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Theorem latcl2 17652
 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 5557 . . 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 17651 . . . . 5 (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom = (𝐵 × 𝐵) ∧ dom = (𝐵 × 𝐵))))
94, 8sylib 221 . . . 4 (𝜑 → (𝐾 ∈ Poset ∧ (dom = (𝐵 × 𝐵) ∧ dom = (𝐵 × 𝐵))))
109simprld 771 . . 3 (𝜑 → dom = (𝐵 × 𝐵))
113, 10eleqtrrd 2893 . 2 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
129simprrd 773 . . 3 (𝜑 → dom = (𝐵 × 𝐵))
133, 12eleqtrrd 2893 . 2 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
1411, 13jca 515 1 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∧ ⟨𝑋, 𝑌⟩ ∈ dom ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ⟨cop 4531   × cxp 5517  dom cdm 5519  ‘cfv 6324  Basecbs 16477  Posetcpo 17544  joincjn 17548  meetcmee 17549  Latclat 17649 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-xp 5525  df-dm 5529  df-iota 6283  df-fv 6332  df-lat 17650 This theorem is referenced by:  latlej1  17664  latlej2  17665  latjle12  17666  latmle1  17680  latmle2  17681  latlem12  17682
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