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 32927
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 2733 . . 3 (glb‘𝑊) = (glb‘𝑊)
3 clatp0cl.0 . . 3 0 = (0.‘𝑊)
41, 2, 3p0val 18474 . 2 (𝑊 ∈ CLat → 0 = ((glb‘𝑊)‘𝐵))
5 ssid 4018 . . 3 𝐵𝐵
61, 2clatglbcl 18552 . . 3 ((𝑊 ∈ CLat ∧ 𝐵𝐵) → ((glb‘𝑊)‘𝐵) ∈ 𝐵)
75, 6mpan2 690 . 2 (𝑊 ∈ CLat → ((glb‘𝑊)‘𝐵) ∈ 𝐵)
84, 7eqeltrd 2837 1 (𝑊 ∈ CLat → 0𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1535  wcel 2104  wss 3963  cfv 6559  Basecbs 17235  glbcglb 18357  0.cp0 18470  CLatccla 18545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3479  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4916  df-iun 5001  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6511  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-riota 7382  df-lub 18393  df-glb 18394  df-p0 18472  df-clat 18546
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator