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Theorem atltcvr 29624
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 5865 . . . . . 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 29558 . . . . . . 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 2337 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  Q  =  R )  ->  ( Q  .\/  R )  =  R )
87breq2d 4035 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  Q  =  R )  ->  ( P  .<  ( Q  .\/  R )  <->  P  .<  R ) )
9 hlatl 29550 . . . . . . . 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 29503 . . . . . . 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 29553 . . . . . . . 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 2283 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2322, 4atbase 29479 . . . . . . . 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 29479 . . . . . . . 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 14156 . . . . . . 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 2283 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
3029, 12pltle 14095 . . . . . 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 29622 . . . . . 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 2524 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .<  ( Q  .\/  R )  ->  P C
( Q  .\/  R
) ) )
4422, 4atbase 29479 . . . 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 29460 . . . 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 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   ltcplt 14075   joincjn 14078   Latclat 14151    <o ccvr 29452   Atomscatm 29453   AtLatcal 29454   HLchlt 29540
This theorem is referenced by:  atlt  29626  2atlt  29628  atexchltN  29630
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541
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