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Theorem atltcvr 33077
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 6097 . . . . . 6  |-  ( Q  =  R  ->  ( Q  .\/  R )  =  ( R  .\/  R
) )
2 simpr3 996 . . . . . . 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 33011 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A )  ->  ( R  .\/  R
)  =  R )
62, 5syldan 470 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( R  .\/  R )  =  R )
71, 6sylan9eqr 2496 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  Q  =  R )  ->  ( Q  .\/  R )  =  R )
87breq2d 4303 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  Q  =  R )  ->  ( P  .<  ( Q  .\/  R )  <->  P  .<  R ) )
9 hlatl 33003 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  AtLat )
109adantr 465 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  AtLat )
11 simpr1 994 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  P  e.  A )
12 atltcvr.s . . . . . . . 8  |-  .<  =  ( lt `  K )
1312, 4atnlt 32956 . . . . . . 7  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  R  e.  A )  ->  -.  P  .<  R )
1410, 11, 2, 13syl3anc 1218 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  -.  P  .<  R )
1514pm2.21d 106 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .<  R  ->  P C ( Q  .\/  R ) ) )
1615adantr 465 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  Q  =  R )  ->  ( P  .<  R  ->  P C ( Q  .\/  R ) ) )
178, 16sylbid 215 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  Q  =  R )  ->  ( P  .<  ( Q  .\/  R )  ->  P C
( Q  .\/  R
) ) )
18 simpl 457 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  HL )
19 hllat 33006 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
2019adantr 465 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  Lat )
21 simpr2 995 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  A )
22 eqid 2442 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2322, 4atbase 32932 . . . . . . . 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 32932 . . . . . . . 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 15220 . . . . . . 7  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( Q  .\/  R )  e.  ( Base `  K
) )
2820, 24, 26, 27syl3anc 1218 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( Q  .\/  R )  e.  ( Base `  K
) )
29 eqid 2442 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
3029, 12pltle 15130 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( Q  .\/  R )  e.  ( Base `  K
) )  ->  ( P  .<  ( Q  .\/  R )  ->  P ( le `  K ) ( Q  .\/  R ) ) )
3118, 11, 28, 30syl3anc 1218 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .<  ( Q  .\/  R )  ->  P ( le `  K ) ( Q  .\/  R ) ) )
3231adantr 465 . . . 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 753 . . . . . . . 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 754 . . . . . . . 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 461 . . . . . . . 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 1168 . . . . . . 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 648 . . . . . 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 33075 . . . . . 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 434 . . . 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 44 . . 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 2688 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .<  ( Q  .\/  R )  ->  P C
( Q  .\/  R
) ) )
4422, 4atbase 32932 . . . 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 32913 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
) )  /\  P C ( Q  .\/  R ) )  ->  P  .<  ( Q  .\/  R
) )
4746ex 434 . . 3  |-  ( ( K  e.  HL  /\  P  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
) )  ->  ( P C ( Q  .\/  R )  ->  P  .<  ( Q  .\/  R ) ) )
4818, 45, 28, 47syl3anc 1218 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P C ( Q  .\/  R )  ->  P  .<  ( Q  .\/  R ) ) )
4943, 48impbid 191 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .<  ( Q  .\/  R )  <->  P C ( Q 
.\/  R ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2605   class class class wbr 4291   ` cfv 5417  (class class class)co 6090   Basecbs 14173   lecple 14244   ltcplt 15110   joincjn 15113   Latclat 15214    <o ccvr 32905   Atomscatm 32906   AtLatcal 32907   HLchlt 32993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-poset 15115  df-plt 15127  df-lub 15143  df-glb 15144  df-join 15145  df-meet 15146  df-p0 15208  df-lat 15215  df-clat 15277  df-oposet 32819  df-ol 32821  df-oml 32822  df-covers 32909  df-ats 32910  df-atl 32941  df-cvlat 32965  df-hlat 32994
This theorem is referenced by:  atlt  33079  2atlt  33081  atexchltN  33083
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