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Theorem cvlexch2 30127
Description: An atomic covering lattice has the exchange property. (Contributed by NM, 6-May-2012.)
Hypotheses
Ref Expression
cvlexch.b  |-  B  =  ( Base `  K
)
cvlexch.l  |-  .<_  =  ( le `  K )
cvlexch.j  |-  .\/  =  ( join `  K )
cvlexch.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cvlexch2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( Q  .\/  X )  ->  Q  .<_  ( P 
.\/  X ) ) )

Proof of Theorem cvlexch2
StepHypRef Expression
1 cvlexch.b . . 3  |-  B  =  ( Base `  K
)
2 cvlexch.l . . 3  |-  .<_  =  ( le `  K )
3 cvlexch.j . . 3  |-  .\/  =  ( join `  K )
4 cvlexch.a . . 3  |-  A  =  ( Atoms `  K )
51, 2, 3, 4cvlexch1 30126 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q )  ->  Q  .<_  ( X 
.\/  P ) ) )
6 cvllat 30124 . . . . 5  |-  ( K  e.  CvLat  ->  K  e.  Lat )
763ad2ant1 978 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  K  e.  Lat )
8 simp22 991 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  Q  e.  A
)
91, 4atbase 30087 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  B )
108, 9syl 16 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  Q  e.  B
)
11 simp23 992 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  X  e.  B
)
121, 3latjcom 14488 . . . 4  |-  ( ( K  e.  Lat  /\  Q  e.  B  /\  X  e.  B )  ->  ( Q  .\/  X
)  =  ( X 
.\/  Q ) )
137, 10, 11, 12syl3anc 1184 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( Q  .\/  X )  =  ( X 
.\/  Q ) )
1413breq2d 4224 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( Q  .\/  X )  <-> 
P  .<_  ( X  .\/  Q ) ) )
15 simp21 990 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  P  e.  A
)
161, 4atbase 30087 . . . . 5  |-  ( P  e.  A  ->  P  e.  B )
1715, 16syl 16 . . . 4  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  P  e.  B
)
181, 3latjcom 14488 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  ( P  .\/  X
)  =  ( X 
.\/  P ) )
197, 17, 11, 18syl3anc 1184 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .\/  X )  =  ( X 
.\/  P ) )
2019breq2d 4224 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( Q  .<_  ( P  .\/  X )  <-> 
Q  .<_  ( X  .\/  P ) ) )
215, 14, 203imtr4d 260 1  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( Q  .\/  X )  ->  Q  .<_  ( P 
.\/  X ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   joincjn 14401   Latclat 14474   Atomscatm 30061   CvLatclc 30063
This theorem is referenced by:  hlexch2  30180
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-join 14433  df-lat 14475  df-ats 30065  df-atl 30096  df-cvlat 30120
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