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Theorem clatglbcl2 17047
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)
2 fvex 6163 . . . . . 6 (Base‘𝐾) ∈ V
31, 2eqeltri 2694 . . . . 5 𝐵 ∈ V
43elpw2 4793 . . . 4 (𝑆 ∈ 𝒫 𝐵𝑆𝐵)
54biimpri 218 . . 3 (𝑆𝐵𝑆 ∈ 𝒫 𝐵)
65adantl 482 . 2 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → 𝑆 ∈ 𝒫 𝐵)
7 eqid 2621 . . . . 5 (lub‘𝐾) = (lub‘𝐾)
8 clatglbcl.g . . . . 5 𝐺 = (glb‘𝐾)
91, 7, 8isclat 17041 . . . 4 (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)))
10 simprr 795 . . . 4 ((𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)) → dom 𝐺 = 𝒫 𝐵)
119, 10sylbi 207 . . 3 (𝐾 ∈ CLat → dom 𝐺 = 𝒫 𝐵)
1211adantr 481 . 2 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → dom 𝐺 = 𝒫 𝐵)
136, 12eleqtrrd 2701 1 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → 𝑆 ∈ dom 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3189  wss 3559  𝒫 cpw 4135  dom cdm 5079  cfv 5852  Basecbs 15792  Posetcpo 16872  lubclub 16874  glbcglb 16875  CLatccla 17039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-dm 5089  df-iota 5815  df-fv 5860  df-clat 17040
This theorem is referenced by:  isglbd  17049  clatglb  17056  clatglble  17057
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