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Theorem 4atexlemcnd 30806
Description: Lemma for 4atexlem7 30809. (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 30801 . . 3  |-  ( ph  ->  T  .<_  W )
10 4thatlem0.c . . . 4  |-  C  =  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )
111, 2, 3, 4, 5, 6, 7, 8, 104atexlemnclw 30804 . . 3  |-  ( ph  ->  -.  C  .<_  W )
12 nbrne2 4222 . . 3  |-  ( ( T  .<_  W  /\  -.  C  .<_  W )  ->  T  =/=  C
)
139, 11, 12syl2anc 643 . 2  |-  ( ph  ->  T  =/=  C )
1414atexlemk 30781 . . . . . . . . 9  |-  ( ph  ->  K  e.  HL )
1514atexlemq 30785 . . . . . . . . 9  |-  ( ph  ->  Q  e.  A )
1614atexlemt 30787 . . . . . . . . 9  |-  ( ph  ->  T  e.  A )
173, 5hlatjcom 30102 . . . . . . . . 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 30102 . . . . . . . . 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 6091 . . . . . . 7  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( R  .\/  T ) )  =  ( ( T  .\/  Q )  ./\  ( T  .\/  R ) ) )
2414atexlemkc 30792 . . . . . . . . 9  |-  ( ph  ->  K  e.  CvLat )
2514atexlemp 30784 . . . . . . . . 9  |-  ( ph  ->  P  e.  A )
2614atexlempnq 30789 . . . . . . . . 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 30082 . . . . . . . . . 10  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  R  =/=  Q )
3029necomd 2681 . . . . . . . . 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 30802 . . . . . . . . 9  |-  ( ph  ->  -.  T  .<_  ( P 
.\/  Q ) )
335, 3cvlsupr7 30083 . . . . . . . . . . . 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 30102 . . . . . . . . . . . 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 2470 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  Q
)  =  ( Q 
.\/  R ) )
3837breq2d 4216 . . . . . . . . 9  |-  ( ph  ->  ( T  .<_  ( P 
.\/  Q )  <->  T  .<_  ( Q  .\/  R ) ) )
3932, 38mtbid 292 . . . . . . . 8  |-  ( ph  ->  -.  T  .<_  ( Q 
.\/  R ) )
402, 3, 4, 52llnma2 30523 . . . . . . . 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 2468 . . . . . 6  |-  ( ph  ->  T  =  ( ( Q  .\/  T ) 
./\  ( R  .\/  T ) ) )
4342adantr 452 . . . . 5  |-  ( (
ph  /\  C  =  D )  ->  T  =  ( ( Q 
.\/  T )  ./\  ( R  .\/  T ) ) )
4414atexlemkl 30791 . . . . . . . . . 10  |-  ( ph  ->  K  e.  Lat )
451, 3, 54atexlemqtb 30795 . . . . . . . . . 10  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
461, 3, 54atexlempsb 30794 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
47 eqid 2435 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
4847, 2, 4latmle1 14497 . . . . . . . . . 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 4237 . . . . . . . 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 30101 . . . . . . . . . . . 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 14497 . . . . . . . . . . 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 4237 . . . . . . . . 9  |-  ( ph  ->  D  .<_  ( R  .\/  T ) )
5958adantr 452 . . . . . . . 8  |-  ( (
ph  /\  C  =  D )  ->  D  .<_  ( R  .\/  T
) )
6052, 59eqbrtrd 4224 . . . . . . 7  |-  ( (
ph  /\  C  =  D )  ->  C  .<_  ( R  .\/  T
) )
611, 2, 3, 4, 5, 6, 7, 8, 104atexlemc 30803 . . . . . . . . . 10  |-  ( ph  ->  C  e.  A )
6247, 5atbase 30024 . . . . . . . . . 10  |-  ( C  e.  A  ->  C  e.  ( Base `  K
) )
6361, 62syl 16 . . . . . . . . 9  |-  ( ph  ->  C  e.  ( Base `  K ) )
6447, 2, 4latlem12 14499 . . . . . . . . 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 30095 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  AtLat )
6914, 68syl 16 . . . . . . . 8  |-  ( ph  ->  K  e.  AtLat )
7042, 16eqeltrrd 2510 . . . . . . . 8  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( R  .\/  T ) )  e.  A )
712, 5atcmp 30046 . . . . . . . 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 2470 . . . 4  |-  ( (
ph  /\  C  =  D )  ->  T  =  C )
7675ex 424 . . 3  |-  ( ph  ->  ( C  =  D  ->  T  =  C ) )
7776necon3d 2636 . 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 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   meetcmee 14394   Latclat 14466   Atomscatm 29998   AtLatcal 29999   CvLatclc 30000   HLchlt 30085   LHypclh 30718
This theorem is referenced by:  4atexlemex4  30807
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lhyp 30722
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