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Theorem atcvrj2b 35569
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 1049 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =  R )  ->  Q  =/=  R )
21necomd 2653 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =  R )  ->  R  =/=  Q )
3 simpl1 997 . . . . . . 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 1074 . . . . . . 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 1073 . . . . . . 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 35555 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  Q  e.  A )  ->  ( R  =/=  Q  <->  R C ( Q  .\/  R ) ) )
103, 4, 5, 9syl3anc 1226 . . . . . 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 4370 . . . . . 6  |-  ( P  =  R  ->  ( P C ( Q  .\/  R )  <->  R C ( Q 
.\/  R ) ) )
1312adantl 464 . . . . 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 997 . . . . 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 998 . . . . 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 459 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  /\  P  =/=  R )  ->  P  =/=  R )
18 simpl3r 1050 . . . . 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 35568 . . . . 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 1230 . . . 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 2700 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  ->  P C ( Q  .\/  R ) )
23223expia 1196 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( Q  =/=  R  /\  P  .<_  ( Q 
.\/  R ) )  ->  P C ( Q  .\/  R ) ) )
24 hlatl 35498 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  AtLat )
2524ad2antrr 723 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  K  e.  AtLat )
26 simplr1 1036 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  P  e.  A )
27 eqid 2382 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
2827, 8atn0 35446 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  P  =/=  ( 0. `  K
) )
2925, 26, 28syl2anc 659 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  P  =/=  ( 0. `  K
) )
30 simpll 751 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  K  e.  HL )
31 eqid 2382 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
3231, 8atbase 35427 . . . . . . . 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 1037 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  Q  e.  A )
35 simplr3 1038 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  R  e.  A )
36 simpr 459 . . . . . . 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 35565 . . . . . . 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 1239 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  ( P  =  ( 0. `  K )  <->  Q  =  R ) )
3938necon3bid 2640 . . . . 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 35501 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
4241ad2antrr 723 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  K  e.  Lat )
4331, 8atbase 35427 . . . . . . . 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 35427 . . . . . . . 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 15798 . . . . . . 7  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( Q  .\/  R )  e.  ( Base `  K
) )
4842, 44, 46, 47syl3anc 1226 . . . . . 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 1174 . . . . 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 35416 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
) )  /\  P C ( Q  .\/  R ) )  ->  P  .<_  ( Q  .\/  R
) )
5149, 50sylancom 665 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  P  .<_  ( Q  .\/  R
) )
5240, 51jca 530 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  P C ( Q  .\/  R ) )  ->  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R
) ) )
5352ex 432 . 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   class class class wbr 4367   ` cfv 5496  (class class class)co 6196   Basecbs 14634   lecple 14709   joincjn 15690   0.cp0 15784   Latclat 15792    <o ccvr 35400   Atomscatm 35401   AtLatcal 35402   HLchlt 35488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-preset 15674  df-poset 15692  df-plt 15705  df-lub 15721  df-glb 15722  df-join 15723  df-meet 15724  df-p0 15786  df-lat 15793  df-clat 15855  df-oposet 35314  df-ol 35316  df-oml 35317  df-covers 35404  df-ats 35405  df-atl 35436  df-cvlat 35460  df-hlat 35489
This theorem is referenced by:  atcvrj2  35570
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