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Theorem 4atexlemc 30258
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 ) )
Assertion
Ref Expression
4atexlemc  |-  ( ph  ->  C  e.  A )

Proof of Theorem 4atexlemc
StepHypRef Expression
1 4thatlem0.c . . 3  |-  C  =  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )
2 4thatlem.ph . . . . 5  |-  ( 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 ) ) ) )
324atexlemkl 30246 . . . 4  |-  ( ph  ->  K  e.  Lat )
4 4thatlem0.j . . . . 5  |-  .\/  =  ( join `  K )
5 4thatlem0.a . . . . 5  |-  A  =  ( Atoms `  K )
62, 4, 54atexlemqtb 30250 . . . 4  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
72, 4, 54atexlempsb 30249 . . . 4  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
8 eqid 2283 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
9 4thatlem0.m . . . . 5  |-  ./\  =  ( meet `  K )
108, 9latmcom 14181 . . . 4  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  =  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) ) )
113, 6, 7, 10syl3anc 1182 . . 3  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  =  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) ) )
121, 11syl5eq 2327 . 2  |-  ( ph  ->  C  =  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) ) )
1324atexlemk 30236 . . 3  |-  ( ph  ->  K  e.  HL )
1424atexlemp 30239 . . 3  |-  ( ph  ->  P  e.  A )
1524atexlems 30241 . . 3  |-  ( ph  ->  S  e.  A )
1624atexlemq 30240 . . 3  |-  ( ph  ->  Q  e.  A )
1724atexlemt 30242 . . 3  |-  ( ph  ->  T  e.  A )
18 4thatlem0.l . . . 4  |-  .<_  =  ( le `  K )
192, 18, 4, 54atexlempns 30251 . . 3  |-  ( ph  ->  P  =/=  S )
20 4thatlem0.h . . . . 5  |-  H  =  ( LHyp `  K
)
21 4thatlem0.u . . . . 5  |-  U  =  ( ( P  .\/  Q )  ./\  W )
22 4thatlem0.v . . . . 5  |-  V  =  ( ( P  .\/  S )  ./\  W )
232, 18, 4, 9, 5, 20, 21, 224atexlemntlpq 30257 . . . 4  |-  ( ph  ->  -.  T  .<_  ( P 
.\/  Q ) )
2418, 4, 5atnlej2 29569 . . . . 5  |-  ( ( K  e.  HL  /\  ( T  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  T  .<_  ( P  .\/  Q
) )  ->  T  =/=  Q )
2524necomd 2529 . . . 4  |-  ( ( K  e.  HL  /\  ( T  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  T  .<_  ( P  .\/  Q
) )  ->  Q  =/=  T )
2613, 17, 14, 16, 23, 25syl131anc 1195 . . 3  |-  ( ph  ->  Q  =/=  T )
2724atexlempnq 30244 . . . 4  |-  ( ph  ->  P  =/=  Q )
2824atexlemnslpq 30245 . . . 4  |-  ( ph  ->  -.  S  .<_  ( P 
.\/  Q ) )
2918, 4, 54atlem0ae 29783 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  -.  Q  .<_  ( P 
.\/  S ) )
3013, 14, 16, 15, 27, 28, 29syl132anc 1200 . . 3  |-  ( ph  ->  -.  Q  .<_  ( P 
.\/  S ) )
318, 5atbase 29479 . . . . 5  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
3217, 31syl 15 . . . 4  |-  ( ph  ->  T  e.  ( Base `  K ) )
332, 18, 4, 9, 5, 20, 214atexlemu 30253 . . . . 5  |-  ( ph  ->  U  e.  A )
342, 18, 4, 9, 5, 20, 21, 224atexlemv 30254 . . . . 5  |-  ( ph  ->  V  e.  A )
358, 4, 5hlatjcl 29556 . . . . 5  |-  ( ( K  e.  HL  /\  U  e.  A  /\  V  e.  A )  ->  ( U  .\/  V
)  e.  ( Base `  K ) )
3613, 33, 34, 35syl3anc 1182 . . . 4  |-  ( ph  ->  ( U  .\/  V
)  e.  ( Base `  K ) )
378, 5atbase 29479 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
3816, 37syl 15 . . . . 5  |-  ( ph  ->  Q  e.  ( Base `  K ) )
398, 4latjcl 14156 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  Q  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  .\/  Q )  e.  ( Base `  K ) )
403, 7, 38, 39syl3anc 1182 . . . 4  |-  ( ph  ->  ( ( P  .\/  S )  .\/  Q )  e.  ( Base `  K
) )
4124atexlemkc 30247 . . . . 5  |-  ( ph  ->  K  e.  CvLat )
422, 18, 4, 9, 5, 20, 21, 224atexlemunv 30255 . . . . 5  |-  ( ph  ->  U  =/=  V )
4324atexlemutvt 30243 . . . . 5  |-  ( ph  ->  ( U  .\/  T
)  =  ( V 
.\/  T ) )
445, 18, 4cvlsupr4 29535 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( U  e.  A  /\  V  e.  A  /\  T  e.  A )  /\  ( U  =/=  V  /\  ( U  .\/  T
)  =  ( V 
.\/  T ) ) )  ->  T  .<_  ( U  .\/  V ) )
