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Theorem clatp0cl 26276
Description: The poset zero of a complete lattice belongs to its base. (Contributed by Thierry Arnoux, 17-Feb-2018.)
Hypotheses
Ref Expression
clatp0cl.b  |-  B  =  ( Base `  W
)
clatp0cl.0  |-  .0.  =  ( 0. `  W )
Assertion
Ref Expression
clatp0cl  |-  ( W  e.  CLat  ->  .0.  e.  B )

Proof of Theorem clatp0cl
StepHypRef Expression
1 clatp0cl.b . . 3  |-  B  =  ( Base `  W
)
2 eqid 2454 . . 3  |-  ( glb `  W )  =  ( glb `  W )
3 clatp0cl.0 . . 3  |-  .0.  =  ( 0. `  W )
41, 2, 3p0val 15329 . 2  |-  ( W  e.  CLat  ->  .0.  =  ( ( glb `  W
) `  B )
)
5 ssid 3482 . . 3  |-  B  C_  B
61, 2clatglbcl 15402 . . 3  |-  ( ( W  e.  CLat  /\  B  C_  B )  ->  (
( glb `  W
) `  B )  e.  B )
75, 6mpan2 671 . 2  |-  ( W  e.  CLat  ->  ( ( glb `  W ) `
 B )  e.  B )
84, 7eqeltrd 2542 1  |-  ( W  e.  CLat  ->  .0.  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    C_ wss 3435   ` cfv 5525   Basecbs 14291   glbcglb 15231   0.cp0 15325   CLatccla 15395
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 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-lub 15262  df-glb 15263  df-p0 15327  df-clat 15396
This theorem is referenced by: (None)
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