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Theorem clatglbcl2 18459
Description: Any subset of the base set has a GLB in a complete lattice. (Contributed by NM, 13-Sep-2018.)
Hypotheses
Ref Expression
clatglbcl.b 𝐵 = (Base‘𝐾)
clatglbcl.g 𝐺 = (glb‘𝐾)
Assertion
Ref Expression
clatglbcl2 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → 𝑆 ∈ dom 𝐺)

Proof of Theorem clatglbcl2
StepHypRef Expression
1 clatglbcl.b . . . . . 6 𝐵 = (Base‘𝐾)
21fvexi 6906 . . . . 5 𝐵 ∈ V
32elpw2 5346 . . . 4 (𝑆 ∈ 𝒫 𝐵𝑆𝐵)
43biimpri 227 . . 3 (𝑆𝐵𝑆 ∈ 𝒫 𝐵)
54adantl 483 . 2 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → 𝑆 ∈ 𝒫 𝐵)
6 eqid 2733 . . . . 5 (lub‘𝐾) = (lub‘𝐾)
7 clatglbcl.g . . . . 5 𝐺 = (glb‘𝐾)
81, 6, 7isclat 18453 . . . 4 (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)))
9 simprr 772 . . . 4 ((𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)) → dom 𝐺 = 𝒫 𝐵)
108, 9sylbi 216 . . 3 (𝐾 ∈ CLat → dom 𝐺 = 𝒫 𝐵)
1110adantr 482 . 2 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → dom 𝐺 = 𝒫 𝐵)
125, 11eleqtrrd 2837 1 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → 𝑆 ∈ dom 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wss 3949  𝒫 cpw 4603  dom cdm 5677  cfv 6544  Basecbs 17144  Posetcpo 18260  lubclub 18262  glbcglb 18263  CLatccla 18451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-dm 5687  df-iota 6496  df-fv 6552  df-clat 18452
This theorem is referenced by:  isglbd  18462  clatglb  18469  clatglble  18470  glbconN  38247
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