Theorem latcl2 18388

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 5715	. . 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 18385	. . . . 5 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵))))
9	4, 8	sylib 217	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵))))
10	9	simprld 770	. . 3 ⊢ (𝜑 → dom ∨ = (𝐵 × 𝐵))
11	3, 10	eleqtrrd 2836	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∨ )
12	9	simprrd 772	. . 3 ⊢ (𝜑 → dom ∧ = (𝐵 × 𝐵))
13	3, 12	eleqtrrd 2836	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∧ )
14	11, 13	jca 512	1 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∨ ∧ ⟨𝑋, 𝑌⟩ ∈ dom ∧ ))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ⟨cop 4634   × cxp 5674  dom cdm 5676  ‘cfv 6543  Basecbs 17143  Posetcpo 18259  joincjn 18263  meetcmee 18264  Latclat 18383

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-dm 5686  df-iota 6495  df-fv 6551  df-lat 18384

This theorem is referenced by:  latlej1  18400  latlej2  18401  latjle12  18402  latmle1  18416  latmle2  18417  latlem12  18418