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Theorem 4atexlemcnd 30186
Description: Lemma for 4atexlem7 30189. (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 ) )
4thatlem0.d  |-  D  =  ( ( R  .\/  T )  ./\  ( P  .\/  S ) )
Assertion
Ref Expression
4atexlemcnd  |-  ( ph  ->  C  =/=  D )

Proof of Theorem 4atexlemcnd
StepHypRef Expression
1 4thatlem.ph . . . 4  |-  ( 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 ) ) ) )
2 4thatlem0.l . . . 4  |-  .<_  =  ( le `  K )
3 4thatlem0.j . . . 4  |-  .\/  =  ( join `  K )
4 4thatlem0.m . . . 4  |-  ./\  =  ( meet `  K )
5 4thatlem0.a . . . 4  |-  A  =  ( Atoms `  K )
6 4thatlem0.h . . . 4  |-  H  =  ( LHyp `  K
)
7 4thatlem0.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
8 4thatlem0.v . . . 4  |-  V  =  ( ( P  .\/  S )  ./\  W )
91, 2, 3, 4, 5, 6, 7, 84atexlemtlw 30181 . . 3  |-  ( ph  ->  T  .<_  W )
10 4thatlem0.c . . . 4  |-  C  =  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )
111, 2, 3, 4, 5, 6, 7, 8, 104atexlemnclw 30184 . . 3  |-  ( ph  ->  -.  C  .<_  W )
12 nbrne2 4171 . . 3  |-  ( ( T  .<_  W  /\  -.  C  .<_  W )  ->  T  =/=  C
)
139, 11, 12syl2anc 643 . 2  |-  ( ph  ->  T  =/=  C )
1414atexlemk 30161 . . . . . . . . 9  |-  ( ph  ->  K  e.  HL )
1514atexlemq 30165 . . . . . . . . 9  |-  ( ph  ->  Q  e.  A )
1614atexlemt 30167 . . . . . . . . 9  |-  ( ph  ->  T  e.  A )
173, 5hlatjcom 29482 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  ( Q  .\/  T
)  =  ( T 
.\/  Q ) )
1814, 15, 16, 17syl3anc 1184 . . . . . . . 8  |-  ( ph  ->  ( Q  .\/  T
)  =  ( T 
.\/  Q ) )
19 simp221 1098 . . . . . . . . . 10  |-  ( ( ( ( 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 ) ) )  ->  R  e.  A )
201, 19sylbi 188 . . . . . . . . 9  |-  ( ph  ->  R  e.  A )
213, 5hlatjcom 29482 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  R  e.  A  /\  T  e.  A )  ->  ( R  .\/  T
)  =  ( T 
.\/  R ) )
2214, 20, 16, 21syl3anc 1184 . . . . . . . 8  |-  ( ph  ->  ( R  .\/  T
)  =  ( T 
.\/  R ) )
2318, 22oveq12d 6038 . . . . . . 7  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( R  .\/  T ) )  =  ( ( T  .\/  Q )  ./\  ( T  .\/  R ) ) )
2414atexlemkc 30172 . . . . . . . . 9  |-  ( ph  ->  K  e.  CvLat )
2514atexlemp 30164 . . . . . . . . 9  |-  ( ph  ->  P  e.  A )
2614atexlempnq 30169 . . . . . . . . 9  |-  ( ph  ->  P  =/=  Q )
27 simp223 1100 . . . . . . . . . 10  |-  ( ( ( ( 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 ) ) )  ->  ( P  .\/  R )  =  ( Q  .\/  R ) )
281, 27sylbi 188 . . . . . . . . 9  |-  ( ph  ->  ( P  .\/  R
)  =  ( Q 
.\/  R ) )
295, 3cvlsupr6 29462 . . . . . . . . . 10  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  R  =/=  Q )
3029necomd 2633 . . . . . . . . 9  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  Q  =/=  R )
3124, 25, 15, 20, 26, 28, 30syl132anc 1202 . . . . . . . 8  |-  ( ph  ->  Q  =/=  R )
