Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  4atexlemc Structured version   Unicode version

Theorem 4atexlemc 34740
Description: Lemma for 4atexlem7 34746. (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 34728 . . . 4  |-  ( ph  ->  K  e.  Lat )
4 4thatlem0.j . . . . 5  |-  .\/  =  ( join `  K )
5 4thatlem0.a . . . . 5  |-  A  =  ( Atoms `  K )
62, 4, 54atexlemqtb 34732 . . . 4  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
72, 4, 54atexlempsb 34731 . . . 4  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
8 eqid 2460 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
9 4thatlem0.m . . . . 5  |-  ./\  =  ( meet `  K )
108, 9latmcom 15551 . . . 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 1223 . . 3  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  =  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) ) )
121, 11syl5eq 2513 . 2  |-  ( ph  ->  C  =  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) ) )
1324atexlemk 34718 . . 3  |-  ( ph  ->  K  e.  HL )
1424atexlemp 34721 . . 3  |-  ( ph  ->  P  e.  A )
1524atexlems 34723 . . 3  |-  ( ph  ->  S  e.  A )
1624atexlemq 34722 . . 3  |-  ( ph  ->  Q  e.  A )
1724atexlemt 34724 . . 3  |-  ( ph  ->  T  e.  A )
18 4thatlem0.l . . . 4  |-  .<_  =  ( le `  K )
192, 18, 4, 54atexlempns 34733 . . 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 34739 . . . 4  |-  ( ph  ->  -.  T  .<_  ( P 
.\/  Q ) )
2418, 4, 5atnlej2 34051 . . . . 5  |-  ( ( K  e.  HL  /\  ( T  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  T  .<_  ( P  .\/  Q
) )  ->  T  =/=  Q )
2524necomd 2731 . . . 4  |-  ( ( K  e.  HL  /\  ( T  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  T  .<_  ( P  .\/  Q
) )  ->  Q  =/=  T )
2613, 17, 14, 16, 23, 25syl131anc 1236 . . 3  |-  ( ph  ->  Q  =/=  T )
2724atexlempnq 34726 . . . 4  |-  ( ph  ->  P  =/=  Q )
2824atexlemnslpq 34727 . . . 4  |-  ( ph  ->  -.  S  .<_  ( P 
.\/  Q ) )
2918, 4, 54atlem0ae 34265 . . . 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 1241 . . 3  |-  ( ph  ->  -.  Q  .<_  ( P 
.\/  S ) )
318, 5atbase 33961 . . . . 5  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
3217, 31syl 16 . . . 4  |-  ( ph  ->  T  e.  ( Base `  K ) )
332, 18, 4, 9, 5, 20, 214atexlemu 34735 . . . . 5  |-  ( ph  ->  U  e.  A )
342, 18, 4, 9, 5, 20, 21, 224atexlemv 34736 . . . . 5  |-  ( ph  ->  V  e.  A )
358, 4, 5hlatjcl 34038 . . . . 5  |-  ( ( K  e.  HL  /\  U  e.  A  /\  V  e.  A )  ->  ( U  .\/  V
)  e.  ( Base `  K ) )
3613, 33, 34, 35syl3anc 1223 . . . 4  |-  ( ph  ->  ( U  .\/  V
)  e.  ( Base `  K ) )
378, 5atbase 33961 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
3816, 37syl 16 . . . . 5  |-  ( ph  ->  Q  e.  ( Base `  K ) )
398, 4latjcl 15527 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  Q  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  .\/  Q )  e.  ( Base `  K ) )
403, 7, 38, 39syl3anc 1223 . . . 4  |-  ( ph  ->  ( ( P  .\/  S )  .\/  Q )  e.  ( Base `  K
) )
4124atexlemkc 34729 . . . . 5  |-  ( ph  ->  K  e.  CvLat )
422, 18, 4, 9, 5, 20, 21, 224atexlemunv 34737 . . . . 5  |-  ( ph  ->  U  =/=  V )
4324atexlemutvt 34725 . . . . 5  |-  ( ph  ->  ( U  .\/  T
)  =  ( V 
.\/  T ) )
445, 18, 4cvlsupr4 34017 . . . . 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 1241 . . . 4  |-  ( ph  ->  T  .<_  ( U  .\/  V ) )
