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Theorem clatglbcl2 18564
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 6921 . . . . 5 𝐵 ∈ V
32elpw2 5340 . . . 4 (𝑆 ∈ 𝒫 𝐵𝑆𝐵)
43biimpri 228 . . 3 (𝑆𝐵𝑆 ∈ 𝒫 𝐵)
54adantl 481 . 2 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → 𝑆 ∈ 𝒫 𝐵)
6 eqid 2735 . . . . 5 (lub‘𝐾) = (lub‘𝐾)
7 clatglbcl.g . . . . 5 𝐺 = (glb‘𝐾)
81, 6, 7isclat 18558 . . . 4 (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)))
9 simprr 773 . . . 4 ((𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)) → dom 𝐺 = 𝒫 𝐵)
108, 9sylbi 217 . . 3 (𝐾 ∈ CLat → dom 𝐺 = 𝒫 𝐵)
1110adantr 480 . 2 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → dom 𝐺 = 𝒫 𝐵)
125, 11eleqtrrd 2842 1 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → 𝑆 ∈ dom 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wss 3963  𝒫 cpw 4605  dom cdm 5689  cfv 6563  Basecbs 17245  Posetcpo 18365  lubclub 18367  glbcglb 18368  CLatccla 18556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  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 4913  df-br 5149  df-dm 5699  df-iota 6516  df-fv 6571  df-clat 18557
This theorem is referenced by:  isglbd  18567  clatglb  18574  clatglble  18575  glbconN  39359
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