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Theorem 1cvrjat 34939
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 757 . . . . . 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 34874 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  ->  ( -.  P  .<_  X  <-> 
X C ( X 
.\/  P ) ) )
87adantr 465 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  ( -.  P  .<_  X  <->  X C
( X  .\/  P
) ) )
91, 8mpbid 210 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  X C ( X  .\/  P ) )
10 simpl1 1000 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  K  e.  HL )
11 hlop 34827 . . . . . . 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 1001 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  X  e.  B )
14 hllat 34828 . . . . . . . 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 1002 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  P  e.  A )
172, 6atbase 34754 . . . . . . . 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 15555 . . . . . . 7  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  ( X  .\/  P
)  e.  B )
2015, 13, 18, 19syl3anc 1229 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  ( X  .\/  P )  e.  B )
21 eqid 2443 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
222, 21, 5cvrcon3b 34742 . . . . . 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 1229 . . . . 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 210 . . . 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 34825 . . . . . 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 34659 . . . . . 6  |-  ( ( K  e.  OP  /\  ( X  .\/  P )  e.  B )  -> 
( ( oc `  K ) `  ( X  .\/  P ) )  e.  B )
2812, 20, 27syl2anc 661 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  (
( oc `  K
) `  ( X  .\/  P ) )  e.  B )
292, 21opoccl 34659 . . . . . . 7  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
3012, 13, 29syl2anc 661 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  (
( oc `  K
) `  X )  e.  B )
31 eqid 2443 . . . . . . . . 9  |-  ( 0.
`  K )  =  ( 0. `  K
)
32 1cvrjat.u . . . . . . . . 9  |-  .1.  =  ( 1. `  K )
3331, 32, 21opoc1 34667 . . . . . . . 8  |-  ( K  e.  OP  ->  (
( oc `  K
) `  .1.  )  =  ( 0. `  K ) )
3410, 11, 333syl 20 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  (
( oc `  K
) `  .1.  )  =  ( 0. `  K ) )
35 simprl 756 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  X C  .1.  )
362, 32op1cl 34650 . . . . . . . . . 10  |-  ( K  e.  OP  ->  .1.  e.  B )
3710, 11, 363syl 20 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  .1.  e.  B )
382, 21, 5cvrcon3b 34742 . . . . . . . . 9  |-  ( ( K  e.  OP  /\  X  e.  B  /\  .1.  e.  B )  -> 
( X C  .1.  <->  ( ( oc `  K
) `  .1.  ) C ( ( oc
`  K ) `  X ) ) )
3912, 13, 37, 38syl3anc 1229 . . . . . . . 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 210 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  (
( oc `  K
) `  .1.  ) C ( ( oc
`  K ) `  X ) )
4134, 40eqbrtrrd 4459 . . . . . 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 34751 . . . . . . 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 922 . . . . 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 34779 . . . . 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 1229 . . . 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 210 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  (
( oc `  K
) `  ( X  .\/  P ) )  =  ( 0. `  K
) )
4847fveq2d 5860 . 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 34660 . . 3  |-  ( ( K  e.  OP  /\  ( X  .\/  P )  e.  B )  -> 
( ( oc `  K ) `  (
( oc `  K
) `  ( X  .\/  P ) ) )  =  ( X  .\/  P ) )
5012, 20, 49syl2anc 661 . 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 34668 . . 3  |-  ( K  e.  OP  ->  (
( oc `  K
) `  ( 0. `  K ) )  =  .1.  )
5210, 11, 513syl 20 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  (
( oc `  K
) `  ( 0. `  K ) )  =  .1.  )
5348, 50, 523eqtr3d 2492 1  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  ( X  .\/  P )  =  .1.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   Basecbs 14509   lecple 14581   occoc 14582   joincjn 15447   0.cp0 15541   1.cp1 15542   Latclat 15549   OPcops 34637    <o ccvr 34727   Atomscatm 34728   AtLatcal 34729   HLchlt 34815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-preset 15431  df-poset 15449  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-oposet 34641  df-ol 34643  df-oml 34644  df-covers 34731  df-ats 34732  df-atl 34763  df-cvlat 34787  df-hlat 34816
This theorem is referenced by:  1cvrat  34940  lhpjat1  35484
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