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

Theorem 4atexlemtlw 30765
Description: Lemma for 4atexlem7 30773. (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 )
Assertion
Ref Expression
4atexlemtlw  |-  ( ph  ->  T  .<_  W )

Proof of Theorem 4atexlemtlw
StepHypRef Expression
1 eqid 2435 . 2  |-  ( Base `  K )  =  (
Base `  K )
2 4thatlem0.l . 2  |-  .<_  =  ( le `  K )
3 4thatlem.ph . . 3  |-  ( 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 ) ) ) )
434atexlemkl 30755 . 2  |-  ( ph  ->  K  e.  Lat )
534atexlemt 30751 . . 3  |-  ( ph  ->  T  e.  A )
6 4thatlem0.a . . . 4  |-  A  =  ( Atoms `  K )
71, 6atbase 29988 . . 3  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
85, 7syl 16 . 2  |-  ( ph  ->  T  e.  ( Base `  K ) )
934atexlemk 30745 . . 3  |-  ( ph  ->  K  e.  HL )
10 4thatlem0.j . . . 4  |-  .\/  =  ( join `  K )
11 4thatlem0.m . . . 4  |-  ./\  =  ( meet `  K )
12 4thatlem0.h . . . 4  |-  H  =  ( LHyp `  K
)
13 4thatlem0.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
143, 2, 10, 11, 6, 12, 134atexlemu 30762 . . 3  |-  ( ph  ->  U  e.  A )
15 4thatlem0.v . . . 4  |-  V  =  ( ( P  .\/  S )  ./\  W )
163, 2, 10, 11, 6, 12, 13, 154atexlemv 30763 . . 3  |-  ( ph  ->  V  e.  A )
171, 10, 6hlatjcl 30065 . . 3  |-  ( ( K  e.  HL  /\  U  e.  A  /\  V  e.  A )  ->  ( U  .\/  V
)  e.  ( Base `  K ) )
189, 14, 16, 17syl3anc 1184 . 2  |-  ( ph  ->  ( U  .\/  V
)  e.  ( Base `  K ) )
193, 124atexlemwb 30757 . 2  |-  ( ph  ->  W  e.  ( Base `  K ) )
2034atexlemkc 30756 . . 3  |-  ( ph  ->  K  e.  CvLat )
213, 2, 10, 11, 6, 12, 13, 154atexlemunv 30764 . . 3  |-  ( ph  ->  U  =/=  V )
2234atexlemutvt 30752 . . 3  |-  ( ph  ->  ( U  .\/  T
)  =  ( V 
.\/  T ) )
236, 2, 10cvlsupr4 30044 . . 3  |-  ( ( K  e.  CvLat  /\  ( U  e.  A  /\  V  e.  A  /\  T  e.  A )  /\  ( U  =/=  V  /\  ( U  .\/  T
)  =  ( V 
.\/  T ) ) )  ->  T  .<_  ( U  .\/  V ) )
2420, 14, 16, 5, 21, 22, 23syl132anc 1202 . 2  |-  ( ph  ->  T  .<_  ( U  .\/  V ) )
2534atexlemp 30748 . . . . . 6  |-  ( ph  ->  P  e.  A )
2634atexlemq 30749 . . . . . 6  |-  ( ph  ->  Q  e.  A )
271, 10, 6hlatjcl 30065 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
289, 25, 26, 27syl3anc 1184 . . . . 5  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
291, 2, 11latmle2 14496 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
304, 28, 19, 29syl3anc 1184 . . . 4  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
3113, 30syl5eqbr 4237 . . 3  |-  ( ph  ->  U  .<_  W )
323, 10, 64atexlempsb 30758 . . . . 5  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
331, 2, 11latmle2 14496 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  W )
344, 32, 19, 33syl3anc 1184 . . . 4  |-  ( ph  ->  ( ( P  .\/  S )  ./\  W )  .<_  W )
3515, 34syl5eqbr 4237 . . 3  |-  ( ph  ->  V  .<_  W )
361, 6atbase 29988 . . . . 5  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
3714, 36syl 16 . . . 4  |-  ( ph  ->  U  e.  ( Base `  K ) )
381, 6atbase 29988 . . . . 5  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
3916, 38syl 16 . . . 4  |-  ( ph  ->  V  e.  ( Base `  K ) )
401, 2, 10latjle12 14481 . . . 4  |-  ( ( K  e.  Lat  /\  ( U  e.  ( Base `  K )  /\  V  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) ) )  -> 
( ( U  .<_  W  /\  V  .<_  W )  <-> 
( U  .\/  V
)  .<_  W ) )
414, 37, 39, 19, 40syl13anc 1186 . . 3  |-  ( ph  ->  ( ( U  .<_  W  /\  V  .<_  W )  <-> 
( U  .\/  V
)  .<_  W ) )
4231, 35, 41mpbi2and 888 . 2  |-  ( ph  ->  ( U  .\/  V
)  .<_  W )
431, 2, 4, 8, 18, 19, 24, 42lattrd 14477 1  |-  ( ph  ->  T  .<_  W )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13459   lecple 13526   joincjn 14391   meetcmee 14392   Latclat 14464   Atomscatm 29962   CvLatclc 29964   HLchlt 30049   LHypclh 30682
This theorem is referenced by:  4atexlemntlpq  30766  4atexlemnclw  30768  4atexlemcnd  30770
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14393  df-plt 14405  df-lub 14421  df-glb 14422  df-join 14423  df-meet 14424  df-p0 14458  df-p1 14459  df-lat 14465  df-clat 14527  df-oposet 29875  df-ol 29877  df-oml 29878  df-covers 29965  df-ats 29966  df-atl 29997  df-cvlat 30021  df-hlat 30050  df-lhyp 30686
  Copyright terms: Public domain W3C validator