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Theorem atltcvr 30232
Description: An equivalence of less-than ordering and covers relation. (Contributed by NM, 7-Feb-2012.)
Hypotheses
Ref Expression
atltcvr.s  |-  .<  =  ( lt `  K )
atltcvr.j  |-  .\/  =  ( join `  K )
atltcvr.a  |-  A  =  ( Atoms `  K )
atltcvr.c  |-  C  =  (  <o  `  K )
Assertion
Ref Expression
atltcvr  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .<  ( Q  .\/  R )  <->  P C ( Q 
.\/  R ) ) )

Proof of Theorem atltcvr
StepHypRef Expression
1 oveq1 6088 . . . . . 6  |-  ( Q  =  R  ->  ( Q  .\/  R )  =  ( R  .\/  R
) )
2 simpr3 965 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  A )
3 atltcvr.j . . . . . . . 8  |-  .\/  =  ( join `  K )
4 atltcvr.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
53, 4hlatjidm 30166 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A )  ->  ( R  .\/  R
)  =  R )
62, 5syldan 457 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( R  .\/  R )  =  R )
71, 6sylan9eqr 2490 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  Q  =  R )  ->  ( Q  .\/  R )  =  R )
87breq2d 4224 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  Q  =  R )  ->  ( P  .<  ( Q  .\/  R )  <->  P  .<  R ) )
9 hlatl 30158 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  AtLat )
109adantr 452 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  AtLat )
11 simpr1 963 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  P  e.  A )
12 atltcvr.s . . . . . . . 8  |-  .<  =  ( lt `  K )
1312, 4atnlt 30111 . . . . . . 7  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  R  e.  A )  ->  -.  P  .<  R )
1410, 11, 2, 13syl3anc 1184 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  -.  P  .<  R )
1514pm2.21d 100 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .<  R  ->  P C ( Q  .\/  R ) ) )
1615adantr 452 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  Q  =  R )  ->  ( P  .<  R  ->  P C ( Q  .\/  R ) ) )
178, 16sylbid 207 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  Q  =  R )  ->  ( P  .<  ( Q  .\/  R )  ->  P C
( Q  .\/  R
) ) )
18 simpl 444 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  HL )
19 hllat 30161 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
2019adantr 452 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  Lat )
21 simpr2 964 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  A )
22 eqid 2436 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2322, 4atbase 30087 . . . . . . . 8  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
2421, 23syl 16 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  ( Base `  K
) )
2522, 4atbase 30087 . . . . . . . 8  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
262, 25syl 16 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  ( Base `  K
) )
2722, 3latjcl 14479 . . . . . . 7  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( Q  .\/  R )  e.  ( Base `  K
) )
2820, 24, 26, 27syl3anc 1184 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( Q  .\/  R )  e.  ( Base `  K
) )
29 eqid 2436 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
3029, 12pltle 14418 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( Q  .\/  R )  e.  ( Base `  K
) )  ->  ( P  .<  ( Q  .\/  R )  ->  P ( le `  K ) ( Q  .\/  R ) ) )
3118, 11, 28, 30syl3anc 1184 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .<  ( Q  .\/  R )  ->  P ( le `  K ) ( Q  .\/  R ) ) )
3231adantr 452 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  Q  =/=  R )  ->  ( P  .<  ( Q  .\/  R )  ->  P ( le `  K ) ( Q  .\/  R ) ) )
33 simpll 731 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  ( Q  =/=  R  /\  P
( le `  K
) ( Q  .\/  R ) ) )  ->  K  e.  HL )
34 simplr 732 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  ( Q  =/=  R  /\  P
( le `  K
) ( Q  .\/  R ) ) )  -> 
( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )
35 simpr 448 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  ( Q  =/=  R  /\  P
( le `  K
) ( Q  .\/  R ) ) )  -> 
( Q  =/=  R  /\  P ( le `  K ) ( Q 
.\/  R ) ) )
3633, 34, 353jca 1134 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  ( Q  =/=  R  /\  P
( le `  K
) ( Q  .\/  R ) ) )  -> 
( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P ( le `  K ) ( Q  .\/  R
) ) ) )
3736anassrs 630 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  /\  Q  =/=  R )  /\  P ( le `  K ) ( Q 
.\/  R ) )  ->  ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( Q  =/=  R  /\  P
( le `  K
) ( Q  .\/  R ) ) ) )
38 atltcvr.c . . . . . . 7  |-  C  =  (  <o  `  K )
3929, 3, 38, 4atcvrj2 30230 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P ( le `  K ) ( Q  .\/  R
) ) )  ->  P C ( Q  .\/  R ) )
4037, 39syl 16 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  /\  Q  =/=  R )  /\  P ( le `  K ) ( Q 
.\/  R ) )  ->  P C ( Q  .\/  R ) )
4140ex 424 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  Q  =/=  R )  ->  ( P ( le `  K ) ( Q 
.\/  R )  ->  P C ( Q  .\/  R ) ) )
4232, 41syld 42 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  Q  =/=  R )  ->  ( P  .<  ( Q  .\/  R )  ->  P C
( Q  .\/  R
) ) )
4317, 42pm2.61dane 2682 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .<  ( Q  .\/  R )  ->  P C
( Q  .\/  R
) ) )
4422, 4atbase 30087 . . . 4  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
4511, 44syl 16 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  P  e.  ( Base `  K
) )
4622, 12, 38cvrlt 30068 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
) )  /\  P C ( Q  .\/  R ) )  ->  P  .<  ( Q  .\/  R
) )
4746ex 424 . . 3  |-  ( ( K  e.  HL  /\  P  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
) )  ->  ( P C ( Q  .\/  R )  ->  P  .<  ( Q  .\/  R ) ) )
4818, 45, 28, 47syl3anc 1184 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P C ( Q  .\/  R )  ->  P  .<  ( Q  .\/  R ) ) )
4943, 48impbid 184 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .<  ( Q  .\/  R )  <->  P C ( Q 
.\/  R ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   ltcplt 14398   joincjn 14401   Latclat 14474    <o ccvr 30060   Atomscatm 30061   AtLatcal 30062   HLchlt 30148
This theorem is referenced by:  atlt  30234  2atlt  30236  atexchltN  30238
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-nel 2602  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-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149
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