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Theorem latcl2 16871
 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)
3 opelxpi 5072 . . . 4 ((𝑋𝐵𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
41, 2, 3syl2anc 691 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
5 latcl2.k . . . . 5 (𝜑𝐾 ∈ Lat)
6 latcl2.b . . . . . 6 𝐵 = (Base‘𝐾)
7 latcl2.j . . . . . 6 = (join‘𝐾)
8 latcl2.m . . . . . 6 = (meet‘𝐾)
96, 7, 8islat 16870 . . . . 5 (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom = (𝐵 × 𝐵) ∧ dom = (𝐵 × 𝐵))))
105, 9sylib 207 . . . 4 (𝜑 → (𝐾 ∈ Poset ∧ (dom = (𝐵 × 𝐵) ∧ dom = (𝐵 × 𝐵))))
11 simprl 790 . . . 4 ((𝐾 ∈ Poset ∧ (dom = (𝐵 × 𝐵) ∧ dom = (𝐵 × 𝐵))) → dom = (𝐵 × 𝐵))
1210, 11syl 17 . . 3 (𝜑 → dom = (𝐵 × 𝐵))
134, 12eleqtrrd 2691 . 2 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
1410simprrd 793 . . 3 (𝜑 → dom = (𝐵 × 𝐵))
154, 14eleqtrrd 2691 . 2 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
1613, 15jca 553 1 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∧ ⟨𝑋, 𝑌⟩ ∈ dom ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   × cxp 5036  dom cdm 5038  ‘cfv 5804  Basecbs 15695  Posetcpo 16763  joincjn 16767  meetcmee 16768  Latclat 16868 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-dm 5048  df-iota 5768  df-fv 5812  df-lat 16869 This theorem is referenced by:  latlej1  16883  latlej2  16884  latjle12  16885  latmle1  16899  latmle2  16900  latlem12  16901
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