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Theorem 4atexlemcnd 34868
Description: Lemma for 4atexlem7 34871. (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 34863 . . 3  |-  ( ph  ->  T  .<_  W )
10 4thatlem0.c . . . 4  |-  C  =  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )
111, 2, 3, 4, 5, 6, 7, 8, 104atexlemnclw 34866 . . 3  |-  ( ph  ->  -.  C  .<_  W )
12 nbrne2 4465 . . 3  |-  ( ( T  .<_  W  /\  -.  C  .<_  W )  ->  T  =/=  C
)
139, 11, 12syl2anc 661 . 2  |-  ( ph  ->  T  =/=  C )
1414atexlemk 34843 . . . . . . . . 9  |-  ( ph  ->  K  e.  HL )
1514atexlemq 34847 . . . . . . . . 9  |-  ( ph  ->  Q  e.  A )
1614atexlemt 34849 . . . . . . . . 9  |-  ( ph  ->  T  e.  A )
173, 5hlatjcom 34164 . . . . . . . . 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 34164 . . . . . . . . 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 6300 . . . . . . 7  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( R  .\/  T ) )  =  ( ( T  .\/  Q )  ./\  ( T  .\/  R ) ) )
2414atexlemkc 34854 . . . . . . . . 9  |-  ( ph  ->  K  e.  CvLat )
2514atexlemp 34846 . . . . . . . . 9  |-  ( ph  ->  P  e.  A )
2614atexlempnq 34851 . . . . . . . . 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 34144 . . . . . . . . . 10  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  R  =/=  Q )
3029necomd 2738 . . . . . . . . 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 34864 . . . . . . . . 9  |-  ( ph  ->  -.  T  .<_  ( P 
.\/  Q ) )
335, 3cvlsupr7 34145 . . . . . . . . . . . 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 34164 . . . . . . . . . . . 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 2511 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  Q
)  =  ( Q 
.\/  R ) )
3837breq2d 4459 . . . . . . . . 9  |-  ( ph  ->  ( T  .<_  ( P 
.\/  Q )  <->  T  .<_  ( Q  .\/  R ) ) )
3932, 38mtbid 300 . . . . . . . 8  |-  ( ph  ->  -.  T  .<_  ( Q 
.\/  R ) )
402, 3, 4, 52llnma2 34585 . . . . . . . 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 2509 . . . . . 6  |-  ( ph  ->  T  =  ( ( Q  .\/  T ) 
./\  ( R  .\/  T ) ) )
4342adantr 465 . . . . 5  |-  ( (
ph  /\  C  =  D )  ->  T  =  ( ( Q 
.\/  T )  ./\  ( R  .\/  T ) ) )
4414atexlemkl 34853 . . . . . . . . . 10  |-  ( ph  ->  K  e.  Lat )
451, 3, 54atexlemqtb 34857 . . . . . . . . . 10  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
461, 3, 54atexlempsb 34856 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
47 eqid 2467 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
4847, 2, 4latmle1 15559 . . . . . . . . . 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 4480 . . . . . . . 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 34163 . . . . . . . . . . . 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 15559 . . . . . . . . . . 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 4480 . . . . . . . . 9  |-  ( ph  ->  D  .<_  ( R  .\/  T ) )
5958adantr 465 . . . . . . . 8  |-  ( (
ph  /\  C  =  D )  ->  D  .<_  ( R  .\/  T
) )
6052, 59eqbrtrd 4467 . . . . . . 7  |-  ( (
ph  /\  C  =  D )  ->  C  .<_  ( R  .\/  T
) )
611, 2, 3, 4, 5, 6, 7, 8, 104atexlemc 34865 . . . . . . . . . 10  |-  ( ph  ->  C  e.  A )
6247, 5atbase 34086 . . . . . . . . . 10  |-  ( C  e.  A  ->  C  e.  ( Base `  K
) )
6361, 62syl 16 . . . . . . . . 9  |-  ( ph  ->  C  e.  ( Base `  K ) )
6447, 2, 4latlem12 15561 . . . . . . . . 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 919 . . . . . 6  |-  ( (
ph  /\  C  =  D )  ->  C  .<_  ( ( Q  .\/  T )  ./\  ( R  .\/  T ) ) )
68 hlatl 34157 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  AtLat )
6914, 68syl 16 . . . . . . . 8  |-  ( ph  ->  K  e.  AtLat )
7042, 16eqeltrrd 2556 . . . . . . . 8  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( R  .\/  T ) )  e.  A )
712, 5atcmp 34108 . . . . . . . 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 2511 . . . 4  |-  ( (
ph  /\  C  =  D )  ->  T  =  C )
7675ex 434 . . 3  |-  ( ph  ->  ( C  =  D  ->  T  =  C ) )
7776necon3d 2691 . 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 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14486   lecple 14558   joincjn 15427   meetcmee 15428   Latclat 15528   Atomscatm 34060   AtLatcal 34061   CvLatclc 34062   HLchlt 34147   LHypclh 34780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-p1 15523  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294  df-lplanes 34295  df-lhyp 34784
This theorem is referenced by:  4atexlemex4  34869
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