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Theorem 4atexlemtlw 34738
Description: Lemma for 4atexlem7 34746. (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 2460 . 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 34728 . 2  |-  ( ph  ->  K  e.  Lat )
534atexlemt 34724 . . 3  |-  ( ph  ->  T  e.  A )
6 4thatlem0.a . . . 4  |-  A  =  ( Atoms `  K )
71, 6atbase 33961 . . 3  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
85, 7syl 16 . 2  |-  ( ph  ->  T  e.  ( Base `  K ) )
934atexlemk 34718 . . 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 34735 . . 3  |-  ( ph  ->  U  e.  A )
15 4thatlem0.v . . . 4  |-  V  =  ( ( P  .\/  S )  ./\  W )
163, 2, 10, 11, 6, 12, 13, 154atexlemv 34736 . . 3  |-  ( ph  ->  V  e.  A )
171, 10, 6hlatjcl 34038 . . 3  |-  ( ( K  e.  HL  /\  U  e.  A  /\  V  e.  A )  ->  ( U  .\/  V
)  e.  ( Base `  K ) )
189, 14, 16, 17syl3anc 1223 . 2  |-  ( ph  ->  ( U  .\/  V
)  e.  ( Base `  K ) )
193, 124atexlemwb 34730 . 2  |-  ( ph  ->  W  e.  ( Base `  K ) )
2034atexlemkc 34729 . . 3  |-  ( ph  ->  K  e.  CvLat )
213, 2, 10, 11, 6, 12, 13, 154atexlemunv 34737 . . 3  |-  ( ph  ->  U  =/=  V )
2234atexlemutvt 34725 . . 3  |-  ( ph  ->  ( U  .\/  T
)  =  ( V 
.\/  T ) )
236, 2, 10cvlsupr4 34017 . . 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 1241 . 2  |-  ( ph  ->  T  .<_  ( U  .\/  V ) )
2534atexlemp 34721 . . . . . 6  |-  ( ph  ->  P  e.  A )
2634atexlemq 34722 . . . . . 6  |-  ( ph  ->  Q  e.  A )
271, 10, 6hlatjcl 34038 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
289, 25, 26, 27syl3anc 1223 . . . . 5  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
291, 2, 11latmle2 15553 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
304, 28, 19, 29syl3anc 1223 . . . 4  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
3113, 30syl5eqbr 4473 . . 3  |-  ( ph  ->  U  .<_  W )
323, 10, 64atexlempsb 34731 . . . . 5  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
331, 2, 11latmle2 15553 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  W )
344, 32, 19, 33syl3anc 1223 . . . 4  |-  ( ph  ->  ( ( P  .\/  S )  ./\  W )  .<_  W )
3515, 34syl5eqbr 4473 . . 3  |-  ( ph  ->  V  .<_  W )
361, 6atbase 33961 . . . . 5  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
3714, 36syl 16 . . . 4  |-  ( ph  ->  U  e.  ( Base `  K ) )
381, 6atbase 33961 . . . . 5  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
3916, 38syl 16 . . . 4  |-  ( ph  ->  V  e.  ( Base `  K ) )
401, 2, 10latjle12 15538 . . . 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 1225 . . 3  |-  ( ph  ->  ( ( U  .<_  W  /\  V  .<_  W )  <-> 
( U  .\/  V
)  .<_  W ) )
4231, 35, 41mpbi2and 914 . 2  |-  ( ph  ->  ( U  .\/  V
)  .<_  W )
431, 2, 4, 8, 18, 19, 24, 42lattrd 15534 1  |-  ( ph  ->  T  .<_  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   lecple 14551   joincjn 15420   meetcmee 15421   Latclat 15521   Atomscatm 33935   CvLatclc 33937   HLchlt 34022   LHypclh 34655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-p1 15516  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-lhyp 34659
This theorem is referenced by:  4atexlemntlpq  34739  4atexlemnclw  34741  4atexlemcnd  34743
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