321, 2, 3, 4, 5, 6, 7, 84atexlemntlpq 30182 . . . . . . . . 9  |-  ( ph  ->  -.  T  .<_  ( P 
.\/  Q ) )
335, 3cvlsupr7 29463 . . . . . . . . . . . 12  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  ( P  .\/  Q )  =  ( R  .\/  Q ) )
3424, 25, 15, 20, 26, 28, 33syl132anc 1202 . . . . . . . . . . 11  |-  ( ph  ->  ( P  .\/  Q
)  =  ( R 
.\/  Q ) )
353, 5hlatjcom 29482 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  =  ( R 
.\/  Q ) )
3614, 15, 20, 35syl3anc 1184 . . . . . . . . . . 11  |-  ( ph  ->  ( Q  .\/  R
)  =  ( R 
.\/  Q ) )
3734, 36eqtr4d 2422 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  Q
)  =  ( Q 
.\/  R ) )
3837breq2d 4165 . . . . . . . . 9  |-  ( ph  ->  ( T  .<_  ( P 
.\/  Q )  <->  T  .<_  ( Q  .\/  R ) ) )
3932, 38mtbid 292 . . . . . . . 8  |-  ( ph  ->  -.  T  .<_  ( Q 
.\/  R ) )
402, 3, 4, 52llnma2 29903 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  T  e.  A
)  /\  ( Q  =/=  R  /\  -.  T  .<_  ( Q  .\/  R
) ) )  -> 
( ( T  .\/  Q )  ./\  ( T  .\/  R ) )  =  T )
4114, 15, 20, 16, 31, 39, 40syl132anc 1202 . . . . . . 7  |-  ( ph  ->  ( ( T  .\/  Q )  ./\  ( T  .\/  R ) )  =  T )
4223, 41eqtr2d 2420 . . . . . 6  |-  ( ph  ->  T  =  ( ( Q  .\/  T ) 
./\  ( R  .\/  T ) ) )
4342adantr 452 . . . . 5  |-  ( (
ph  /\  C  =  D )  ->  T  =  ( ( Q 
.\/  T )  ./\  ( R  .\/  T ) ) )
4414atexlemkl 30171 . . . . . . . . . 10  |-  ( ph  ->  K  e.  Lat )
451, 3, 54atexlemqtb 30175 . . . . . . . . . 10  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
461, 3, 54atexlempsb 30174 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
47 eqid 2387 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
4847, 2, 4latmle1 14432 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
4944, 45, 46, 48syl3anc 1184 . . . . . . . . 9  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
5010, 49syl5eqbr 4186 . . . . . . . 8  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
5150adantr 452 . . . . . . 7  |-  ( (
ph  /\  C  =  D )  ->  C  .<_  ( Q  .\/  T
) )
52 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  C  =  D )  ->  C  =  D )
53 4thatlem0.d . . . . . . . . . 10  |-  D  =  ( ( R  .\/  T )  ./\  ( P  .\/  S ) )
5447, 3, 5hlatjcl 29481 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  R  e.  A  /\  T  e.  A )  ->  ( R  .\/  T
)  e.  ( Base `  K ) )
5514, 20, 16, 54syl3anc 1184 . . . . . . . . . . 11  |-  ( ph  ->  ( R  .\/  T
)  e.  ( Base `  K ) )
5647, 2, 4latmle1 14432 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( R  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( R  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( R  .\/  T ) )
5744, 55, 46, 56syl3anc 1184 . . . . . . . . . 10  |-  ( ph  ->  ( ( R  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( R  .\/  T ) )
5853, 57syl5eqbr 4186 . . . . . . . . 9  |-  ( ph  ->  D  .<_  ( R  .\/  T ) )
