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Theorem atltcvr 30246
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 5881 . . . . . 6  |-  ( Q  =  R  ->  ( Q  .\/  R )  =  ( R  .\/  R
) )
2 simpr3 963 . . . . . . 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 30180 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A )  ->  ( R  .\/  R
)  =  R )
62, 5syldan 456 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( R  .\/  R )  =  R )
71, 6sylan9eqr 2350 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  Q  =  R )  ->  ( Q  .\/  R )  =  R )
87breq2d 4051 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  Q  =  R )  ->  ( P  .<  ( Q  .\/  R )  <->  P  .<  R ) )
9 hlatl 30172 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  AtLat )
109adantr 451 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  AtLat )
11 simpr1 961 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  P  e.  A )
12 atltcvr.s . . . . . . . 8  |-  .<  =  ( lt `  K )
1312, 4atnlt 30125 . . . . . . 7  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  R  e.  A )  ->  -.  P  .<  R )
1410, 11, 2, 13syl3anc 1182 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  -.  P  .<  R )
1514pm2.21d 98 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .<  R  ->  P C ( Q  .\/  R ) ) )
1615adantr 451 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  Q  =  R )  ->  ( P  .<  R  ->  P C ( Q  .\/  R ) ) )
178, 16sylbid 206 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  Q  =  R )  ->  ( P  .<  ( Q  .\/  R )  ->  P C
( Q  .\/  R
) ) )
18 simpl 443 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  HL )
19 hllat 30175 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
2019adantr 451 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  Lat )
21 simpr2 962 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  A )
22 eqid 2296 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2322, 4atbase 30101 . . . . . . . 8  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
2421, 23syl 15 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  ( Base `  K
) )
2522, 4atbase 30101 . . . . . . . 8  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
262, 25syl 15 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  ( Base `  K
) )
2722, 3latjcl 14172 . . . . . . 7  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( Q  .\/  R )  e.  ( Base `  K
) )
2820, 24, 26, 27syl3anc 1182 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( Q  .\/  R )  e.  ( Base `  K
) )
29 eqid 2296 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
3029, 12pltle 14111 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( Q  .\/  R )  e.  ( Base `  K
) )  ->  ( P  .<  ( Q  .\/  R )  ->  P ( le `  K ) ( Q  .\/  R ) ) )
3118, 11, 28, 30syl3anc 1182 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .<  ( Q  .\/  R )  ->  P ( le `  K ) ( Q  .\/  R ) ) )
3231adantr 451 . . . 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 730 . . . . . . . 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 731 . . . . . . . 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 447 . . . . . . . 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 1132 . . . . . . 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 629 . . . . . 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 30244 . . . . . 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 15 . . . . 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 423 . . . 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 40 . . 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 2537 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .<  ( Q  .\/  R )  ->  P C
( Q  .\/  R
) ) )
4422, 4atbase 30101 . . . 4  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
4511, 44syl 15 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  P  e.  ( Base `  K
) )
4622, 12, 38cvrlt 30082 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
) )  /\  P C ( Q  .\/  R ) )  ->  P  .<  ( Q  .\/  R
) )
4746ex 423 . . 3  |-  ( ( K  e.  HL  /\  P  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
) )  ->  ( P C ( Q  .\/  R )  ->  P  .<  ( Q  .\/  R ) ) )
4818, 45, 28, 47syl3anc 1182 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P C ( Q  .\/  R )  ->  P  .<  ( Q  .\/  R ) ) )
4943, 48impbid 183 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   ltcplt 14091   joincjn 14094   Latclat 14167    <o ccvr 30074   Atomscatm 30075   AtLatcal 30076   HLchlt 30162
This theorem is referenced by:  atlt  30248  2atlt  30250  atexchltN  30252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163
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