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Theorem 1cvrjat 29957
Description: An element covered by the lattice unit, when joined with an atom not under it, equals the lattice unit. (Contributed by NM, 30-Apr-2012.)
Hypotheses
Ref Expression
1cvrjat.b  |-  B  =  ( Base `  K
)
1cvrjat.l  |-  .<_  =  ( le `  K )
1cvrjat.j  |-  .\/  =  ( join `  K )
1cvrjat.u  |-  .1.  =  ( 1. `  K )
1cvrjat.c  |-  C  =  (  <o  `  K )
1cvrjat.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
1cvrjat  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  ( X  .\/  P )  =  .1.  )

Proof of Theorem 1cvrjat
StepHypRef Expression
1 simprr 734 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  -.  P  .<_  X )
2 1cvrjat.b . . . . . . . 8  |-  B  =  ( Base `  K
)
3 1cvrjat.l . . . . . . . 8  |-  .<_  =  ( le `  K )
4 1cvrjat.j . . . . . . . 8  |-  .\/  =  ( join `  K )
5 1cvrjat.c . . . . . . . 8  |-  C  =  (  <o  `  K )
6 1cvrjat.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
72, 3, 4, 5, 6cvr1 29892 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  ->  ( -.  P  .<_  X  <-> 
X C ( X 
.\/  P ) ) )
87adantr 452 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  ( -.  P  .<_  X  <->  X C
( X  .\/  P
) ) )
91, 8mpbid 202 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  X C ( X  .\/  P ) )
10 simpl1 960 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  K  e.  HL )
11 hlop 29845 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OP )
1210, 11syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  K  e.  OP )
13 simpl2 961 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  X  e.  B )
14 hllat 29846 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
1510, 14syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  K  e.  Lat )
16 simpl3 962 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  P  e.  A )
172, 6atbase 29772 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  B )
1816, 17syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  P  e.  B )
192, 4latjcl 14434 . . . . . . 7  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  ( X  .\/  P
)  e.  B )
2015, 13, 18, 19syl3anc 1184 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  ( X  .\/  P )  e.  B )
21 eqid 2404 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
222, 21, 5cvrcon3b 29760 . . . . . 6  |-  ( ( K  e.  OP  /\  X  e.  B  /\  ( X  .\/  P )  e.  B )  -> 
( X C ( X  .\/  P )  <-> 
( ( oc `  K ) `  ( X  .\/  P ) ) C ( ( oc
`  K ) `  X ) ) )
2312, 13, 20, 22syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  ( X C ( X  .\/  P )  <->  ( ( oc
`  K ) `  ( X  .\/  P ) ) C ( ( oc `  K ) `
 X ) ) )
249, 23mpbid 202 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  (
( oc `  K
) `  ( X  .\/  P ) ) C ( ( oc `  K ) `  X
) )
25 hlatl 29843 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  AtLat )
2610, 25syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  K  e.  AtLat )
272, 21opoccl 29677 . . . . . 6  |-  ( ( K  e.  OP  /\  ( X  .\/  P )  e.  B )  -> 
( ( oc `  K ) `  ( X  .\/  P ) )  e.  B )
2812, 20, 27syl2anc 643 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  (
( oc `  K
) `  ( X  .\/  P ) )  e.  B )
292, 21opoccl 29677 . . . . . . 7  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
3012, 13, 29syl2anc 643 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  (
( oc `  K
) `  X )  e.  B )
31 eqid 2404 . . . . . . . . 9  |-  ( 0.
