Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  atnle0 Structured version   Unicode version

Theorem atnle0 30107
Description: An atom is not less than or equal to zero. (Contributed by NM, 17-Oct-2011.)
Hypotheses
Ref Expression
atnle0.l  |-  .<_  =  ( le `  K )
atnle0.z  |-  .0.  =  ( 0. `  K )
atnle0.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atnle0  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  -.  P  .<_  .0.  )

Proof of Theorem atnle0
StepHypRef Expression
1 atlpos 30099 . . 3  |-  ( K  e.  AtLat  ->  K  e.  Poset
)
21adantr 452 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  K  e.  Poset )
3 eqid 2436 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
4 atnle0.z . . . 4  |-  .0.  =  ( 0. `  K )
53, 4atl0cl 30101 . . 3  |-  ( K  e.  AtLat  ->  .0.  e.  ( Base `  K )
)
65adantr 452 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  .0.  e.  ( Base `  K
) )
7 atnle0.a . . . 4  |-  A  =  ( Atoms `  K )
83, 7atbase 30087 . . 3  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
98adantl 453 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  P  e.  ( Base `  K
) )
10 eqid 2436 . . 3  |-  (  <o  `  K )  =  ( 
<o  `  K )
114, 10, 7atcvr0 30086 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  .0.  (  <o  `  K ) P )
12 atnle0.l . . 3  |-  .<_  =  ( le `  K )
133, 12, 10cvrnle 30078 . 2  |-  ( ( ( K  e.  Poset  /\  .0.  e.  ( Base `  K )  /\  P  e.  ( Base `  K
) )  /\  .0.  (  <o  `  K ) P )  ->  -.  P  .<_  .0.  )
142, 6, 9, 11, 13syl31anc 1187 1  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  -.  P  .<_  .0.  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454   Basecbs 13469   lecple 13536   Posetcpo 14397   0.cp0 14466    <o ccvr 30060   Atomscatm 30061   AtLatcal 30062
This theorem is referenced by:  pmap0  30562  trlnle  30983  cdlemg27b  31493
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-poset 14403  df-plt 14415  df-lat 14475  df-covers 30064  df-ats 30065  df-atl 30096
  Copyright terms: Public domain W3C validator