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Theorem clatglbcl2 17475
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 6451 . . . . 5 𝐵 ∈ V
32elpw2 5052 . . . 4 (𝑆 ∈ 𝒫 𝐵𝑆𝐵)
43biimpri 220 . . 3 (𝑆𝐵𝑆 ∈ 𝒫 𝐵)
54adantl 475 . 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 17469 . . . 4 (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)))
9 simprr 789 . . . 4 ((𝐾 ∈ Poset ∧ (dom (lub‘𝐾) = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)) → dom 𝐺 = 𝒫 𝐵)
108, 9sylbi 209 . . 3 (𝐾 ∈ CLat → dom 𝐺 = 𝒫 𝐵)
1110adantr 474 . 2 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → dom 𝐺 = 𝒫 𝐵)
125, 11eleqtrrd 2909 1 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → 𝑆 ∈ dom 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  wss 3798  𝒫 cpw 4380  dom cdm 5346  cfv 6127  Basecbs 16229  Posetcpo 17300  lubclub 17302  glbcglb 17303  CLatccla 17467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-dm 5356  df-iota 6090  df-fv 6135  df-clat 17468
This theorem is referenced by:  isglbd  17477  clatglb  17484  clatglble  17485
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