4541, 33, 34, 17, 42, 43, 44syl132anc 1200 . . . 4  |-  ( ph  ->  T  .<_  ( U  .\/  V ) )
468, 4, 5hlatjcl 29556 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
4713, 14, 16, 46syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
482, 204atexlemwb 30248 . . . . . . . 8  |-  ( ph  ->  W  e.  ( Base `  K ) )
498, 18, 9latmle1 14182 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q ) )
503, 47, 48, 49syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q
) )
5121, 50syl5eqbr 4056 . . . . . 6  |-  ( ph  ->  U  .<_  ( P  .\/  Q ) )
528, 18, 9latmle1 14182 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S ) )
533, 7, 48, 52syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S
) )
5422, 53syl5eqbr 4056 . . . . . 6  |-  ( ph  ->  V  .<_  ( P  .\/  S ) )
558, 5atbase 29479 . . . . . . . 8  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
5633, 55syl 15 . . . . . . 7  |-  ( ph  ->  U  e.  ( Base `  K ) )
578, 5atbase 29479 . . . . . . . 8  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
5834, 57syl 15 . . . . . . 7  |-  ( ph  ->  V  e.  ( Base `  K ) )
598, 18, 4latjlej12 14173 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( U  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  /\  ( V  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
) ) )  -> 
( ( U  .<_  ( P  .\/  Q )  /\  V  .<_  ( P 
.\/  S ) )  ->  ( U  .\/  V )  .<_  ( ( P  .\/  Q )  .\/  ( P  .\/  S ) ) ) )
603, 56, 47, 58, 7, 59syl122anc 1191 . . . . . 6  |-  ( ph  ->  ( ( U  .<_  ( P  .\/  Q )  /\  V  .<_  ( P 
.\/  S ) )  ->  ( U  .\/  V )  .<_  ( ( P  .\/  Q )  .\/  ( P  .\/  S ) ) ) )
6151, 54, 60mp2and 660 . . . . 5  |-  ( ph  ->  ( U  .\/  V
)  .<_  ( ( P 
.\/  Q )  .\/  ( P  .\/  S ) ) )
624, 5hlatjass 29559 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  S  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  S )  =  ( P  .\/  ( Q  .\/  S ) ) )
6313, 14, 16, 15, 62syl13anc 1184 . . . . . 6  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  S )  =  ( P  .\/  ( Q  .\/  S ) ) )
648, 5atbase 29479 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
6514, 64syl 15 . . . . . . 7  |-  ( ph  ->  P  e.  ( Base `  K ) )
668, 5atbase 29479 . . . . . . . 8  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
6715, 66syl 15 . . . . . . 7  |-  ( ph  ->  S  e.  ( Base `  K ) )
688, 4latj32 14203 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) ) )  -> 
( ( P  .\/  Q )  .\/  S )  =  ( ( P 
.\/  S )  .\/  Q ) )
693, 65, 38, 67, 68syl13anc 1184 . . . . . 6  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  S )  =  ( ( P 
.\/  S )  .\/  Q ) )
708, 4latjjdi 14209 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) ) )  -> 
( P  .\/  ( Q  .\/  S ) )  =  ( ( P 
.\/  Q )  .\/  ( P  .\/  S ) ) )
713, 65, 38, 67, 70syl13anc 1184 . . . . . 6  |-  ( ph  ->  ( P  .\/  ( Q  .\/  S ) )  =  ( ( P 
.\/  Q )  .\/  ( P  .\/  S ) ) )
7263, 69, 713eqtr3rd 2324 . . . . 5  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  ( P 
.\/  S ) )  =  ( ( P 
.\/  S )  .\/  Q ) )
7361, 72breqtrd 4047 . . . 4  |-  ( ph  ->  ( U  .\/  V
)  .<_  ( ( P 
.\/  S )  .\/  Q ) )
748, 18, 3, 32, 36, 40, 45, 73lattrd 14164 . . 3  |-  ( ph  ->  T  .<_  ( ( P  .\/  S )  .\/  Q ) )
7518, 4, 9, 52atmat 29750 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  /\  ( Q  e.  A  /\  T  e.  A  /\  P  =/=  S
)  /\  ( Q  =/=  T  /\  -.  Q  .<_  ( P  .\/  S
)  /\  T  .<_  ( ( P  .\/  S
)  .\/  Q )
) )  ->  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  e.  A )
7613, 14, 15, 16, 17, 19, 26, 30, 74, 75syl333anc 1214 . 2  |-  ( ph  ->  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  e.  A )
7712, 76eqeltrd 2357 1  |-  ( ph  ->  C  e.  A )
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   CvLatclc 29455   HLchlt 29540   LHypclh 30173
This theorem is referenced by:  4atexlemnclw  30259  4atexlemex2  30260  4atexlemcnd  30261
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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