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Theorem 1cvrjat 29664
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 733 . . . . . 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 29599 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  ->  ( -.  P  .<_  X  <-> 
X C ( X 
.\/  P ) ) )
87adantr 451 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  ( -.  P  .<_  X  <->  X C
( X  .\/  P
) ) )
91, 8mpbid 201 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  X C ( X  .\/  P ) )
10 simpl1 958 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  K  e.  HL )
11 hlop 29552 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OP )
1210, 11syl 15 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  K  e.  OP )
13 simpl2 959 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  X  e.  B )
14 hllat 29553 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
1510, 14syl 15 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  K  e.  Lat )
16 simpl3 960 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  P  e.  A )
172, 6atbase 29479 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  B )
1816, 17syl 15 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  P  e.  B )
192, 4latjcl 14156 . . . . . . 7  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  ( X  .\/  P
)  e.  B )
2015, 13, 18, 19syl3anc 1182 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  ( X  .\/  P )  e.  B )
21 eqid 2283 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
222, 21, 5cvrcon3b 29467 . . . . . 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 1182 . . . . 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 201 . . . 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 29550 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  AtLat )
2610, 25syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  K  e.  AtLat )
272, 21opoccl 29384 . . . . . 6  |-  ( ( K  e.  OP  /\  ( X  .\/  P )  e.  B )  -> 
( ( oc `  K ) `  ( X  .\/  P ) )  e.  B )
2812, 20, 27syl2anc 642 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  (
( oc `  K
) `  ( X  .\/  P ) )  e.  B )
292, 21opoccl 29384 . . . . . . 7  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
3012, 13, 29syl2anc 642 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  (
( oc `  K
) `  X )  e.  B )
31 eqid 2283 . . . . . . . . 9  |-  ( 0.
`  K )  =  ( 0. `  K
)
32 1cvrjat.u . . . . . . . . 9  |-  .1.  =  ( 1. `  K )
3331, 32, 21opoc1 29392 . . . . . . . 8  |-  ( K  e.  OP  ->  (
( oc `  K
) `  .1.  )  =  ( 0. `  K ) )
3410, 11, 333syl 18 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  (
( oc `  K
) `  .1.  )  =  ( 0. `  K ) )
35 simprl 732 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  X C  .1.  )
362, 32op1cl 29375 . . . . . . . . . 10  |-  ( K  e.  OP  ->  .1.  e.  B )
3710, 11, 363syl 18 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  .1.  e.  B )
382, 21, 5cvrcon3b 29467 . . . . . . . . 9  |-  ( ( K  e.  OP  /\  X  e.  B  /\  .1.  e.  B )  -> 
( X C  .1.  <->  ( ( oc `  K
) `  .1.  ) C ( ( oc
`  K ) `  X ) ) )
3912, 13, 37, 38syl3anc 1182 . . . . . . . 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 201 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  (
( oc `  K
) `  .1.  ) C ( ( oc
`  K ) `  X ) )
4134, 40eqbrtrrd 4045 . . . . . 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 29476 . . . . . . 7  |-  ( K  e.  HL  ->  (
( ( oc `  K ) `  X
)  e.  A  <->  ( (
( oc `  K
) `  X )  e.  B  /\  ( 0. `  K ) C ( ( oc `  K ) `  X
) ) ) )
4310, 42syl 15 . . . . . 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 888 . . . . 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 29504 . . . . 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 1182 . . . 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 201 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  (
( oc `  K
) `  ( X  .\/  P ) )  =  ( 0. `  K
) )
4847fveq2d 5529 . 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 29385 . . 3  |-  ( ( K  e.  OP  /\  ( X  .\/  P )  e.  B )  -> 
( ( oc `  K ) `  (
( oc `  K
) `  ( X  .\/  P ) ) )  =  ( X  .\/  P ) )
5012, 20, 49syl2anc 642 . 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 29393 . . 3  |-  ( K  e.  OP  ->  (
( oc `  K
) `  ( 0. `  K ) )  =  .1.  )
5210, 11, 513syl 18 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  (
( oc `  K
) `  ( 0. `  K ) )  =  .1.  )
5348, 50, 523eqtr3d 2323 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   occoc 13216   joincjn 14078   0.cp0 14143   1.cp1 14144   Latclat 14151   OPcops 29362    <o ccvr 29452   Atomscatm 29453   AtLatcal 29454   HLchlt 29540
This theorem is referenced by:  1cvrat  29665  lhpjat1  30209
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541
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