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Theorem clatglbcl2 17717
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 6680 . . . . 5 𝐵 ∈ V
32elpw2 5244 . . . 4 (𝑆 ∈ 𝒫 𝐵𝑆𝐵)
43biimpri 229 . . 3 (𝑆𝐵𝑆 ∈ 𝒫 𝐵)
54adantl 482 . 2 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → 𝑆 ∈ 𝒫 𝐵)
6 eqid 2825 . . . . 5 (lub‘𝐾) = (lub‘𝐾)
7 clatglbcl.g . . . . 5 𝐺 = (glb‘𝐾)
81, 6, 7isclat 17711 . . . 4 (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)))
9 simprr 769 . . . 4 ((𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)) → dom 𝐺 = 𝒫 𝐵)
108, 9sylbi 218 . . 3 (𝐾 ∈ CLat → dom 𝐺 = 𝒫 𝐵)
1110adantr 481 . 2 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → dom 𝐺 = 𝒫 𝐵)
125, 11eleqtrrd 2920 1 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → 𝑆 ∈ dom 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wss 3939  𝒫 cpw 4541  dom cdm 5553  cfv 6351  Basecbs 16475  Posetcpo 17542  lubclub 17544  glbcglb 17545  CLatccla 17709
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-dm 5563  df-iota 6311  df-fv 6359  df-clat 17710
This theorem is referenced by:  isglbd  17719  clatglb  17726  clatglble  17727
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