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Theorem 4atexlemtlw 30183
Description: Lemma for 4atexlem7 30191. (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 2389 . 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 30173 . 2  |-  ( ph  ->  K  e.  Lat )
534atexlemt 30169 . . 3  |-  ( ph  ->  T  e.  A )
6 4thatlem0.a . . . 4  |-  A  =  ( Atoms `  K )
71, 6atbase 29406 . . 3  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
85, 7syl 16 . 2  |-  ( ph  ->  T  e.  ( Base `  K ) )
934atexlemk 30163 . . 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 30180 . . 3  |-  ( ph  ->  U  e.  A )
15 4thatlem0.v . . . 4  |-  V  =  ( ( P  .\/  S )  ./\  W )
163, 2, 10, 11, 6, 12, 13, 154atexlemv 30181 . . 3  |-  ( ph  ->  V  e.  A )
171, 10, 6hlatjcl 29483 . . 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 30175 . 2  |-  ( ph  ->  W  e.  ( Base `  K ) )
2034atexlemkc 30174 . . 3  |-  ( ph  ->  K  e.  CvLat )
213, 2, 10, 11, 6, 12, 13, 154atexlemunv 30182 . . 3  |-  ( ph  ->  U  =/=  V )
2234atexlemutvt 30170 . . 3  |-  ( ph  ->  ( U  .\/  T
)  =  ( V 
.\/  T ) )
236, 2, 10cvlsupr4 29462 . . 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 30166 . . . . . 6  |-  ( ph  ->  P  e.  A )
2634atexlemq 30167 . . . . . 6  |-  ( ph  ->  Q  e.  A )
271, 10, 6hlatjcl 29483 . . . . . 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 14435 . . . . 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 4188 . . 3  |-  ( ph  ->  U  .<_  W )
323, 10, 64atexlempsb 30176 . . . . 5  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
331, 2, 11latmle2 14435 . . . . 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 4188 . . 3  |-  ( ph  ->  V  .<_  W )
361, 6atbase 29406 . . . . 5  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
3714, 36syl 16 . . . 4  |-  ( ph  ->  U  e.  ( Base `  K ) )
381, 6atbase 29406 . . . . 5  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
3916, 38syl 16 . . . 4  |-  ( ph  ->  V  e.  ( Base `  K ) )
401, 2, 10latjle12 14420 . . . 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 14416 1  |-  ( ph  ->  T  .<_  W )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2552   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   Basecbs 13398   lecple 13465   joincjn 14330   meetcmee 14331   Latclat 14403   Atomscatm 29380   CvLatclc 29382   HLchlt 29467   LHypclh 30100
This theorem is referenced by:  4atexlemntlpq  30184  4atexlemnclw  30186  4atexlemcnd  30188
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-poset 14332  df-plt 14344  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-p0 14397  df-p1 14398  df-lat 14404  df-clat 14466  df-oposet 29293  df-ol 29295  df-oml 29296  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468  df-lhyp 30104
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