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Theorem 4atexlemtlw 33344
Description: Lemma for 4atexlem7 33352. (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 2429 . 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 33334 . 2  |-  ( ph  ->  K  e.  Lat )
534atexlemt 33330 . . 3  |-  ( ph  ->  T  e.  A )
6 4thatlem0.a . . . 4  |-  A  =  ( Atoms `  K )
71, 6atbase 32567 . . 3  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
85, 7syl 17 . 2  |-  ( ph  ->  T  e.  ( Base `  K ) )
934atexlemk 33324 . . 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 33341 . . 3  |-  ( ph  ->  U  e.  A )
15 4thatlem0.v . . . 4  |-  V  =  ( ( P  .\/  S )  ./\  W )
163, 2, 10, 11, 6, 12, 13, 154atexlemv 33342 . . 3  |-  ( ph  ->  V  e.  A )
171, 10, 6hlatjcl 32644 . . 3  |-  ( ( K  e.  HL  /\  U  e.  A  /\  V  e.  A )  ->  ( U  .\/  V
)  e.  ( Base `  K ) )
189, 14, 16, 17syl3anc 1264 . 2  |-  ( ph  ->  ( U  .\/  V
)  e.  ( Base `  K ) )
193, 124atexlemwb 33336 . 2  |-  ( ph  ->  W  e.  ( Base `  K ) )
2034atexlemkc 33335 . . 3  |-  ( ph  ->  K  e.  CvLat )
213, 2, 10, 11, 6, 12, 13, 154atexlemunv 33343 . . 3  |-  ( ph  ->  U  =/=  V )
2234atexlemutvt 33331 . . 3  |-  ( ph  ->  ( U  .\/  T
)  =  ( V 
.\/  T ) )
236, 2, 10cvlsupr4 32623 . . 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 1282 . 2  |-  ( ph  ->  T  .<_  ( U  .\/  V ) )
2534atexlemp 33327 . . . . . 6  |-  ( ph  ->  P  e.  A )
2634atexlemq 33328 . . . . . 6  |-  ( ph  ->  Q  e.  A )
271, 10, 6hlatjcl 32644 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
289, 25, 26, 27syl3anc 1264 . . . . 5  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
291, 2, 11latmle2 16274 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
304, 28, 19, 29syl3anc 1264 . . . 4  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
3113, 30syl5eqbr 4459 . . 3  |-  ( ph  ->  U  .<_  W )
323, 10, 64atexlempsb 33337 . . . . 5  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
331, 2, 11latmle2 16274 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  W )
344, 32, 19, 33syl3anc 1264 . . . 4  |-  ( ph  ->  ( ( P  .\/  S )  ./\  W )  .<_  W )
3515, 34syl5eqbr 4459 . . 3  |-  ( ph  ->  V  .<_  W )
361, 6atbase 32567 . . . . 5  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
3714, 36syl 17 . . . 4  |-  ( ph  ->  U  e.  ( Base `  K ) )
381, 6atbase 32567 . . . . 5  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
3916, 38syl 17 . . . 4  |-  ( ph  ->  V  e.  ( Base `  K ) )
401, 2, 10latjle12 16259 . . . 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 1266 . . 3  |-  ( ph  ->  ( ( U  .<_  W  /\  V  .<_  W )  <-> 
( U  .\/  V
)  .<_  W ) )
4231, 35, 41mpbi2and 929 . 2  |-  ( ph  ->  ( U  .\/  V
)  .<_  W )
431, 2, 4, 8, 18, 19, 24, 42lattrd 16255 1  |-  ( ph  ->  T  .<_  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   lecple 15159   joincjn 16140   meetcmee 16141   Latclat 16242   Atomscatm 32541   CvLatclc 32543   HLchlt 32628   LHypclh 33261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-p1 16237  df-lat 16243  df-clat 16305  df-oposet 32454  df-ol 32456  df-oml 32457  df-covers 32544  df-ats 32545  df-atl 32576  df-cvlat 32600  df-hlat 32629  df-lhyp 33265
This theorem is referenced by:  4atexlemntlpq  33345  4atexlemnclw  33347  4atexlemcnd  33349
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