Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  1cvrjat Structured version   Unicode version

Theorem 1cvrjat 33427
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 756 . . . . . 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 33362 . . . . . . 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 991 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  K  e.  HL )
11 hlop 33315 . . . . . . 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 992 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  X  e.  B )
14 hllat 33316 . . . . . . . 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 993 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  P  e.  A )
172, 6atbase 33242 . . . . . . . 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 15325 . . . . . . 7  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  ( X  .\/  P
)  e.  B )
2015, 13, 18, 19syl3anc 1219 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  ( X  .\/  P )  e.  B )
21 eqid 2451 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
222, 21, 5cvrcon3b 33230 . . . . . 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 1219 . . . . 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 33313 . . . . . 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 33147 . . . . . 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 33147 . . . . . . 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 2451 . . . . . . . . 9  |-  ( 0.
`  K )  =  ( 0. `  K
)
32 1cvrjat.u . . . . . . . . 9  |-  .1.  =  ( 1. `  K )
3331, 32, 21opoc1 33155 . . . . . . . 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 755 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1. 
/\  -.  P  .<_  X ) )  ->  X C  .1.  )
362, 32op1cl 33138 . . . . . . . . . 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 33230 . . . . . . . . 9  |-  ( ( K  e.  OP  /\  X  e.  B  /\  .1.  e.  B )  -> 
( X C  .1.  <->  ( ( oc `  K
) `  .1.  ) C ( ( oc
`  K ) `  X ) ) )
3912, 13, 37, 38syl3anc 1219 . . . . . . . 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 4414 . . . . . 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 33239 . . . . . . 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 913 . . . . 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 33267 . . . . 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 1219 . . . 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 5795 . 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 33148 . . 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 33156 . . 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 2500 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 965    = wceq 1370    e. wcel 1758   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   Basecbs 14278   lecple 14349   occoc 14350   joincjn 15218   0.cp0 15311   1.cp1 15312   Latclat 15319   OPcops 33125    <o ccvr 33215   Atomscatm 33216   AtLatcal 33217   HLchlt 33303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-poset 15220  df-plt 15232  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-p0 15313  df-p1 15314  df-lat 15320  df-clat 15382  df-oposet 33129  df-ol 33131  df-oml 33132  df-covers 33219  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304
This theorem is referenced by:  1cvrat  33428  lhpjat1  33972
  Copyright terms: Public domain W3C validator