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Theorem 4atexlemcnd 35897
Description: Lemma for 4atexlem7 35900. (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 35892 . . 3  |-  ( ph  ->  T  .<_  W )
10 4thatlem0.c . . . 4  |-  C  =  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )
111, 2, 3, 4, 5, 6, 7, 8, 104atexlemnclw 35895 . . 3  |-  ( ph  ->  -.  C  .<_  W )
12 nbrne2 4474 . . 3  |-  ( ( T  .<_  W  /\  -.  C  .<_  W )  ->  T  =/=  C
)
139, 11, 12syl2anc 661 . 2  |-  ( ph  ->  T  =/=  C )
1414atexlemk 35872 . . . . . . . . 9  |-  ( ph  ->  K  e.  HL )
1514atexlemq 35876 . . . . . . . . 9  |-  ( ph  ->  Q  e.  A )
1614atexlemt 35878 . . . . . . . . 9  |-  ( ph  ->  T  e.  A )
173, 5hlatjcom 35193 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  ( Q  .\/  T
)  =  ( T 
.\/  Q ) )
1814, 15, 16, 17syl3anc 1228 . . . . . . . 8  |-  ( ph  ->  ( Q  .\/  T
)  =  ( T 
.\/  Q ) )
19 simp221 1137 . . . . . . . . . 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 195 . . . . . . . . 9  |-  ( ph  ->  R  e.  A )
213, 5hlatjcom 35193 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  R  e.  A  /\  T  e.  A )  ->  ( R  .\/  T
)  =  ( T 
.\/  R ) )
2214, 20, 16, 21syl3anc 1228 . . . . . . . 8  |-  ( ph  ->  ( R  .\/  T
)  =  ( T 
.\/  R ) )
2318, 22oveq12d 6314 . . . . . . 7  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( R  .\/  T ) )  =  ( ( T  .\/  Q )  ./\  ( T  .\/  R ) ) )
2414atexlemkc 35883 . . . . . . . . 9  |-  ( ph  ->  K  e.  CvLat )
2514atexlemp 35875 . . . . . . . . 9  |-  ( ph  ->  P  e.  A )
2614atexlempnq 35880 . . . . . . . . 9  |-  ( ph  ->  P  =/=  Q )
27 simp223 1139 . . . . . . . . . 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 195 . . . . . . . . 9  |-  ( ph  ->  ( P  .\/  R
)  =  ( Q 
.\/  R ) )
295, 3cvlsupr6 35173 . . . . . . . . . 10  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  R  =/=  Q )
3029necomd 2728 . . . . . . . . 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 1246 . . . . . . . 8  |-  ( ph  ->  Q  =/=  R )
321, 2, 3, 4, 5, 6, 7, 84atexlemntlpq 35893 . . . . . . . . 9  |-  ( ph  ->  -.  T  .<_  ( P 
.\/  Q ) )
335, 3cvlsupr7 35174 . . . . . . . . . . . 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 1246 . . . . . . . . . . 11  |-  ( ph  ->  ( P  .\/  Q
)  =  ( R 
.\/  Q ) )
353, 5hlatjcom 35193 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  =  ( R 
.\/  Q ) )
3614, 15, 20, 35syl3anc 1228 . . . . . . . . . . 11  |-  ( ph  ->  ( Q  .\/  R
)  =  ( R 
.\/  Q ) )
3734, 36eqtr4d 2501 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  Q
)  =  ( Q 
.\/  R ) )
3837breq2d 4468 . . . . . . . . 9  |-  ( ph  ->  ( T  .<_  ( P 
.\/  Q )  <->  T  .<_  ( Q  .\/  R ) ) )
3932, 38mtbid 300 . . . . . . . 8  |-  ( ph  ->  -.  T  .<_  ( Q 
.\/  R ) )
402, 3, 4, 52llnma2 35614 . . . . . . . 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 1246 . . . . . . 7  |-  ( ph  ->  ( ( T  .\/  Q )  ./\  ( T  .\/  R ) )  =  T )
4223, 41eqtr2d 2499 . . . . . 6  |-  ( ph  ->  T  =  ( ( Q  .\/  T ) 
./\  ( R  .\/  T ) ) )
4342adantr 465 . . . . 5  |-  ( (
ph  /\  C  =  D )  ->  T  =  ( ( Q 
.\/  T )  ./\  ( R  .\/  T ) ) )
4414atexlemkl 35882 . . . . . . . . . 10  |-  ( ph  ->  K  e.  Lat )
451, 3, 54atexlemqtb 35886 . . . . . . . . . 10  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
461, 3, 54atexlempsb 35885 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
47 eqid 2457 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
4847, 2, 4latmle1 15832 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
4944, 45, 46, 48syl3anc 1228 . . . . . . . . 9  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
