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Theorem latcl2 18399
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 5715 . . 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 18396 . . . . 5 (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom = (𝐵 × 𝐵) ∧ dom = (𝐵 × 𝐵))))
94, 8sylib 217 . . . 4 (𝜑 → (𝐾 ∈ Poset ∧ (dom = (𝐵 × 𝐵) ∧ dom = (𝐵 × 𝐵))))
109simprld 769 . . 3 (𝜑 → dom = (𝐵 × 𝐵))
113, 10eleqtrrd 2835 . 2 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
129simprrd 771 . . 3 (𝜑 → dom = (𝐵 × 𝐵))
133, 12eleqtrrd 2835 . 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 1540  wcel 2105  cop 4634   × cxp 5674  dom cdm 5676  cfv 6543  Basecbs 17151  Posetcpo 18270  joincjn 18274  meetcmee 18275  Latclat 18394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-dm 5686  df-iota 6495  df-fv 6551  df-lat 18395
This theorem is referenced by:  latlej1  18411  latlej2  18412  latjle12  18413  latmle1  18427  latmle2  18428  latlem12  18429
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