Theorem latjidm

Description: Lattice join is idempotent. (Contributed by NM, 8-Oct-2011)

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

Proof

Step	Hyp	Ref	Expression
1		latidm.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		latidm.j	⊢ ∨ = ( join ‘ 𝐾 )
3		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
4		simpl	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ) → 𝐾 ∈ Lat )
5	1 2	latjcl	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑋 ) ∈ 𝐵 )
6	5	3anidm23	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑋 ) ∈ 𝐵 )
7		simpr	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 )
8	1 3	latref	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ( le ‘ 𝐾 ) 𝑋 )
9	1 3 2	latjle12	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ) → ( ( 𝑋 ( le ‘ 𝐾 ) 𝑋 ∧ 𝑋 ( le ‘ 𝐾 ) 𝑋 ) ↔ ( 𝑋 ∨ 𝑋 ) ( le ‘ 𝐾 ) 𝑋 ) )
10	4 7 7 7 9	syl13anc	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑋 ( le ‘ 𝐾 ) 𝑋 ∧ 𝑋 ( le ‘ 𝐾 ) 𝑋 ) ↔ ( 𝑋 ∨ 𝑋 ) ( le ‘ 𝐾 ) 𝑋 ) )
11	8 8 10	mpbi2and	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑋 ) ( le ‘ 𝐾 ) 𝑋 )
12	1 3 2	latlej1	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ( le ‘ 𝐾 ) ( 𝑋 ∨ 𝑋 ) )
13	12	3anidm23	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ( le ‘ 𝐾 ) ( 𝑋 ∨ 𝑋 ) )
14	1 3 4 6 7 11 13	latasymd	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑋 ) = 𝑋 )