468, 4, 5hlatjcl 34038 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
4713, 14, 16, 46syl3anc 1223 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
482, 204atexlemwb 34730 . . . . . . . 8  |-  ( ph  ->  W  e.  ( Base `  K ) )
498, 18, 9latmle1 15552 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q ) )
503, 47, 48, 49syl3anc 1223 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q
) )
5121, 50syl5eqbr 4473 . . . . . 6  |-  ( ph  ->  U  .<_  ( P  .\/  Q ) )
528, 18, 9latmle1 15552 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S ) )
533, 7, 48, 52syl3anc 1223 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S
) )
5422, 53syl5eqbr 4473 . . . . . 6  |-  ( ph  ->  V  .<_  ( P  .\/  S ) )
558, 5atbase 33961 . . . . . . . 8  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
5633, 55syl 16 . . . . . . 7  |-  ( ph  ->  U  e.  ( Base `  K ) )
578, 5atbase 33961 . . . . . . . 8  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
5834, 57syl 16 . . . . . . 7  |-  ( ph  ->  V  e.  ( Base `  K ) )
598, 18, 4latjlej12 15543 . . . . . . 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 1232 . . . . . 6  |-  ( ph  ->  ( ( U  .<_  ( P  .\/  Q )  /\  V  .<_  ( P 
.\/  S ) )  ->  ( U  .\/  V )  .<_  ( ( P  .\/  Q )  .\/  ( P  .\/  S ) ) ) )
6151, 54, 60mp2and 679 . . . . 5  |-  ( ph  ->  ( U  .\/  V
)  .<_  ( ( P 
.\/  Q )  .\/  ( P  .\/  S ) ) )
624, 5hlatjass 34041 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  S  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  S )  =  ( P  .\/  ( Q  .\/  S ) ) )
6313, 14, 16, 15, 62syl13anc 1225 . . . . . 6  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  S )  =  ( P  .\/  ( Q  .\/  S ) ) )
648, 5atbase 33961 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
6514, 64syl 16 . . . . . . 7  |-  ( ph  ->  P  e.  ( Base `  K ) )
668, 5atbase 33961 . . . . . . . 8  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
6715, 66syl 16 . . . . . . 7  |-  ( ph  ->  S  e.  ( Base `  K ) )
688, 4latj32 15573 . . . . . . 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 1225 . . . . . 6  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  S )  =  ( ( P 
.\/  S )  .\/  Q ) )
708, 4latjjdi 15579 . . . . . . 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 1225 . . . . . 6  |-  ( ph  ->  ( P  .\/  ( Q  .\/  S ) )  =  ( ( P 
.\/  Q )  .\/  ( P  .\/  S ) ) )
7263, 69, 713eqtr3rd 2510 . . . . 5  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  ( P 
.\/  S ) )  =  ( ( P 
.\/  S )  .\/  Q ) )
7361, 72breqtrd 4464 . . . 4  |-  ( ph  ->  ( U  .\/  V
)  .<_  ( ( P 
.\/  S )  .\/  Q ) )
748, 18, 3, 32, 36, 40, 45, 73lattrd 15534 . . 3  |-  ( ph  ->  T  .<_  ( ( P  .\/  S )  .\/  Q ) )
7518, 4, 9, 52atmat 34232 . . 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 1255 . 2  |-  ( ph  ->  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  e.  A )
7712, 76eqeltrd 2548 1  |-  ( ph  ->  C  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   lecple 14551   joincjn 15420   meetcmee 15421   Latclat 15521   Atomscatm 33935   CvLatclc 33937   HLchlt 34022   LHypclh 34655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-p1 15516  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-llines 34169  df-lplanes 34170  df-lhyp 34659
This theorem is referenced by:  4atexlemnclw  34741  4atexlemex2  34742  4atexlemcnd  34743
  Copyright terms: Public domain W3C validator