Theorem latcl2 18395

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	⊢ (𝜑 → 𝐾 ∈ Lat)
latcl2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
latcl2.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
latcl2	⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∨ ∧ ⟨𝑋, 𝑌⟩ ∈ dom ∧ ))

Proof of Theorem latcl2

Step	Hyp	Ref	Expression
1		latcl2.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
2		latcl2.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
3	1, 2	opelxpd 5677	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
4		latcl2.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat)
5		latcl2.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
6		latcl2.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
7		latcl2.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
8	5, 6, 7	islat 18392	. . . . 5 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵))))
9	4, 8	sylib 218	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵))))
10	9	simprld 771	. . 3 ⊢ (𝜑 → dom ∨ = (𝐵 × 𝐵))
11	3, 10	eleqtrrd 2831	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∨ )
12	9	simprrd 773	. . 3 ⊢ (𝜑 → dom ∧ = (𝐵 × 𝐵))
13	3, 12	eleqtrrd 2831	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∧ )
14	11, 13	jca 511	1 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∨ ∧ ⟨𝑋, 𝑌⟩ ∈ dom ∧ ))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ⟨cop 4595   × cxp 5636  dom cdm 5638  ‘cfv 6511  Basecbs 17179  Posetcpo 18268  joincjn 18272  meetcmee 18273  Latclat 18390

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-xp 5644  df-dm 5648  df-iota 6464  df-fv 6519  df-lat 18391

This theorem is referenced by:  latlej1  18407  latlej2  18408  latjle12  18409  latmle1  18423  latmle2  18424  latlem12  18425