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Theorem 4atexlemnclw 34033
Description: Lemma for 4atexlem7 34038. (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 34020 . . . . 5  |-  ( ph  ->  K  e.  Lat )
4 4thatlem0.j . . . . . 6  |-  .\/  =  ( join `  K )
5 4thatlem0.a . . . . . 6  |-  A  =  ( Atoms `  K )
62, 4, 54atexlemqtb 34024 . . . . 5  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
72, 4, 54atexlempsb 34023 . . . . 5  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
8 eqid 2452 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
9 4thatlem0.l . . . . . 6  |-  .<_  =  ( le `  K )
10 4thatlem0.m . . . . . 6  |-  ./\  =  ( meet `  K )
118, 9, 10latmle1 15360 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
123, 6, 7, 11syl3anc 1219 . . . 4  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
131, 12syl5eqbr 4428 . . 3  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
14 simp13r 1104 . . . . 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 34021 . . . . . 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 34028 . . . . . 6  |-  ( ph  ->  V  e.  A )
2124atexlemq 34014 . . . . . 6  |-  ( ph  ->  Q  e.  A )
2224atexlemt 34016 . . . . . 6  |-  ( ph  ->  T  e.  A )
232, 9, 4, 10, 5, 17, 184atexlemu 34027 . . . . . . 7  |-  ( ph  ->  U  e.  A )
242, 9, 4, 10, 5, 17, 18, 194atexlemunv 34029 . . . . . . 7  |-  ( ph  ->  U  =/=  V )
2524atexlemutvt 34017 . . . . . . 7  |-  ( ph  ->  ( U  .\/  T
)  =  ( V 
.\/  T ) )
265, 4cvlsupr6 33311 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( U  e.  A  /\  V  e.  A  /\  T  e.  A )  /\  ( U  =/=  V  /\  ( U  .\/  T
)  =  ( V 
.\/  T ) ) )  ->  T  =/=  V )
2726necomd 2720 . . . . . . 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 1237 . . . . . 6  |-  ( ph  ->  V  =/=  T )
299, 4, 5cvlatexch2 33301 . . . . . 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 1232 . . . . 5  |-  ( ph  ->  ( V  .<_  ( Q 
.\/  T )  ->  Q  .<_  ( V  .\/  T ) ) )
312, 174atexlemwb 34022 . . . . . . . . 9  |-  ( ph  ->  W  e.  ( Base `  K ) )
328, 9, 10latmle2 15361 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  W )
333, 7, 31, 32syl3anc 1219 . . . . . . . 8  |-  ( ph  ->  ( ( P  .\/  S )  ./\  W )  .<_  W )
3419, 33syl5eqbr 4428 . . . . . . 7  |-  ( ph  ->  V  .<_  W )
352, 9, 4, 10, 5, 17, 18, 194atexlemtlw 34030 . . . . . . 7  |-  ( ph  ->  T  .<_  W )
368, 5atbase 33253 . . . . . . . . 9  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
3720, 36syl 16 . . . . . . . 8  |-  ( ph  ->  V  e.  ( Base `  K ) )
388, 5atbase 33253 . . . . . . . . 9  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
3922, 38syl 16 . . . . . . . 8  |-  ( ph  ->  T  e.  ( Base `  K ) )
408, 9, 4latjle12 15346 . . . . . . . 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 1221 . . . . . . 7  |-  ( ph  ->  ( ( V  .<_  W  /\  T  .<_  W )  <-> 
( V  .\/  T
)  .<_  W ) )
4234, 35, 41mpbi2and 912 . . . . . 6  |-  ( ph  ->  ( V  .\/  T
)  .<_  W )
438, 5atbase 33253 . . . . . . . 8  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
4421, 43syl 16 . . . . . . 7  |-  ( ph  ->  Q  e.  ( Base `  K ) )
4524atexlemk 34010 . . . . . . . 8  |-  ( ph  ->  K  e.  HL )
468, 4, 5hlatjcl 33330 . . . . . . . 8  |-  ( ( K  e.  HL  /\  V  e.  A  /\  T  e.  A )  ->  ( V  .\/  T
)  e.  ( Base `  K ) )
4745, 20, 22, 46syl3anc 1219 . . . . . . 7  |-  ( ph  ->  ( V  .\/  T
)  e.  ( Base `  K ) )
488, 9lattr 15340 . . . . . . 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 1221 . . . . . 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 4413 . . 3  |-  ( ( C  .<_  ( Q  .\/  T )  /\  -.  V  .<_  ( Q  .\/  T ) )  ->  C  =/=  V )
5413, 52, 53syl2anc 661 . 2  |-  ( ph  ->  C  =/=  V )
5524atexlemw 34011 . . . 4  |-  ( ph  ->  W  e.  H )
5645, 55jca 532 . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
5724atexlempw 34012 . . 3  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
5824atexlems 34015 . . 3  |-  ( ph  ->  S  e.  A )
592, 9, 4, 10, 5, 17, 18, 19, 14atexlemc 34032 . . 3  |-  ( ph  ->  C  e.  A )
602, 9, 4, 54atexlempns 34025 . . 3  |-  ( ph  ->  P  =/=  S )
618, 9, 10latmle2 15361 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( P  .\/  S ) )
623, 6, 7, 61syl3anc 1219 . . . 4  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( P  .\/  S ) )
631, 62syl5eqbr 4428 . . 3  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
649, 4, 10, 5, 17, 19lhpat3 34009 . . 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 1235 . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2645   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   Basecbs 14287   lecple 14359   joincjn 15228   meetcmee 15229   Latclat 15329   Atomscatm 33227   CvLatclc 33229   HLchlt 33314   LHypclh 33947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-poset 15230  df-plt 15242  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-p0 15323  df-p1 15324  df-lat 15330  df-clat 15392  df-oposet 33140  df-ol 33142  df-oml 33143  df-covers 33230  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315  df-llines 33461  df-lplanes 33462  df-lhyp 33951
This theorem is referenced by:  4atexlemex2  34034  4atexlemcnd  34035
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