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Metamath Proof Explorer

Theorem latjcom

Description: The join of a lattice commutes. ( chjcom analog.) (Contributed by NM, 16-Sep-2011)

Ref Expression
Hypotheses latjcom.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
latjcom.j ⊢ ∨ = ( join ‘ 𝐾 )
Assertion latjcom ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑌 ) = ( 𝑌 ∨ 𝑋 ) )

Proof

Step Hyp Ref Expression
1 latjcom.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 latjcom.j ⊢ ∨ = ( join ‘ 𝐾 )
3 opelxpi ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ⟨ 𝑋 , 𝑌 ⟩ ∈ ( 𝐵 × 𝐵 ) )
4 3 3adant1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ⟨ 𝑋 , 𝑌 ⟩ ∈ ( 𝐵 × 𝐵 ) )
5 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
6 1 2 5 islat ⊢ ( 𝐾 ∈ Lat ↔ ( 𝐾 ∈ Poset ∧ ( dom ∨ = ( 𝐵 × 𝐵 ) ∧ dom ( meet ‘ 𝐾 ) = ( 𝐵 × 𝐵 ) ) ) )
7 simprl ⊢ ( ( 𝐾 ∈ Poset ∧ ( dom ∨ = ( 𝐵 × 𝐵 ) ∧ dom ( meet ‘ 𝐾 ) = ( 𝐵 × 𝐵 ) ) ) → dom ∨ = ( 𝐵 × 𝐵 ) )
8 6 7 sylbi ⊢ ( 𝐾 ∈ Lat → dom ∨ = ( 𝐵 × 𝐵 ) )
9 8 3ad2ant1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → dom ∨ = ( 𝐵 × 𝐵 ) )
10 4 9 eleqtrrd ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∨ )
11 opelxpi ⊢ ( ( 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ⟨ 𝑌 , 𝑋 ⟩ ∈ ( 𝐵 × 𝐵 ) )
12 11 ancoms ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ⟨ 𝑌 , 𝑋 ⟩ ∈ ( 𝐵 × 𝐵 ) )
13 12 3adant1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ⟨ 𝑌 , 𝑋 ⟩ ∈ ( 𝐵 × 𝐵 ) )
14 13 9 eleqtrrd ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ⟨ 𝑌 , 𝑋 ⟩ ∈ dom ∨ )
15 10 14 jca ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∨ ∧ ⟨ 𝑌 , 𝑋 ⟩ ∈ dom ∨ ) )
16 latpos ⊢ ( 𝐾 ∈ Lat → 𝐾 ∈ Poset )
17 1 2 joincom ⊢ ( ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∨ ∧ ⟨ 𝑌 , 𝑋 ⟩ ∈ dom ∨ ) ) → ( 𝑋 ∨ 𝑌 ) = ( 𝑌 ∨ 𝑋 ) )
18 16 17 syl3anl1 ⊢ ( ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∨ ∧ ⟨ 𝑌 , 𝑋 ⟩ ∈ dom ∨ ) ) → ( 𝑋 ∨ 𝑌 ) = ( 𝑌 ∨ 𝑋 ) )
19 15 18 mpdan ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑌 ) = ( 𝑌 ∨ 𝑋 ) )