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Theorem latcl2 18494
Description: The join and meet of any two elements exist. (Contributed by NM, 14-Sep-2018.)
Hypotheses
Ref Expression
latcl2.b 𝐵 = (Base‘𝐾)
latcl2.j = (join‘𝐾)
latcl2.m = (meet‘𝐾)
latcl2.k (𝜑𝐾 ∈ Lat)
latcl2.x (𝜑𝑋𝐵)
latcl2.y (𝜑𝑌𝐵)
Assertion
Ref Expression
latcl2 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∧ ⟨𝑋, 𝑌⟩ ∈ dom ))

Proof of Theorem latcl2
StepHypRef Expression
1 latcl2.x . . . 4 (𝜑𝑋𝐵)
2 latcl2.y . . . 4 (𝜑𝑌𝐵)
31, 2opelxpd 5728 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
4 latcl2.k . . . . 5 (𝜑𝐾 ∈ Lat)
5 latcl2.b . . . . . 6 𝐵 = (Base‘𝐾)
6 latcl2.j . . . . . 6 = (join‘𝐾)
7 latcl2.m . . . . . 6 = (meet‘𝐾)
85, 6, 7islat 18491 . . . . 5 (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom = (𝐵 × 𝐵) ∧ dom = (𝐵 × 𝐵))))
94, 8sylib 218 . . . 4 (𝜑 → (𝐾 ∈ Poset ∧ (dom = (𝐵 × 𝐵) ∧ dom = (𝐵 × 𝐵))))
109simprld 772 . . 3 (𝜑 → dom = (𝐵 × 𝐵))
113, 10eleqtrrd 2842 . 2 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
129simprrd 774 . . 3 (𝜑 → dom = (𝐵 × 𝐵))
133, 12eleqtrrd 2842 . 2 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
1411, 13jca 511 1 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∧ ⟨𝑋, 𝑌⟩ ∈ dom ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cop 4637   × cxp 5687  dom cdm 5689  cfv 6563  Basecbs 17245  Posetcpo 18365  joincjn 18369  meetcmee 18370  Latclat 18489
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  ax-pr 5438
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-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-dm 5699  df-iota 6516  df-fv 6571  df-lat 18490
This theorem is referenced by:  latlej1  18506  latlej2  18507  latjle12  18508  latmle1  18522  latmle2  18523  latlem12  18524
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