
4 4thatlem0.j . . . . . 6  |-  .\/  =  ( join `  K )
5 4thatlem0.a . . . . . 6  |-  A  =  ( Atoms `  K )
62, 4, 54atexlemqtb 33059 . . . . 5  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
72, 4, 54atexlempsb 33058 . . . . 5  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
8 eqid 2402 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
9 4thatlem0.l . . . . . 6  |-  .<_  =  ( le `  K )
10 4thatlem0.m . . . . . 6  |-  ./\  =  ( meet `  K )
118, 9, 10latmle1 15922 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
123, 6, 7, 11syl3anc 1230 . . . 4  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
131, 12syl5eqbr 4427 . . 3  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
14 simp13r 1113 . . . . 5  |-  ( ( ( ( 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 ) ) )  ->  -.  Q  .<_  W )
152, 14sylbi 195 . . . 4  |-  ( ph  ->  -.  Q  .<_  W )
1624atexlemkc 33056 . . . . . 6  |-  ( ph  ->  K  e.  CvLat )
17 4thatlem0.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
18 4thatlem0.u . . . . . . 7  |-  U  =  ( ( P  .\/  Q )  ./\  W )
19 4thatlem0.v . . . . . . 7  |-  V  =  ( ( P  .\/  S )  ./\  W )
202, 9, 4, 10, 5, 17, 18, 194atexlemv 33063 . . . . . 6  |-  ( ph  ->  V  e.  A )
2124atexlemq 33049 . . . . . 6  |-  ( ph  ->  Q  e.  A )
2224atexlemt 33051 . . . . . 6  |-  ( ph  ->  T  e.  A )
232, 9, 4, 10, 5, 17, 184atexlemu 33062 . . . . . . 7  |-  ( ph  ->  U  e.  A )
242, 9, 4, 10, 5, 17, 18, 194atexlemunv 33064 . . . . . . 7  |-  ( ph  ->  U  =/=  V )
2524atexlemutvt 33052 . . . . . . 7  |-  ( ph  ->  ( U  .\/  T
)  =  ( V 
.\/  T ) )
265, 4cvlsupr6 32346 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( U  e.  A  /\  V  e.  A  /\  T  e.  A )  /\  ( U  =/=  V  /\  ( U  .\/  T
)  =  ( V 
.\/  T ) ) )  ->  T  =/=  V )
2726necomd 2674 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( U  e.  A  /\  V  e.  A  /\  T  e.  A )  /\  ( U  =/=  V  /\  ( U  .\/  T
)  =  ( V 
.\/  T ) ) )  ->  V  =/=  T )
2816, 23, 20, 22, 24, 25, 27syl132anc 1248 . . . . . 6  |-  ( ph  ->  V  =/=  T )
299, 4, 5cvlatexch2 32336 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( V  e.  A  /\  Q  e.  A  /\  T  e.  A )  /\  V  =/=  T
)  ->  ( V  .<_  ( Q  .\/  T
)  ->  Q  .<_  ( V  .\/  T ) ) )
3016, 20, 21, 22, 28, 29syl131anc 1243 . . . . 5  |-  ( ph  ->  ( V  .<_  ( Q 
.\/  T )  ->  Q  .<_  ( V  .\/  T ) ) )
312, 174atexlemwb 33057 . . . . . . . . 9  |-  ( ph  ->  W  e.  ( Base `  K ) )
328, 9, 10latmle2 15923 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  W )
333, 7, 31, 32syl3anc 1230 . . . . . . . 8  |-  ( ph  ->  ( ( P  .\/  S )  ./\  W )  .<_  W )
3419, 33syl5eqbr 4427 . . . . . . 7  |-  ( ph  ->  V  .<_  W )
352, 9, 4, 10, 5, 17, 18, 194atexlemtlw 33065 . . . . . . 7  |-  ( ph  ->  T  .<_  W )
368, 5atbase 32288 . . . . . . . . 9  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
3720, 36syl 17 . . . . . . . 8  |-  ( ph  ->  V  e.  ( Base `  K ) )
388, 5atbase 32288 . . . . . . . . 9  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
3922, 38syl 17 . . . . . . . 8  |-  ( ph  ->  T  e.  ( Base `  K ) )
408, 9, 4latjle12 15908 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( V  e.  ( Base `  K )  /\  T  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) ) )  -> 