5958adantr 452 . . . . . . . 8  |-  ( (
ph  /\  C  =  D )  ->  D  .<_  ( R  .\/  T
) )
6052, 59eqbrtrd 4173 . . . . . . 7  |-  ( (
ph  /\  C  =  D )  ->  C  .<_  ( R  .\/  T
) )
611, 2, 3, 4, 5, 6, 7, 8, 104atexlemc 30183 . . . . . . . . . 10  |-  ( ph  ->  C  e.  A )
6247, 5atbase 29404 . . . . . . . . . 10  |-  ( C  e.  A  ->  C  e.  ( Base `  K
) )
6361, 62syl 16 . . . . . . . . 9  |-  ( ph  ->  C  e.  ( Base `  K ) )
6447, 2, 4latlem12 14434 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( C  e.  ( Base `  K )  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( R  .\/  T )  e.  (
Base `  K )
) )  ->  (
( C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  T ) )  <->  C  .<_  ( ( Q  .\/  T
)  ./\  ( R  .\/  T ) ) ) )
6544, 63, 45, 55, 64syl13anc 1186 . . . . . . . 8  |-  ( ph  ->  ( ( C  .<_  ( Q  .\/  T )  /\  C  .<_  ( R 
.\/  T ) )  <-> 
C  .<_  ( ( Q 
.\/  T )  ./\  ( R  .\/  T ) ) ) )
6665adantr 452 . . . . . . 7  |-  ( (
ph  /\  C  =  D )  ->  (
( C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  T ) )  <->  C  .<_  ( ( Q  .\/  T
)  ./\  ( R  .\/  T ) ) ) )
6751, 60, 66mpbi2and 888 . . . . . 6  |-  ( (
ph  /\  C  =  D )  ->  C  .<_  ( ( Q  .\/  T )  ./\  ( R  .\/  T ) ) )
68 hlatl 29475 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  AtLat )
6914, 68syl 16 . . . . . . . 8  |-  ( ph  ->  K  e.  AtLat )
7042, 16eqeltrrd 2462 . . . . . . . 8  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( R  .\/  T ) )  e.  A )
712, 5atcmp 29426 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  C  e.  A  /\  (
( Q  .\/  T
)  ./\  ( R  .\/  T ) )  e.  A )  ->  ( C  .<_  ( ( Q 
.\/  T )  ./\  ( R  .\/  T ) )  <->  C  =  (
( Q  .\/  T
)  ./\  ( R  .\/  T ) ) ) )
7269, 61, 70, 71syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( C  .<_  ( ( Q  .\/  T ) 
./\  ( R  .\/  T ) )  <->  C  =  ( ( Q  .\/  T )  ./\  ( R  .\/  T ) ) ) )
7372adantr 452 . . . . . 6  |-  ( (
ph  /\  C  =  D )  ->  ( C  .<_  ( ( Q 
.\/  T )  ./\  ( R  .\/  T ) )  <->  C  =  (
( Q  .\/  T
)  ./\  ( R  .\/  T ) ) ) )
7467, 73mpbid 202 . . . . 5  |-  ( (
ph  /\  C  =  D )  ->  C  =  ( ( Q 
.\/  T )  ./\  ( R  .\/  T ) ) )
7543, 74eqtr4d 2422 . . . 4  |-  ( (
ph  /\  C  =  D )  ->  T  =  C )
7675ex 424 . . 3  |-  ( ph  ->  ( C  =  D  ->  T  =  C ) )
7776necon3d 2588 . 2  |-  ( ph  ->  ( T  =/=  C  ->  C  =/=  D ) )
7813, 77mpd 15 1  |-  ( ph  ->  C  =/=  D )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   Basecbs 13396   lecple 13463   joincjn 14328   meetcmee 14329   Latclat 14401   Atomscatm 29378   AtLatcal 29379   CvLatclc 29380   HLchlt 29465   LHypclh 30098
This theorem is referenced by:  4atexlemex4  30187
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-p1 14396  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-llines 29612  df-lplanes 29613  df-lhyp 30102
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