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Theorem 2atm2atN 29974
Description: Two joins with a common atom have a nonzero meet. (Contributed by NM, 4-Jul-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
2atm2at.j  |-  .\/  =  ( join `  K )
2atm2at.m  |-  ./\  =  ( meet `  K )
2atm2at.z  |-  .0.  =  ( 0. `  K )
2atm2at.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
2atm2atN  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( R  .\/  P
)  ./\  ( R  .\/  Q ) )  =/= 
.0.  )

Proof of Theorem 2atm2atN
StepHypRef Expression
1 hlop 29552 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
21adantr 451 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  OP )
3 simpr3 963 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  A )
4 2atm2at.z . . . . 5  |-  .0.  =  ( 0. `  K )
5 eqid 2283 . . . . 5  |-  ( lt
`  K )  =  ( lt `  K
)
6 2atm2at.a . . . . 5  |-  A  =  ( Atoms `  K )
74, 5, 60ltat 29481 . . . 4  |-  ( ( K  e.  OP  /\  R  e.  A )  ->  .0.  ( lt `  K ) R )
82, 3, 7syl2anc 642 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  .0.  ( lt `  K ) R )
9 simpl 443 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  HL )
10 simpr1 961 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  P  e.  A )
11 eqid 2283 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
12 2atm2at.j . . . . . 6  |-  .\/  =  ( join `  K )
1311, 12, 6hlatlej1 29564 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  P  e.  A )  ->  R ( le `  K ) ( R 
.\/  P ) )
149, 3, 10, 13syl3anc 1182 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R
( le `  K
) ( R  .\/  P ) )
15 simpr2 962 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  A )
1611, 12, 6hlatlej1 29564 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  Q  e.  A )  ->  R ( le `  K ) ( R 
.\/  Q ) )
179, 3, 15, 16syl3anc 1182 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R
( le `  K
) ( R  .\/  Q ) )
18 hllat 29553 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
1918adantr 451 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  Lat )
20 eqid 2283 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2120, 6atbase 29479 . . . . . 6  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
223, 21syl 15 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  ( Base `  K
) )
2320, 12, 6hlatjcl 29556 . . . . . 6  |-  ( ( K  e.  HL  /\  R  e.  A  /\  P  e.  A )  ->  ( R  .\/  P
)  e.  ( Base `  K ) )
249, 3, 10, 23syl3anc 1182 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( R  .\/  P )  e.  ( Base `  K
) )
2520, 12, 6hlatjcl 29556 . . . . . 6  |-  ( ( K  e.  HL  /\  R  e.  A  /\  Q  e.  A )  ->  ( R  .\/  Q
)  e.  ( Base `  K ) )
269, 3, 15, 25syl3anc 1182 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( R  .\/  Q )  e.  ( Base `  K
) )
27 2atm2at.m . . . . . 6  |-  ./\  =  ( meet `  K )
2820, 11, 27latlem12 14184 . . . . 5  |-  ( ( K  e.  Lat  /\  ( R  e.  ( Base `  K )  /\  ( R  .\/  P )  e.  ( Base `  K
)  /\  ( R  .\/  Q )  e.  (
Base `  K )
) )  ->  (
( R ( le
`  K ) ( R  .\/  P )  /\  R ( le
`  K ) ( R  .\/  Q ) )  <->  R ( le `  K ) ( ( R  .\/  P ) 
./\  ( R  .\/  Q ) ) ) )
2919, 22, 24, 26, 28syl13anc 1184 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( R ( le
`  K ) ( R  .\/  P )  /\  R ( le
`  K ) ( R  .\/  Q ) )  <->  R ( le `  K ) ( ( R  .\/  P ) 
./\  ( R  .\/  Q ) ) ) )
3014, 17, 29mpbi2and 887 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R
( le `  K
) ( ( R 
.\/  P )  ./\  ( R  .\/  Q ) ) )
31 hlpos 29555 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Poset )
3231adantr 451 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  Poset )
3320, 4op0cl 29374 . . . . 5  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
342, 33syl 15 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  .0.  e.  ( Base `  K
) )
3520, 27latmcl 14157 . . . . 5  |-  ( ( K  e.  Lat  /\  ( R  .\/  P )  e.  ( Base `  K
)  /\  ( R  .\/  Q )  e.  (
Base `  K )
)  ->  ( ( R  .\/  P )  ./\  ( R  .\/  Q ) )  e.  ( Base `  K ) )
3619, 24, 26, 35syl3anc 1182 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( R  .\/  P
)  ./\  ( R  .\/  Q ) )  e.  ( Base `  K
) )
3720, 11, 5pltletr 14105 . . . 4  |-  ( ( K  e.  Poset  /\  (  .0.  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )  /\  ( ( R  .\/  P )  ./\  ( R  .\/  Q ) )  e.  ( Base `  K
) ) )  -> 
( (  .0.  ( lt `  K ) R  /\  R ( le
`  K ) ( ( R  .\/  P
)  ./\  ( R  .\/  Q ) ) )  ->  .0.  ( lt `  K ) ( ( R  .\/  P ) 
./\  ( R  .\/  Q ) ) ) )
3832, 34, 22, 36, 37syl13anc 1184 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
(  .0.  ( lt
`  K ) R  /\  R ( le
`  K ) ( ( R  .\/  P
)  ./\  ( R  .\/  Q ) ) )  ->  .0.  ( lt `  K ) ( ( R  .\/  P ) 
./\  ( R  .\/  Q ) ) ) )
398, 30, 38mp2and 660 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  .0.  ( lt `  K ) ( ( R  .\/  P )  ./\  ( R  .\/  Q ) ) )
4020, 5, 4opltn0 29380 . . 3  |-  ( ( K  e.  OP  /\  ( ( R  .\/  P )  ./\  ( R  .\/  Q ) )  e.  ( Base `  K
) )  ->  (  .0.  ( lt `  K
) ( ( R 
.\/  P )  ./\  ( R  .\/  Q ) )  <->  ( ( R 
.\/  P )  ./\  ( R  .\/  Q ) )  =/=  .0.  )
)
412, 36, 40syl2anc 642 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (  .0.  ( lt `  K
) ( ( R 
.\/  P )  ./\  ( R  .\/  Q ) )  <->  ( ( R 
.\/  P )  ./\  ( R  .\/  Q ) )  =/=  .0.  )
)
4239, 41mpbid 201 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( R  .\/  P
)  ./\  ( R  .\/  Q ) )  =/= 
.0.  )
Colors of variables: wff set class
Syntax hints:    -> 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   Posetcpo 14074   ltcplt 14075   joincjn 14078   meetcmee 14079   0.cp0 14143   Latclat 14151   OPcops 29362   Atomscatm 29453   HLchlt 29540
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-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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