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Theorem 4atexlemcnd 30261
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 ) )
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 30256 . . 3  |-  ( ph  ->  T  .<_  W )
10 4thatlem0.c . . . 4  |-  C  =  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )
111, 2, 3, 4, 5, 6, 7, 8, 104atexlemnclw 30259 . . 3  |-  ( ph  ->  -.  C  .<_  W )
12 nbrne2 4041 . . 3  |-  ( ( T  .<_  W  /\  -.  C  .<_  W )  ->  T  =/=  C
)
139, 11, 12syl2anc 642 . 2  |-  ( ph  ->  T  =/=  C )
1414atexlemk 30236 . . . . . . . . 9  |-  ( ph  ->  K  e.  HL )
1514atexlemq 30240 . . . . . . . . 9  |-  ( ph  ->  Q  e.  A )
1614atexlemt 30242 . . . . . . . . 9  |-  ( ph  ->  T  e.  A )
173, 5hlatjcom 29557 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  ( Q  .\/  T
)  =  ( T 
.\/  Q ) )
1814, 15, 16, 17syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( Q  .\/  T
)  =  ( T 
.\/  Q ) )
19 simp221 1096 . . . . . . . . . 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 187 . . . . . . . . 9  |-  ( ph  ->  R  e.  A )
213, 5hlatjcom 29557 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  R  e.  A  /\  T  e.  A )  ->  ( R  .\/  T
)  =  ( T 
.\/  R ) )
2214, 20, 16, 21syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( R  .\/  T
)  =  ( T 
.\/  R ) )
2318, 22oveq12d 5876 . . . . . . 7  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( R  .\/  T ) )  =  ( ( T  .\/  Q )  ./\  ( T  .\/  R ) ) )
2414atexlemkc 30247 . . . . . . . . 9  |-  ( ph  ->  K  e.  CvLat )
2514atexlemp 30239 . . . . . . . . 9  |-  ( ph  ->  P  e.  A )
2614atexlempnq 30244 . . . . . . . . 9  |-  ( ph  ->  P  =/=  Q )
27 simp223 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 ) ) )  ->  ( P  .\/  R )  =  ( Q  .\/  R ) )
281, 27sylbi 187 . . . . . . . . 9  |-  ( ph  ->  ( P  .\/  R
)  =  ( Q 
.\/  R ) )
295, 3cvlsupr6 29537 . . . . . . . . . 10  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  ( P  .\/  R
)  =  ( Q 
.\/  R ) ) )  ->  R  =/=  Q )
3029necomd 2529 . . . . . . . . 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 1200 . . . . . . . 8  |-  ( ph  ->  Q  =/=  R )
321, 2, 3, 4, 5, 6, 7, 84atexlemntlpq 30257 . . . . . . . . 9  |-  ( ph  ->  -.  T  .<_  ( P 
.\/  Q ) )
335, 3cvlsupr7 29538 . . . . . . . . . . . 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 1200 . . . . . . . . . . 11  |-  ( ph  ->  ( P  .\/  Q
)  =  ( R 
.\/  Q ) )
353, 5hlatjcom 29557 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  =  ( R 
.\/  Q ) )
3614, 15, 20, 35syl3anc 1182 . . . . . . . . . . 11  |-  ( ph  ->  ( Q  .\/  R
)  =  ( R 
.\/  Q ) )
3734, 36eqtr4d 2318 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  Q
)  =  ( Q 
.\/  R ) )
3837breq2d 4035 . . . . . . . . 9  |-  ( ph  ->  ( T  .<_  ( P 
.\/  Q )  <->  T  .<_  ( Q  .\/  R ) ) )
3932, 38mtbid 291 . . . . . . . 8  |-  ( ph  ->  -.  T  .<_  ( Q 
.\/  R ) )
402, 3, 4, 52llnma2 29978 . . . . . . . 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 1200 . . . . . . 7  |-  ( ph  ->  ( ( T  .\/  Q )  ./\  ( T  .\/  R ) )  =  T )
4223, 41eqtr2d 2316 . . . . . 6  |-  ( ph  ->  T  =  ( ( Q  .\/  T ) 
./\  ( R  .\/  T ) ) )
4342adantr 451 . . . . 5  |-  ( (
ph  /\  C  =  D )  ->  T  =  ( ( Q 
.\/  T )  ./\  ( R  .\/  T ) ) )
4414atexlemkl 30246 . . . . . . . . . 10  |-  ( ph  ->  K  e.  Lat )
451, 3, 54atexlemqtb 30250 . . . . . . . . . 10  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
461, 3, 54atexlempsb 30249 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
