Theorem latcl2 18622

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 5693	. . 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 18619	. . . . 5 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵))))
9	4, 8	sylib 221	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵))))
10	9	simprld 784	. . 3 ⊢ (𝜑 → dom ∨ = (𝐵 × 𝐵))
11	3, 10	eleqtrrd 2866	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∨ )
12	9	simprrd 786	. . 3 ⊢ (𝜑 → dom ∧ = (𝐵 × 𝐵))
13	3, 12	eleqtrrd 2866	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∧ )
14	11, 13	jca 521	1 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∨ ∧ ⟨𝑋, 𝑌⟩ ∈ dom ∧ ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146  ⟨cop 4837   × cxp 5652  dom cdm 5654  ‘cfv 6546  Basecbs 17399  Posetcpo 18493  joincjn 18497  meetcmee 18498  Latclat 18617

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-xp 5660  df-dm 5664  df-iota 6500  df-fv 6554  df-lat 18618

This theorem is used by:  latlej1  18634  latlej2  18635  latjle12  18636  latmle1  18650  latmle2  18651  latlem12  18652