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Theorem 1cvrjat 30286
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 30221 . . . . . . 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 30174 . . . . . . 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 30175 . . . . . . . 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 30101 . . . . . . . 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 14172 . . . . . . 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 2296 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
222, 21, 5cvrcon3b 30089 . . . . . 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 30172 . . . . . 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 30006 . . . . . 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 30006 . . . . . . 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 2296 . . . . . . . . 9  |-  ( 0.
`  K )  =  ( 0. `  K
)
32 1cvrjat.u . . . . . . . . 9  |-  .1.  =  ( 1. `  K )
3331, 32, 21opoc1 30014 . . . . . . . 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 29997 . . . . . . . . . 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 30089 . . . . . . . . 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 4061 . . . . . 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 30098 . . . . . . 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 30126 . . . . 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 5545 . 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 30007 . . 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 30015 . . 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 2336 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 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   occoc 13232   joincjn 14094   0.cp0 14159   1.cp1 14160   Latclat 14167   OPcops 29984    <o ccvr 30074   Atomscatm 30075   AtLatcal 30076   HLchlt 30162
This theorem is referenced by:  1cvrat  30287  lhpjat1  30831
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163
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