Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  atlle0 Structured version   Unicode version

Theorem atlle0 30201
Description: An element less than or equal to zero equals zero. (chle0 22976 analog.) (Contributed by NM, 21-Oct-2011.)
Hypotheses
Ref Expression
atl0le.b  |-  B  =  ( Base `  K
)
atl0le.l  |-  .<_  =  ( le `  K )
atl0le.z  |-  .0.  =  ( 0. `  K )
Assertion
Ref Expression
atlle0  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  ( X  .<_  .0.  <->  X  =  .0.  ) )

Proof of Theorem atlle0
StepHypRef Expression
1 atl0le.b . . . 4  |-  B  =  ( Base `  K
)
2 atl0le.l . . . 4  |-  .<_  =  ( le `  K )
3 atl0le.z . . . 4  |-  .0.  =  ( 0. `  K )
41, 2, 3atl0le 30200 . . 3  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  .0.  .<_  X )
54biantrud 495 . 2  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  ( X  .<_  .0.  <->  ( X  .<_  .0. 
/\  .0.  .<_  X ) ) )
6 atlpos 30197 . . . 4  |-  ( K  e.  AtLat  ->  K  e.  Poset
)
76adantr 453 . . 3  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  K  e.  Poset )
8 simpr 449 . . 3  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  X  e.  B )
91, 3atl0cl 30199 . . . 4  |-  ( K  e.  AtLat  ->  .0.  e.  B )
109adantr 453 . . 3  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  .0.  e.  B )
111, 2posasymb 14440 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .0.  e.  B )  ->  (
( X  .<_  .0.  /\  .0.  .<_  X )  <->  X  =  .0.  ) )
127, 8, 10, 11syl3anc 1185 . 2  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  (
( X  .<_  .0.  /\  .0.  .<_  X )  <->  X  =  .0.  ) )
135, 12bitrd 246 1  |-  ( ( K  e.  AtLat  /\  X  e.  B )  ->  ( X  .<_  .0.  <->  X  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727   class class class wbr 4237   ` cfv 5483   Basecbs 13500   lecple 13567   Posetcpo 14428   0.cp0 14497   AtLatcal 30160
This theorem is referenced by:  dia0  31948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-undef 6572  df-riota 6578  df-poset 14434  df-glb 14463  df-p0 14499  df-lat 14506  df-atl 30194
  Copyright terms: Public domain W3C validator