Theorem latcl2 18399

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 5708	. . 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 18396	. . . . 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 769	. . 3 ⊢ (𝜑 → dom ∨ = (𝐵 × 𝐵))
11	3, 10	eleqtrrd 2830	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∨ )
12	9	simprrd 771	. . 3 ⊢ (𝜑 → dom ∧ = (𝐵 × 𝐵))
13	3, 12	eleqtrrd 2830	. 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 1533   ∈ wcel 2098  ⟨cop 4629   × cxp 5667  dom cdm 5669  ‘cfv 6536  Basecbs 17151  Posetcpo 18270  joincjn 18274  meetcmee 18275  Latclat 18394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-dm 5679  df-iota 6488  df-fv 6544  df-lat 18395

This theorem is referenced by:  latlej1  18411  latlej2  18412  latjle12  18413  latmle1  18427  latmle2  18428  latlem12  18429