47 eqid 2283 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
4847, 2, 4latmle1 14182 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
4944, 45, 46, 48syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
5010, 49syl5eqbr 4056 . . . . . . . 8  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
5150adantr 451 . . . . . . 7  |-  ( (
ph  /\  C  =  D )  ->  C  .<_  ( Q  .\/  T
) )
52 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  C  =  D )  ->  C  =  D )
53 4thatlem0.d . . . . . . . . . 10  |-  D  =  ( ( R  .\/  T )  ./\  ( P  .\/  S ) )
5447, 3, 5hlatjcl 29556 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  R  e.  A  /\  T  e.  A )  ->  ( R  .\/  T
)  e.  ( Base `  K ) )
5514, 20, 16, 54syl3anc 1182 . . . . . . . . . . 11  |-  ( ph  ->  ( R  .\/  T
)  e.  ( Base `  K ) )
5647, 2, 4latmle1 14182 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( R  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( R  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( R  .\/  T ) )
5744, 55, 46, 56syl3anc 1182 . . . . . . . . . 10  |-  ( ph  ->  ( ( R  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( R  .\/  T ) )
5853, 57syl5eqbr 4056 . . . . . . . . 9  |-  ( ph  ->  D  .<_  ( R  .\/  T ) )
5958adantr 451 . . . . . . . 8  |-  ( (
ph  /\  C  =  D )  ->  D  .<_  ( R  .\/  T
) )
6052, 59eqbrtrd 4043 . . . . . . 7  |-  ( (
ph  /\  C  =  D )  ->  C  .<_  ( R  .\/  T
) )
611, 2, 3, 4, 5, 6, 7, 8, 104atexlemc 30258 . . . . . . . . . 10  |-  ( ph  ->  C  e.  A )
6247, 5atbase 29479 . . . . . . . . . 10  |-  ( C  e.  A  ->  C  e.  ( Base `  K
) )
6361, 62syl 15 . . . . . . . . 9  |-  ( ph  ->  C  e.  ( Base `  K ) )
6447, 2, 4latlem12 14184 . . . . . . . . 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 1184 . . . . . . . 8  |-  ( ph  ->  ( ( C  .<_  ( Q  .\/  T )  /\  C  .<_  ( R 
.\/  T ) )  <-> 
C  .<_  ( ( Q 
.\/  T )  ./\  ( R  .\/  T ) ) ) )
6665adantr 451 . . . . . . 7  |-  ( (
ph  /\  C  =  D )  ->  (
( C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  T ) )  <->  C  .<_  ( ( Q  .\/  T
)  ./\  ( R  .\/  T ) ) ) )
6751, 60, 66mpbi2and 887 . . . . . 6  |-  ( (
ph  /\  C  =  D )  ->  C  .<_  ( ( Q  .\/  T )  ./\  ( R  .\/  T ) ) )
68 hlatl 29550 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  AtLat )
6914, 68syl 15 . . . . . . . 8  |-  ( ph  ->  K  e.  AtLat )
7042, 16eqeltrrd 2358 . . . . . . . 8  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( R  .\/  T ) )  e.  A )
712, 5atcmp 29501 . . . . . . . 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 1182 . . . . . . 7  |-  ( ph  ->  ( C  .<_  ( ( Q  .\/  T ) 
./\  ( R  .\/  T ) )  <->  C  =  ( ( Q  .\/  T )  ./\  ( R  .\/  T ) ) ) )
7372adantr 451 . . . . . 6  |-  ( (
ph  /\  C  =  D )  ->  ( C  .<_  ( ( Q 
.\/  T )  ./\  ( R  .\/  T ) )  <->  C  =  (
( Q  .\/  T
)  ./\  ( R  .\/  T ) ) ) )
7467, 73mpbid 201 . . . . 5  |-  ( (
ph  /\  C  =  D )  ->  C  =  ( ( Q 
.\/  T )  ./\  ( R  .\/  T ) ) )
7543, 74eqtr4d 2318 . . . 4  |-  ( (
ph  /\  C  =  D )  ->  T  =  C )
7675ex 423 . . 3  |-  ( ph  ->  ( C  =  D  ->  T  =  C ) )
7776necon3d 2484 . 2  |-  ( ph  ->  ( T  =/=  C  ->  C  =/=  D ) )
7813, 77mpd 14 1  |-  ( ph  ->  C  =/=  D )
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   AtLatcal 29454   CvLatclc 29455   HLchlt 29540   LHypclh 30173
This theorem is referenced by:  4atexlemex4  30262
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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