Theorem latmidm

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

	Ref	Expression
Hypotheses	latmidm.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	latmidm.m	⊢ ∧ = ( meet ‘ 𝐾 )
Assertion	latmidm	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑋 ) = 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		latmidm.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		latmidm.m	⊢ ∧ = ( meet ‘ 𝐾 )
3		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
4		simpl	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ) → 𝐾 ∈ Lat )
5	1 2	latmcl	⊢ ( ( 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 2	latmle1	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑋 ) ( le ‘ 𝐾 ) 𝑋 )
9	8	3anidm23	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑋 ) ( le ‘ 𝐾 ) 𝑋 )
10	1 3	latref	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ( le ‘ 𝐾 ) 𝑋 )
11	1 3 2	latlem12	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ) → ( ( 𝑋 ( le ‘ 𝐾 ) 𝑋 ∧ 𝑋 ( le ‘ 𝐾 ) 𝑋 ) ↔ 𝑋 ( le ‘ 𝐾 ) ( 𝑋 ∧ 𝑋 ) ) )
12	4 7 7 7 11	syl13anc	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑋 ( le ‘ 𝐾 ) 𝑋 ∧ 𝑋 ( le ‘ 𝐾 ) 𝑋 ) ↔ 𝑋 ( le ‘ 𝐾 ) ( 𝑋 ∧ 𝑋 ) ) )
13	10 10 12	mpbi2and	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ( le ‘ 𝐾 ) ( 𝑋 ∧ 𝑋 ) )
14	1 3 4 6 7 9 13	latasymd	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑋 ) = 𝑋 )