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Theorem clatp0cl 32724
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 2728 . . 3 (glb‘𝑊) = (glb‘𝑊)
3 clatp0cl.0 . . 3 0 = (0.‘𝑊)
41, 2, 3p0val 18426 . 2 (𝑊 ∈ CLat → 0 = ((glb‘𝑊)‘𝐵))
5 ssid 4004 . . 3 𝐵𝐵
61, 2clatglbcl 18504 . . 3 ((𝑊 ∈ CLat ∧ 𝐵𝐵) → ((glb‘𝑊)‘𝐵) ∈ 𝐵)
75, 6mpan2 689 . 2 (𝑊 ∈ CLat → ((glb‘𝑊)‘𝐵) ∈ 𝐵)
84, 7eqeltrd 2829 1 (𝑊 ∈ CLat → 0𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wss 3949  cfv 6553  Basecbs 17187  glbcglb 18309  0.cp0 18422  CLatccla 18497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-lub 18345  df-glb 18346  df-p0 18424  df-clat 18498
This theorem is referenced by: (None)
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