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 )