( ( V  .<_  W  /\  T  .<_  W )  <-> 
( V  .\/  T
)  .<_  W ) )
413, 37, 39, 31, 40syl13anc 1232 . . . . . . 7  |-  ( ph  ->  ( ( V  .<_  W  /\  T  .<_  W )  <-> 
( V  .\/  T
)  .<_  W ) )
4234, 35, 41mpbi2and 922 . . . . . 6  |-  ( ph  ->  ( V  .\/  T
)  .<_  W )
438, 5atbase 32288 . . . . . . . 8  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
4421, 43syl 17 . . . . . . 7  |-  ( ph  ->  Q  e.  ( Base `  K ) )
4524atexlemk 33045 . . . . . . . 8  |-  ( ph  ->  K  e.  HL )
468, 4, 5hlatjcl 32365 . . . . . . . 8  |-  ( ( K  e.  HL  /\  V  e.  A  /\  T  e.  A )  ->  ( V  .\/  T
)  e.  ( Base `  K ) )
4745, 20, 22, 46syl3anc 1230 . . . . . . 7  |-  ( ph  ->  ( V  .\/  T
)  e.  ( Base `  K ) )
488, 9lattr 15902 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  ( V  .\/  T )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
) )  ->  (
( Q  .<_  ( V 
.\/  T )  /\  ( V  .\/  T ) 
.<_  W )  ->  Q  .<_  W ) )
493, 44, 47, 31, 48syl13anc 1232 . . . . . 6  |-  ( ph  ->  ( ( Q  .<_  ( V  .\/  T )  /\  ( V  .\/  T )  .<_  W )  ->  Q  .<_  W )
)
5042, 49mpan2d 672 . . . . 5  |-  ( ph  ->  ( Q  .<_  ( V 
.\/  T )  ->  Q  .<_  W ) )
5130, 50syld 42 . . . 4  |-  ( ph  ->  ( V  .<_  ( Q 
.\/  T )  ->  Q  .<_  W ) )
5215, 51mtod 177 . . 3  |-  ( ph  ->  -.  V  .<_  ( Q 
.\/  T ) )
53 nbrne2 4412 . . 3  |-  ( ( C  .<_  ( Q  .\/  T )  /\  -.  V  .<_  ( Q  .\/  T ) )  ->  C  =/=  V )
5413, 52, 53syl2anc 659 . 2  |-  ( ph  ->  C  =/=  V )
5524atexlemw 33046 . . . 4  |-  ( ph  ->  W  e.  H )
5645, 55jca 530 . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
5724atexlempw 33047 . . 3  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
5824atexlems 33050 . . 3  |-  ( ph  ->  S  e.  A )
592, 9, 4, 10, 5, 17, 18, 19, 14atexlemc 33067 . . 3  |-  ( ph  ->  C  e.  A )
602, 9, 4, 54atexlempns 33060 . . 3  |-  ( ph  ->  P  =/=  S )
618, 9, 10latmle2 15923 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( P  .\/  S ) )
623, 6, 7, 61syl3anc 1230 . . . 4  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( P  .\/  S ) )
631, 62syl5eqbr 4427 . . 3  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
649, 4, 10, 5, 17, 19lhpat3 33044 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( S  e.  A  /\  C  e.  A )  /\  ( P  =/=  S  /\  C  .<_  ( P  .\/  S
) ) )  -> 
( -.  C  .<_  W  <-> 
C  =/=  V ) )
6556, 57, 58, 59, 60, 63, 64syl222anc 1246 . 2  |-  ( ph  ->  ( -.  C  .<_  W  <-> 
C  =/=  V ) )
6654, 65mpbird 232 1  |-  ( ph  ->  -.  C  .<_  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   class class class wbr 4394   ` cfv 5525  (class class class)co 6234   Basecbs 14733   lecple 14808   joincjn 15789   meetcmee 15790   Latclat 15891   Atomscatm 32262   CvLatclc 32264   HLchlt 32349   LHypclh 32982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-preset 15773  df-poset 15791  df-plt 15804  df-lub 15820  df-glb 15821  df-join 15822  df-meet 15823  df-p0 15885  df-p1 15886  df-lat 15892  df-clat 15954  df-oposet 32175  df-ol 32177  df-oml 32178  df-covers 32265  df-ats 32266  df-atl 32297  df-cvlat 32321  df-hlat 32350  df-llines 32496  df-lplanes 32497  df-lhyp 32986
This theorem is referenced by:  4atexlemex2  33069  4atexlemcnd  33070
  Copyright terms: Public domain W3C validator