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Metamath Proof Explorer

Theorem posjidm

Description: Poset join is idempotent. latjidm could be shortened by this. (Contributed by Zhi Wang, 27-Sep-2024)

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

Proof

Step Hyp Ref Expression
1 posjidm.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 posjidm.j ⊢ ∨ = ( join ‘ 𝐾 )
3 eqid ⊢ ( lub ‘ 𝐾 ) = ( lub ‘ 𝐾 )
4 simpl ⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ) → 𝐾 ∈ Poset )
5 simpr ⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 )
6 3 2 4 5 5 joinval ⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑋 ) = ( ( lub ‘ 𝐾 ) ‘ { 𝑋 , 𝑋 } ) )
7 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
8 1 7 posref ⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ( le ‘ 𝐾 ) 𝑋 )
9 eqidd ⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ) → { 𝑋 , 𝑋 } = { 𝑋 , 𝑋 } )
10 4 1 5 5 7 8 9 3 lubpr ⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ) → ( ( lub ‘ 𝐾 ) ‘ { 𝑋 , 𝑋 } ) = 𝑋 )
11 6 10 eqtrd ⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑋 ) = 𝑋 )