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Theorem atltcvr 29917
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 6047 . . . . . 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 29851 . . . . . . 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 2458 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  Q  =  R )  ->  ( Q  .\/  R )  =  R )
87breq2d 4184 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  Q  =  R )  ->  ( P  .<  ( Q  .\/  R )  <->  P  .<  R ) )
9 hlatl 29843 . . . . . . . 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 29796 . . . . . . 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 29846 . . . . . . . 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 2404 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2322, 4atbase 29772 . . . . . . . 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 29772 . . . . . . . 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 14434 . . . . . . 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 2404 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
3029, 12pltle 14373 . . . . . 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 29915 . . . . . 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 2645 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .<  ( Q  .\/  R )  ->  P C
( Q  .\/  R
) ) )
4422, 4atbase 29772 . . . 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 29753 . . . 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 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   ltcplt 14353   joincjn 14356   Latclat 14429    <o ccvr 29745   Atomscatm 29746   AtLatcal 29747   HLchlt 29833
This theorem is referenced by:  atlt  29919  2atlt  29921  atexchltN  29923
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834
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