Theorem latcl2 17657

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 5592	. . 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 17656	. . . . 5 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵))))
9	4, 8	sylib 220	. . . 4 ⊢ (𝜑 → (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵))))
10	9	simprld 770	. . 3 ⊢ (𝜑 → dom ∨ = (𝐵 × 𝐵))
11	3, 10	eleqtrrd 2916	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∨ )
12	9	simprrd 772	. . 3 ⊢ (𝜑 → dom ∧ = (𝐵 × 𝐵))
13	3, 12	eleqtrrd 2916	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∧ )
14	11, 13	jca 514	1 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∨ ∧ ⟨𝑋, 𝑌⟩ ∈ dom ∧ ))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ⟨cop 4572   × cxp 5552  dom cdm 5554  ‘cfv 6354  Basecbs 16482  Posetcpo 17549  joincjn 17553  meetcmee 17554  Latclat 17654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-xp 5560  df-dm 5564  df-iota 6313  df-fv 6362  df-lat 17655

This theorem is referenced by:  latlej1  17669  latlej2  17670  latjle12  17671  latmle1  17685  latmle2  17686  latlem12  17687