`  K )  =  ( 0. `  K
)
32 1cvrjat.u . . . . . . . . 9  |-  .1.  =  ( 1. `  K )
3331, 32, 21opoc1 29685 . . . . . . . 8  |-  ( K  e.  OP  ->  (
( oc `  K
) `  .1.  )  =  ( 0. `  K ) )
3410, 11, 333syl 19 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  (
( oc `  K
) `  .1.  )  =  ( 0. `  K ) )
35 simprl 733 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  X C  .1.  )
362, 32op1cl 29668 . . . . . . . . . 10  |-  ( K  e.  OP  ->  .1.  e.  B )
3710, 11, 363syl 19 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  .1.  e.  B )
382, 21, 5cvrcon3b 29760 . . . . . . . . 9  |-  ( ( K  e.  OP  /\  X  e.  B  /\  .1.  e.  B )  -> 
( X C  .1.  <->  ( ( oc `  K
) `  .1.  ) C ( ( oc
`  K ) `  X ) ) )
3912, 13, 37, 38syl3anc 1184 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  ( X C  .1.  <->  ( ( oc `  K ) `  .1.  ) C ( ( oc `  K ) `
 X ) ) )
4035, 39mpbid 202 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  (
( oc `  K
) `  .1.  ) C ( ( oc
`  K ) `  X ) )
4134, 40eqbrtrrd 4194 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  ( 0. `  K ) C ( ( oc `  K ) `  X
) )
422, 31, 5, 6isat 29769 . . . . . . 7  |-  ( K  e.  HL  ->  (
( ( oc `  K ) `  X
)  e.  A  <->  ( (
( oc `  K
) `  X )  e.  B  /\  ( 0. `  K ) C ( ( oc `  K ) `  X
) ) ) )
4310, 42syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  (
( ( oc `  K ) `  X
)  e.  A  <->  ( (
( oc `  K
) `  X )  e.  B  /\  ( 0. `  K ) C ( ( oc `  K ) `  X
) ) ) )
4430, 41, 43mpbir2and 889 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  (
( oc `  K
) `  X )  e.  A )
452, 3, 31, 5, 6atcvreq0 29797 . . . . 5  |-  ( ( K  e.  AtLat  /\  (
( oc `  K
) `  ( X  .\/  P ) )  e.  B  /\  ( ( oc `  K ) `
 X )  e.  A )  ->  (
( ( oc `  K ) `  ( X  .\/  P ) ) C ( ( oc
`  K ) `  X )  <->  ( ( oc `  K ) `  ( X  .\/  P ) )  =  ( 0.
`  K ) ) )
4626, 28, 44, 45syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  (
( ( oc `  K ) `  ( X  .\/  P ) ) C ( ( oc
`  K ) `  X )  <->  ( ( oc `  K ) `  ( X  .\/  P ) )  =  ( 0.
`  K ) ) )
4724, 46mpbid 202 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  (
( oc `  K
) `  ( X  .\/  P ) )  =  ( 0. `  K
) )
4847fveq2d 5691 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  (
( oc `  K
) `  ( ( oc `  K ) `  ( X  .\/  P ) ) )  =  ( ( oc `  K
) `  ( 0. `  K ) ) )
492, 21opococ 29678 . . 3  |-  ( ( K  e.  OP  /\  ( X  .\/  P )  e.  B )  -> 
( ( oc `  K ) `  (
( oc `  K
) `  ( X  .\/  P ) ) )  =  ( X  .\/  P ) )
5012, 20, 49syl2anc 643 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  (
( oc `  K
) `  ( ( oc `  K ) `  ( X  .\/  P ) ) )  =  ( X  .\/  P ) )
5131, 32, 21opoc0 29686 . . 3  |-  ( K  e.  OP  ->  (
( oc `  K
) `  ( 0. `  K ) )  =  .1.  )
5210, 11, 513syl 19 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  (
( oc `  K
) `  ( 0. `  K ) )  =  .1.  )
5348, 50, 523eqtr3d 2444 1  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  ( X  .\/  P )  =  .1.  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   occoc 13492   joincjn 14356   0.cp0 14421   1.cp1 14422   Latclat 14429   OPcops 29655    <o ccvr 29745   Atomscatm 29746   AtLatcal 29747   HLchlt 29833
This theorem is referenced by:  1cvrat  29958  lhpjat1  30502
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834
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