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Theorem clatp0cl 30585
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 ∈ CLat → 0𝐵)

Proof of Theorem clatp0cl
StepHypRef Expression
1 clatp0cl.b . . 3 𝐵 = (Base‘𝑊)
2 eqid 2818 . . 3 (glb‘𝑊) = (glb‘𝑊)
3 clatp0cl.0 . . 3 0 = (0.‘𝑊)
41, 2, 3p0val 17639 . 2 (𝑊 ∈ CLat → 0 = ((glb‘𝑊)‘𝐵))
5 ssid 3986 . . 3 𝐵𝐵
61, 2clatglbcl 17712 . . 3 ((𝑊 ∈ CLat ∧ 𝐵𝐵) → ((glb‘𝑊)‘𝐵) ∈ 𝐵)
75, 6mpan2 687 . 2 (𝑊 ∈ CLat → ((glb‘𝑊)‘𝐵) ∈ 𝐵)
84, 7eqeltrd 2910 1 (𝑊 ∈ CLat → 0𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wss 3933  cfv 6348  Basecbs 16471  glbcglb 17541  0.cp0 17635  CLatccla 17705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-lub 17572  df-glb 17573  df-p0 17637  df-clat 17706
This theorem is referenced by: (None)
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