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Theorem 4atexlemnclw 30259
Description: Lemma for 4atexlem7 30264. (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 30246 . . . . 5  |-  ( ph  ->  K  e.  Lat )
4 4thatlem0.j . . . . . 6  |-  .\/  =  ( join `  K )
5 4thatlem0.a . . . . . 6  |-  A  =  ( Atoms `  K )
62, 4, 54atexlemqtb 30250 . . . . 5  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
72, 4, 54atexlempsb 30249 . . . . 5  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
8 eqid 2283 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
9 4thatlem0.l . . . . . 6  |-  .<_  =  ( le `  K )
10 4thatlem0.m . . . . . 6  |-  ./\  =  ( meet `  K )
118, 9, 10latmle1 14182 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
123, 6, 7, 11syl3anc 1182 . . . 4  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
131, 12syl5eqbr 4056 . . 3  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
14 simp13r 1071 . . . . 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 187 . . . 4  |-  ( ph  ->  -.  Q  .<_  W )
1624atexlemkc 30247 . . . . . 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 30254 . . . . . 6  |-  ( ph  ->  V  e.  A )
2124atexlemq 30240 . . . . . 6  |-  ( ph  ->  Q  e.  A )
2224atexlemt 30242 . . . . . 6  |-  ( ph  ->  T  e.  A )
232, 9, 4, 10, 5, 17, 184atexlemu 30253 . . . . . . 7  |-  ( ph  ->  U  e.  A )
242, 9, 4, 10, 5, 17, 18, 194atexlemunv 30255 . . . . . . 7  |-  ( ph  ->  U  =/=  V )
2524atexlemutvt 30243 . . . . . . 7  |-  ( ph  ->  ( U  .\/  T
)  =  ( V 
.\/  T ) )
265, 4cvlsupr6 29537 . . . . . . . 8  |-  ( ( K  e.  CvLat  /\  ( U  e.  A  /\  V  e.  A  /\  T  e.  A )  /\  ( U  =/=  V  /\  ( U  .\/  T
)  =  ( V 
.\/  T ) ) )  ->  T  =/=  V )
2726necomd 2529 . . . . . . 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 1200 . . . . . 6  |-  ( ph  ->  V  =/=  T )
299, 4, 5cvlatexch2 29527 . . . . . 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 1195 . . . . 5  |-  ( ph  ->  ( V  .<_  ( Q 
.\/  T )  ->  Q  .<_  ( V  .\/  T ) ) )
312, 174atexlemwb 30248 . . . . . . . . 9  |-  ( ph  ->  W  e.  ( Base `  K ) )
328, 9, 10latmle2 14183 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  W )
333, 7, 31, 32syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( ( P  .\/  S )  ./\  W )  .<_  W )
3419, 33syl5eqbr 4056 . . . . . . 7  |-  ( ph  ->  V  .<_  W )
352, 9, 4, 10, 5, 17, 18, 194atexlemtlw 30256 . . . . . . 7  |-  ( ph  ->  T  .<_  W )
368, 5atbase 29479 . . . . . . . . 9  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
3720, 36syl 15 . . . . . . . 8  |-  ( ph  ->  V  e.  ( Base `  K ) )
388, 5atbase 29479 . . . . . . . . 9  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
3922, 38syl 15 . . . . . . . 8  |-  ( ph  ->  T  e.  ( Base `  K ) )
408, 9, 4latjle12 14168 . . . . . . . 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 1184 . . . . . . 7  |-  ( ph  ->  ( ( V  .<_  W  /\  T  .<_  W )  <-> 
( V  .\/  T
)  .<_  W ) )
4234, 35, 41mpbi2and 887 . . . . . 6  |-  ( ph  ->  ( V  .\/  T
)  .<_  W )
438, 5atbase 29479 . . . . . . . 8  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
4421, 43syl 15 . . . . . . 7  |-  ( ph  ->  Q  e.  ( Base `  K ) )
4524atexlemk 30236 . . . . . . . 8  |-  ( ph  ->  K  e.  HL )
468, 4, 5hlatjcl 29556 . . . . . . . 8  |-  ( ( K  e.  HL  /\  V  e.  A  /\  T  e.  A )  ->  ( V  .\/  T
)  e.  ( Base `  K ) )
4745, 20, 22, 46syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( V  .\/  T
)  e.  ( Base `  K ) )
488, 9lattr 14162 . . . . . . 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 1184 . . . . . 6  |-  ( ph  ->  ( ( Q  .<_  ( V  .\/  T )  /\  ( V  .\/  T )  .<_  W )  ->  Q  .<_  W )
)
5042, 49mpan2d 655 . . . . 5  |-  ( ph  ->  ( Q  .<_  ( V 
.\/  T )  ->  Q  .<_  W ) )
5130, 50syld 40 . . . 4  |-  ( ph  ->  ( V  .<_  ( Q 
.\/  T )  ->  Q  .<_  W ) )
5215, 51mtod 168 . . 3  |-  ( ph  ->  -.  V  .<_  ( Q 
.\/  T ) )
53 nbrne2 4041 . . 3  |-  ( ( C  .<_  ( Q  .\/  T )  /\  -.  V  .<_  ( Q  .\/  T ) )  ->  C  =/=  V )
5413, 52, 53syl2anc 642 . 2  |-  ( ph  ->  C  =/=  V )
5524atexlemw 30237 . . . 4  |-  ( ph  ->  W  e.  H )
5645, 55jca 518 . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
5724atexlempw 30238 . . 3  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
5824atexlems 30241 . . 3  |-  ( ph  ->  S  e.  A )
592, 9, 4, 10, 5, 17, 18, 19, 14atexlemc 30258 . . 3  |-  ( ph  ->  C  e.  A )
602, 9, 4, 54atexlempns 30251 . . 3  |-  ( ph  ->  P  =/=  S )
618, 9, 10latmle2 14183 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( P  .\/  S ) )
623, 6, 7, 61syl3anc 1182 . . . 4  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( P  .\/  S ) )
631, 62syl5eqbr 4056 . . 3  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
649, 4, 10, 5, 17, 19lhpat3 30235 . . 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 1198 . 2  |-  ( ph  ->  ( -.  C  .<_  W  <-> 
C  =/=  V ) )
6654, 65mpbird 223 1  |-  ( ph  ->  -.  C  .<_  W )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Latclat 14151   Atomscatm 29453   CvLatclc 29455   HLchlt 29540   LHypclh 30173
This theorem is referenced by:  4atexlemex2  30260  4atexlemcnd  30261
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lhyp 30177
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