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Theorem latcl2 18069
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 5618 . . 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 18066 . . . . 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 768 . . 3 (𝜑 → dom = (𝐵 × 𝐵))
113, 10eleqtrrd 2842 . 2 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
129simprrd 770 . . 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 1539  wcel 2108  cop 4564   × cxp 5578  dom cdm 5580  cfv 6418  Basecbs 16840  Posetcpo 17940  joincjn 17944  meetcmee 17945  Latclat 18064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-dm 5590  df-iota 6376  df-fv 6426  df-lat 18065
This theorem is referenced by:  latlej1  18081  latlej2  18082  latjle12  18083  latmle1  18097  latmle2  18098  latlem12  18099
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