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Theorem 4atexlemnclw 34866
Description: Lemma for 4atexlem7 34871. (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 34853 . . . . 5  |-  ( ph  ->  K  e.  Lat )
4 4thatlem0.j . . . . . 6  |-  .\/  =  ( join `  K )
5 4thatlem0.a . . . . . 6  |-  A  =  ( Atoms `  K )
62, 4, 54atexlemqtb 34857 . . . . 5  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
72, 4, 54atexlempsb 34856 . . . . 5  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
8 eqid 2467 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
9 4thatlem0.l . . . . . 6  |-  .<_  =  ( le `  K )
10 4thatlem0.m . . . . . 6  |-  ./\  =  ( meet `  K )
118, 9, 10latmle1 15559 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
123, 6, 7, 11syl3anc 1228 . . . 4  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
131, 12syl5eqbr 4480 . . 3  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
14 simp13r 1112 . . . . 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 34854 . . . . . 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 34861 . . . . . 6  |-  ( ph  ->  V  e.  A )
2124atexlemq 34847 . . . . . 6  |-  ( ph  ->  Q  e.  A )
2224atexlemt 34849 . . . . . 6  |-  ( ph  ->  T  e.  A )
232, 9, 4, 10, 5, 17, 184atexlemu 34860 . . . . . . 7  |-  ( ph  ->  U  e.  A )
242, 9, 4, 10, 5, 17, 18, 194atexlemunv 34862 . . . . . . 7  |-  ( ph  ->  U  =/=  V )
2524atexlemutvt 34850 . . . . . . 7  |-  ( ph  ->  ( U  .\/  T
)  =  ( V 
.\/  T ) )
265, 4cvlsupr6 34144 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( U  e.  A  /\  V  e.  A  /\  T  e.  A )  /\  ( U  =/=  V  /\  ( U  .\/  T
)  =  ( V 
.\/  T ) ) )  ->  T  =/=  V )
2726necomd 2738 . . . . . . 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 1246 . . . . . 6  |-  ( ph  ->  V  =/=  T )
299, 4, 5cvlatexch2 34134 . . . . . 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 1241 . . . . 5  |-  ( ph  ->  ( V  .<_  ( Q 
.\/  T )  ->  Q  .<_  ( V  .\/  T ) ) )
312, 174atexlemwb 34855 . . . . . . . . 9  |-  ( ph  ->  W  e.  ( Base `  K ) )
328, 9, 10latmle2 15560 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  W )
333, 7, 31, 32syl3anc 1228 . . . . . . . 8  |-  ( ph  ->  ( ( P  .\/  S )  ./\  W )  .<_  W )
3419, 33syl5eqbr 4480 . . . . . . 7  |-  ( ph  ->  V  .<_  W )
352, 9, 4, 10, 5, 17, 18, 194atexlemtlw 34863 . . . . . . 7  |-  ( ph  ->  T  .<_  W )
368, 5atbase 34086 . . . . . . . . 9  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
3720, 36syl 16 . . . . . . . 8  |-  ( ph  ->  V  e.  ( Base `  K ) )
388, 5atbase 34086 . . . . . . . . 9  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
3922, 38syl 16 . . . . . . . 8  |-  ( ph  ->  T  e.  ( Base `  K ) )
408, 9, 4latjle12 15545 . . . . . . . 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 1230 . . . . . . 7  |-  ( ph  ->  ( ( V  .<_  W  /\  T  .<_  W )  <-> 
( V  .\/  T
)  .<_  W ) )
4234, 35, 41mpbi2and 919 . . . . . 6  |-  ( ph  ->  ( V  .\/  T
)  .<_  W )
438, 5atbase 34086 . . . . . . . 8  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
4421, 43syl 16 . . . . . . 7  |-  ( ph  ->  Q  e.  ( Base `  K ) )
4524atexlemk 34843 . . . . . . . 8  |-  ( ph  ->  K  e.  HL )
468, 4, 5hlatjcl 34163 . . . . . . . 8  |-  ( ( K  e.  HL  /\  V  e.  A  /\  T  e.  A )  ->  ( V  .\/  T
)  e.  ( Base `  K ) )
4745, 20, 22, 46syl3anc 1228 . . . . . . 7  |-  ( ph  ->  ( V  .\/  T
)  e.  ( Base `  K ) )
488, 9lattr 15539 . . . . . . 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 1230 . . . . . 6  |-  ( ph  ->  ( ( Q  .<_  ( V  .\/  T )  /\  ( V  .\/  T )  .<_  W )  ->  Q  .<_  W )
)
5042, 49mpan2d 674 . . . . 5  |-  ( ph  ->  ( Q  .<_  ( V 
.\/  T )  ->  Q  .<_  W ) )
5130, 50syld 44 . . . 4  |-  ( ph  ->  ( V  .<_  ( Q 
.\/  T )  ->  Q  .<_  W ) )
5215, 51mtod 177 . . 3  |-  ( ph  ->  -.  V  .<_  ( Q 
.\/  T ) )
53 nbrne2 4465 . . 3  |-  ( ( C  .<_  ( Q  .\/  T )  /\  -.  V  .<_  ( Q  .\/  T ) )  ->  C  =/=  V )
5413, 52, 53syl2anc 661 . 2  |-  ( ph  ->  C  =/=  V )
5524atexlemw 34844 . . . 4  |-  ( ph  ->  W  e.  H )
5645, 55jca 532 . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
5724atexlempw 34845 . . 3  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
5824atexlems 34848 . . 3  |-  ( ph  ->  S  e.  A )
592, 9, 4, 10, 5, 17, 18, 19, 14atexlemc 34865 . . 3  |-  ( ph  ->  C  e.  A )
602, 9, 4, 54atexlempns 34858 . . 3  |-  ( ph  ->  P  =/=  S )
618, 9, 10latmle2 15560 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( P  .\/  S ) )
623, 6, 7, 61syl3anc 1228 . . . 4  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( P  .\/  S ) )
631, 62syl5eqbr 4480 . . 3  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
649, 4, 10, 5, 17, 19lhpat3 34842 . . 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 1244 . 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14486   lecple 14558   joincjn 15427   meetcmee 15428   Latclat 15528   Atomscatm 34060   CvLatclc 34062   HLchlt 34147   LHypclh 34780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-p1 15523  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294  df-lplanes 34295  df-lhyp 34784
This theorem is referenced by:  4atexlemex2  34867  4atexlemcnd  34868
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