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Theorem latcl2 18481
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 5724 . . 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 18478 . . . . 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 2844 . 2 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
129simprrd 774 . . 3 (𝜑 → dom = (𝐵 × 𝐵))
133, 12eleqtrrd 2844 . 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 2108  cop 4632   × cxp 5683  dom cdm 5685  cfv 6561  Basecbs 17247  Posetcpo 18353  joincjn 18357  meetcmee 18358  Latclat 18476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-dm 5695  df-iota 6514  df-fv 6569  df-lat 18477
This theorem is referenced by:  latlej1  18493  latlej2  18494  latjle12  18495  latmle1  18509  latmle2  18510  latlem12  18511
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