Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  atcvrj2b Unicode version

Theorem atcvrj2b 29546
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 1012 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =  R )  ->  Q  =/=  R )
21necomd 2633 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =  R )  ->  R  =/=  Q )
3 simpl1 960 . . . . . . 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 1037 . . . . . . 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 1036 . . . . . . 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 29532 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  Q  e.  A )  ->  ( R  =/=  Q  <->  R C ( Q  .\/  R ) ) )
103, 4, 5, 9syl3anc 1184 . . . . . 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 202 . . . . 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 4156 . . . . . 6  |-  ( P  =  R  ->  ( P C ( Q  .\/  R )  <->  R C ( Q 
.\/  R ) ) )
1312adantl 453 . . . . 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 224 . . . 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 960 . . . . 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 961 . . . . 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 448 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =/=  R )  ->  P  =/=  R )
18 simpl3r 1013 . . . . 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 29545 . . . . 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 1188 . . . 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 2628 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  ->  P C ( Q  .\/  R ) )
23223expia 1155 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( Q  =/=  R  /\  P  .<_  ( Q 
.\/  R ) )  ->  P C ( Q  .\/  R ) ) )
24 hlatl 29475 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  AtLat )
2524ad2antrr 707 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  K  e.  AtLat )
26 simplr1 999 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  P  e.  A )
27 eqid 2387 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
2827, 8atn0 29423 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  P  =/=  ( 0. `  K
) )
2925, 26, 28syl2anc 643 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  P  =/=  ( 0. `  K
) )
30 simpll 731 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  K  e.  HL )
31 eqid 2387 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
3231, 8atbase 29404 . . . . . . . 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 1000 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  Q  e.  A )
35 simplr3 1001 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  R  e.  A )
36 simpr 448 . . . . . . 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 29542 . . . . . . 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 1197 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  ( P  =  ( 0. `  K )  <->  Q  =  R ) )
3938necon3bid 2585 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  ( P  =/=  ( 0. `  K )  <->  Q  =/=  R ) )
4029, 39mpbid 202 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  Q  =/=  R )
41 hllat 29478 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
4241ad2antrr 707 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  K  e.  Lat )
4331, 8atbase 29404 . . . . . . . 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 29404 . . . . . . . 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 14406 . . . . . . 7  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( Q  .\/  R )  e.  ( Base `  K
) )
4842, 44, 46, 47syl3anc 1184 . . . . . 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 1134 . . . . 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 29393 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
) )  /\  P C ( Q  .\/  R ) )  ->  P  .<_  ( Q  .\/  R
) )
5149, 50sylancom 649 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  P  .<_  ( Q  .\/  R
) )
5240, 51jca 519 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R
) ) )
5352ex 424 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P C ( Q  .\/  R )  ->  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) ) )
5423, 53impbid 184 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   Basecbs 13396   lecple 13463   joincjn 14328   0.cp0 14393   Latclat 14401    <o ccvr 29377   Atomscatm 29378   AtLatcal 29379   HLchlt 29465
This theorem is referenced by:  atcvrj2  29547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466
  Copyright terms: Public domain W3C validator