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Theorem atcvrj2b 33399
Description: Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)
Hypotheses
Ref Expression
atcvrj1x.l  |-  .<_  =  ( le `  K )
atcvrj1x.j  |-  .\/  =  ( join `  K )
atcvrj1x.c  |-  C  =  (  <o  `  K )
atcvrj1x.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atcvrj2b  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( Q  =/=  R  /\  P  .<_  ( Q 
.\/  R ) )  <-> 
P C ( Q 
.\/  R ) ) )

Proof of Theorem atcvrj2b
StepHypRef Expression
1 simpl3l 1043 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =  R )  ->  Q  =/=  R )
21necomd 2722 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =  R )  ->  R  =/=  Q )
3 simpl1 991 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =  R )  ->  K  e.  HL )
4 simpl23 1068 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =  R )  ->  R  e.  A )
5 simpl22 1067 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =  R )  ->  Q  e.  A )
6 atcvrj1x.j . . . . . . . 8  |-  .\/  =  ( join `  K )
7 atcvrj1x.c . . . . . . . 8  |-  C  =  (  <o  `  K )
8 atcvrj1x.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
96, 7, 8atcvr2 33385 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  Q  e.  A )  ->  ( R  =/=  Q  <->  R C ( Q  .\/  R ) ) )
103, 4, 5, 9syl3anc 1219 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =  R )  ->  ( R  =/=  Q  <->  R C
( Q  .\/  R
) ) )
112, 10mpbid 210 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =  R )  ->  R C ( Q  .\/  R ) )
12 breq1 4402 . . . . . 6  |-  ( P  =  R  ->  ( P C ( Q  .\/  R )  <->  R C ( Q 
.\/  R ) ) )
1312adantl 466 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =  R )  ->  ( P C ( Q  .\/  R )  <->  R C ( Q 
.\/  R ) ) )
1411, 13mpbird 232 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =  R )  ->  P C ( Q  .\/  R ) )
15 simpl1 991 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =/=  R )  ->  K  e.  HL )
16 simpl2 992 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =/=  R )  ->  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )
)
17 simpr 461 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =/=  R )  ->  P  =/=  R )
18 simpl3r 1044 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =/=  R )  ->  P  .<_  ( Q  .\/  R
) )
19 atcvrj1x.l . . . . . 6  |-  .<_  =  ( le `  K )
2019, 6, 7, 8atcvrj1 33398 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  ->  P C ( Q  .\/  R ) )
2115, 16, 17, 18, 20syl112anc 1223 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =/=  R )  ->  P C ( Q  .\/  R ) )
2214, 21pm2.61dane 2769 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  ->  P C ( Q  .\/  R ) )
23223expia 1190 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( Q  =/=  R  /\  P  .<_  ( Q 
.\/  R ) )  ->  P C ( Q  .\/  R ) ) )
24 hlatl 33328 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  AtLat )
2524ad2antrr 725 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  K  e.  AtLat )
26 simplr1 1030 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  P  e.  A )
27 eqid 2454 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
2827, 8atn0 33276 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  P  =/=  ( 0. `  K
) )
2925, 26, 28syl2anc 661 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  P  =/=  ( 0. `  K
) )
30 simpll 753 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  K  e.  HL )
31 eqid 2454 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
3231, 8atbase 33257 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
3326, 32syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  P  e.  ( Base `  K
) )
34 simplr2 1031 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  Q  e.  A )
35 simplr3 1032 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  R  e.  A )
36 simpr 461 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  P C ( Q  .\/  R ) )
3731, 6, 27, 7, 8atcvrj0 33395 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  ( Base `  K )  /\  Q  e.  A  /\  R  e.  A )  /\  P C ( Q 
.\/  R ) )  ->  ( P  =  ( 0. `  K
)  <->  Q  =  R
) )
3830, 33, 34, 35, 36, 37syl131anc 1232 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  ( P  =  ( 0. `  K )  <->  Q  =  R ) )
3938necon3bid 2709 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  ( P  =/=  ( 0. `  K )  <->  Q  =/=  R ) )
4029, 39mpbid 210 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  Q  =/=  R )
41 hllat 33331 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
4241ad2antrr 725 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  K  e.  Lat )
4331, 8atbase 33257 . . . . . . . 8  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
4434, 43syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  Q  e.  ( Base `  K
) )
4531, 8atbase 33257 . . . . . . . 8  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
4635, 45syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  R  e.  ( Base `  K
) )
4731, 6latjcl 15339 . . . . . . 7  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( Q  .\/  R )  e.  ( Base `  K
) )
4842, 44, 46, 47syl3anc 1219 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  ( Q  .\/  R )  e.  ( Base `  K
) )
4930, 33, 483jca 1168 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  ( K  e.  HL  /\  P  e.  ( Base `  K
)  /\  ( Q  .\/  R )  e.  (
Base `  K )
) )
5031, 19, 7cvrle 33246 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
) )  /\  P C ( Q  .\/  R ) )  ->  P  .<_  ( Q  .\/  R
) )
5149, 50sylancom 667 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  P  .<_  ( Q  .\/  R
) )
5240, 51jca 532 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R
) ) )
5352ex 434 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P C ( Q  .\/  R )  ->  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) ) )
5423, 53impbid 191 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( Q  =/=  R  /\  P  .<_  ( Q 
.\/  R ) )  <-> 
P C ( Q 
.\/  R ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2647   class class class wbr 4399   ` cfv 5525  (class class class)co 6199   Basecbs 14291   lecple 14363   joincjn 15232   0.cp0 15325   Latclat 15333    <o ccvr 33230   Atomscatm 33231   AtLatcal 33232   HLchlt 33318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-poset 15234  df-plt 15246  df-lub 15262  df-glb 15263  df-join 15264  df-meet 15265  df-p0 15327  df-lat 15334  df-clat 15396  df-oposet 33144  df-ol 33146  df-oml 33147  df-covers 33234  df-ats 33235  df-atl 33266  df-cvlat 33290  df-hlat 33319
This theorem is referenced by:  atcvrj2  33400
  Copyright terms: Public domain W3C validator