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Theorem clatp0cl 27321
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 2467 . . 3  |-  ( glb `  W )  =  ( glb `  W )
3 clatp0cl.0 . . 3  |-  .0.  =  ( 0. `  W )
41, 2, 3p0val 15524 . 2  |-  ( W  e.  CLat  ->  .0.  =  ( ( glb `  W
) `  B )
)
5 ssid 3523 . . 3  |-  B  C_  B
61, 2clatglbcl 15597 . . 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 2555 1  |-  ( W  e.  CLat  ->  .0.  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    C_ wss 3476   ` cfv 5586   Basecbs 14486   glbcglb 15426   0.cp0 15520   CLatccla 15590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-lub 15457  df-glb 15458  df-p0 15522  df-clat 15591
This theorem is referenced by: (None)
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