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Theorem 4atexlemnclw 30929
Description: Lemma for 4atexlem7 30934. (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 30916 . . . . 5  |-  ( ph  ->  K  e.  Lat )
4 4thatlem0.j . . . . . 6  |-  .\/  =  ( join `  K )
5 4thatlem0.a . . . . . 6  |-  A  =  ( Atoms `  K )
62, 4, 54atexlemqtb 30920 . . . . 5  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
72, 4, 54atexlempsb 30919 . . . . 5  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
8 eqid 2438 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
9 4thatlem0.l . . . . . 6  |-  .<_  =  ( le `  K )
10 4thatlem0.m . . . . . 6  |-  ./\  =  ( meet `  K )
118, 9, 10latmle1 14507 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
123, 6, 7, 11syl3anc 1185 . . . 4  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
131, 12syl5eqbr 4247 . . 3  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
14 simp13r 1074 . . . . 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 189 . . . 4  |-  ( ph  ->  -.  Q  .<_  W )
1624atexlemkc 30917 . . . . . 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 30924 . . . . . 6  |-  ( ph  ->  V  e.  A )
2124atexlemq 30910 . . . . . 6  |-  ( ph  ->  Q  e.  A )
2224atexlemt 30912 . . . . . 6  |-  ( ph  ->  T  e.  A )
232, 9, 4, 10, 5, 17, 184atexlemu 30923 . . . . . . 7  |-  ( ph  ->  U  e.  A )
242, 9, 4, 10, 5, 17, 18, 194atexlemunv 30925 . . . . . . 7  |-  ( ph  ->  U  =/=  V )
2524atexlemutvt 30913 . . . . . . 7  |-  ( ph  ->  ( U  .\/  T
)  =  ( V 
.\/  T ) )
265, 4cvlsupr6 30207 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( U  e.  A  /\  V  e.  A  /\  T  e.  A )  /\  ( U  =/=  V  /\  ( U  .\/  T
)  =  ( V 
.\/  T ) ) )  ->  T  =/=  V )
2726necomd 2689 . . . . . . 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 1203 . . . . . 6  |-  ( ph  ->  V  =/=  T )
299, 4, 5cvlatexch2 30197 . . . . . 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 1198 . . . . 5  |-  ( ph  ->  ( V  .<_  ( Q 
.\/  T )  ->  Q  .<_  ( V  .\/  T ) ) )
312, 174atexlemwb 30918 . . . . . . . . 9  |-  ( ph  ->  W  e.  ( Base `  K ) )
328, 9, 10latmle2 14508 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  W )
333, 7, 31, 32syl3anc 1185 . . . . . . . 8  |-  ( ph  ->  ( ( P  .\/  S )  ./\  W )  .<_  W )
3419, 33syl5eqbr 4247 . . . . . . 7  |-  ( ph  ->  V  .<_  W )
352, 9, 4, 10, 5, 17, 18, 194atexlemtlw 30926 . . . . . . 7  |-  ( ph  ->  T  .<_  W )
368, 5atbase 30149 . . . . . . . . 9  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
3720, 36syl 16 . . . . . . . 8  |-  ( ph  ->  V  e.  ( Base `  K ) )
388, 5atbase 30149 . . . . . . . . 9  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
3922, 38syl 16 . . . . . . . 8  |-  ( ph  ->  T  e.  ( Base `  K ) )
408, 9, 4latjle12 14493 . . . . . . . 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 1187 . . . . . . 7  |-  ( ph  ->  ( ( V  .<_  W  /\  T  .<_  W )  <-> 
( V  .\/  T
)  .<_  W ) )
4234, 35, 41mpbi2and 889 . . . . . 6  |-  ( ph  ->  ( V  .\/  T
)  .<_  W )
438, 5atbase 30149 . . . . . . . 8  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
4421, 43syl 16 . . . . . . 7  |-  ( ph  ->  Q  e.  ( Base `  K ) )
4524atexlemk 30906 . . . . . . . 8  |-  ( ph  ->  K  e.  HL )
468, 4, 5hlatjcl 30226 . . . . . . . 8  |-  ( ( K  e.  HL  /\  V  e.  A  /\  T  e.  A )  ->  ( V  .\/  T
)  e.  ( Base `  K ) )
4745, 20, 22, 46syl3anc 1185 . . . . . . 7  |-  ( ph  ->  ( V  .\/  T
)  e.  ( Base `  K ) )
488, 9lattr 14487 . . . . . . 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 1187 . . . . . 6  |-  ( ph  ->  ( ( Q  .<_  ( V  .\/  T )  /\  ( V  .\/  T )  .<_  W )  ->  Q  .<_  W )
)
5042, 49mpan2d 657 . . . . 5  |-  ( ph  ->  ( Q  .<_  ( V 
.\/  T )  ->  Q  .<_  W ) )
5130, 50syld 43 . . . 4  |-  ( ph  ->  ( V  .<_  ( Q 
.\/  T )  ->  Q  .<_  W ) )
5215, 51mtod 171 . . 3  |-  ( ph  ->  -.  V  .<_  ( Q 
.\/  T ) )
53 nbrne2 4232 . . 3  |-  ( ( C  .<_  ( Q  .\/  T )  /\  -.  V  .<_  ( Q  .\/  T ) )  ->  C  =/=  V )
5413, 52, 53syl2anc 644 . 2  |-  ( ph  ->  C  =/=  V )
5524atexlemw 30907 . . . 4  |-  ( ph  ->  W  e.  H )
5645, 55jca 520 . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
5724atexlempw 30908 . . 3  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
5824atexlems 30911 . . 3  |-  ( ph  ->  S  e.  A )
592, 9, 4, 10, 5, 17, 18, 19, 14atexlemc 30928 . . 3  |-  ( ph  ->  C  e.  A )
602, 9, 4, 54atexlempns 30921 . . 3  |-  ( ph  ->  P  =/=  S )
618, 9, 10latmle2 14508 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( P  .\/  S ) )
623, 6, 7, 61syl3anc 1185 . . . 4  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( P  .\/  S ) )
631, 62syl5eqbr 4247 . . 3  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
649, 4, 10, 5, 17, 19lhpat3 30905 . . 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 1201 . 2  |-  ( ph  ->  ( -.  C  .<_  W  <-> 
C  =/=  V ) )
6654, 65mpbird 225 1  |-  ( ph  ->  -.  C  .<_  W )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   meetcmee 14404   Latclat 14476   Atomscatm 30123   CvLatclc 30125   HLchlt 30210   LHypclh 30843
This theorem is referenced by:  4atexlemex2  30930  4atexlemcnd  30931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-llines 30357  df-lplanes 30358  df-lhyp 30847
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