5010, 49syl5eqbr 4489 . . . . . . . 8  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
5150adantr 465 . . . . . . 7  |-  ( (
ph  /\  C  =  D )  ->  C  .<_  ( Q  .\/  T
) )
52 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  C  =  D )  ->  C  =  D )
53 4thatlem0.d . . . . . . . . . 10  |-  D  =  ( ( R  .\/  T )  ./\  ( P  .\/  S ) )
5447, 3, 5hlatjcl 35192 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  R  e.  A  /\  T  e.  A )  ->  ( R  .\/  T
)  e.  ( Base `  K ) )
5514, 20, 16, 54syl3anc 1228 . . . . . . . . . . 11  |-  ( ph  ->  ( R  .\/  T
)  e.  ( Base `  K ) )
5647, 2, 4latmle1 15832 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( R  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( R  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( R  .\/  T ) )
5744, 55, 46, 56syl3anc 1228 . . . . . . . . . 10  |-  ( ph  ->  ( ( R  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( R  .\/  T ) )
5853, 57syl5eqbr 4489 . . . . . . . . 9  |-  ( ph  ->  D  .<_  ( R  .\/  T ) )
5958adantr 465 . . . . . . . 8  |-  ( (
ph  /\  C  =  D )  ->  D  .<_  ( R  .\/  T
) )
6052, 59eqbrtrd 4476 . . . . . . 7  |-  ( (
ph  /\  C  =  D )  ->  C  .<_  ( R  .\/  T
) )
611, 2, 3, 4, 5, 6, 7, 8, 104atexlemc 35894 . . . . . . . . . 10  |-  ( ph  ->  C  e.  A )
6247, 5atbase 35115 . . . . . . . . . 10  |-  ( C  e.  A  ->  C  e.  ( Base `  K
) )
6361, 62syl 16 . . . . . . . . 9  |-  ( ph  ->  C  e.  ( Base `  K ) )
6447, 2, 4latlem12 15834 . . . . . . . . 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 1230 . . . . . . . 8  |-  ( ph  ->  ( ( C  .<_  ( Q  .\/  T )  /\  C  .<_  ( R 
.\/  T ) )  <-> 
C  .<_  ( ( Q 
.\/  T )  ./\  ( R  .\/  T ) ) ) )
6665adantr 465 . . . . . . 7  |-  ( (
ph  /\  C  =  D )  ->  (
( C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  T ) )  <->  C  .<_  ( ( Q  .\/  T
)  ./\  ( R  .\/  T ) ) ) )
6751, 60, 66mpbi2and 921 . . . . . 6  |-  ( (
ph  /\  C  =  D )  ->  C  .<_  ( ( Q  .\/  T )  ./\  ( R  .\/  T ) ) )
68 hlatl 35186 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  AtLat )
6914, 68syl 16 . . . . . . . 8  |-  ( ph  ->  K  e.  AtLat )
7042, 16eqeltrrd 2546 . . . . . . . 8  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( R  .\/  T ) )  e.  A )
712, 5atcmp 35137 . . . . . . . 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 1228 . . . . . . 7  |-  ( ph  ->  ( C  .<_  ( ( Q  .\/  T ) 
./\  ( R  .\/  T ) )  <->  C  =  ( ( Q  .\/  T )  ./\  ( R  .\/  T ) ) ) )
7372adantr 465 . . . . . 6  |-  ( (
ph  /\  C  =  D )  ->  ( C  .<_  ( ( Q 
.\/  T )  ./\  ( R  .\/  T ) )  <->  C  =  (
( Q  .\/  T
)  ./\  ( R  .\/  T ) ) ) )
7467, 73mpbid 210 . . . . 5  |-  ( (
ph  /\  C  =  D )  ->  C  =  ( ( Q 
.\/  T )  ./\  ( R  .\/  T ) ) )
7543, 74eqtr4d 2501 . . . 4  |-  ( (
ph  /\  C  =  D )  ->  T  =  C )
7675ex 434 . . 3  |-  ( ph  ->  ( C  =  D  ->  T  =  C ) )
7776necon3d 2681 . 2  |-  ( ph  ->  ( T  =/=  C  ->  C  =/=  D ) )
7813, 77mpd 15 1  |-  ( ph  ->  C  =/=  D )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Basecbs 14643   lecple 14718   joincjn 15699   meetcmee 15700   Latclat 15801   Atomscatm 35089   AtLatcal 35090   CvLatclc 35091   HLchlt 35176   LHypclh 35809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-preset 15683  df-poset 15701  df-plt 15714  df-lub 15730  df-glb 15731  df-join 15732  df-meet 15733  df-p0 15795  df-p1 15796  df-lat 15802  df-clat 15864  df-oposet 35002  df-ol 35004  df-oml 35005  df-covers 35092  df-ats 35093  df-atl 35124  df-cvlat 35148  df-hlat 35177  df-llines 35323  df-lplanes 35324  df-lhyp 35813
This theorem is referenced by:  4atexlemex4  35898
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