Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  clatp0cl Structured version   Visualization version   GIF version

Theorem clatp0cl 32966
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 2737 . . 3 (glb‘𝑊) = (glb‘𝑊)
3 clatp0cl.0 . . 3 0 = (0.‘𝑊)
41, 2, 3p0val 18472 . 2 (𝑊 ∈ CLat → 0 = ((glb‘𝑊)‘𝐵))
5 ssid 4006 . . 3 𝐵𝐵
61, 2clatglbcl 18550 . . 3 ((𝑊 ∈ CLat ∧ 𝐵𝐵) → ((glb‘𝑊)‘𝐵) ∈ 𝐵)
75, 6mpan2 691 . 2 (𝑊 ∈ CLat → ((glb‘𝑊)‘𝐵) ∈ 𝐵)
84, 7eqeltrd 2841 1 (𝑊 ∈ CLat → 0𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wss 3951  cfv 6561  Basecbs 17247  glbcglb 18356  0.cp0 18468  CLatccla 18543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-lub 18391  df-glb 18392  df-p0 18470  df-clat 18544
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator