Theorem latmlem2

Description: Add meet to both sides of a lattice ordering. ( sslin analog.) (Contributed by NM, 10-Nov-2011)

	Ref	Expression
Hypotheses	latmle.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	latmle.l	⊢ ≤ = ( le ‘ 𝐾 )
	latmle.m	⊢ ∧ = ( meet ‘ 𝐾 )
Assertion	latmlem2	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ≤ 𝑌 → ( 𝑍 ∧ 𝑋 ) ≤ ( 𝑍 ∧ 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		latmle.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		latmle.l	⊢ ≤ = ( le ‘ 𝐾 )
3		latmle.m	⊢ ∧ = ( meet ‘ 𝐾 )
4	1 2 3	latmlem1	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ≤ 𝑌 → ( 𝑋 ∧ 𝑍 ) ≤ ( 𝑌 ∧ 𝑍 ) ) )
5	1 3	latmcom	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑍 ) = ( 𝑍 ∧ 𝑋 ) )
6	5	3adant3r2	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ∧ 𝑍 ) = ( 𝑍 ∧ 𝑋 ) )
7	1 3	latmcom	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑌 ∧ 𝑍 ) = ( 𝑍 ∧ 𝑌 ) )
8	7	3adant3r1	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑌 ∧ 𝑍 ) = ( 𝑍 ∧ 𝑌 ) )
9	6 8	breq12d	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ∧ 𝑍 ) ≤ ( 𝑌 ∧ 𝑍 ) ↔ ( 𝑍 ∧ 𝑋 ) ≤ ( 𝑍 ∧ 𝑌 ) ) )
10	4 9	sylibd	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ≤ 𝑌 → ( 𝑍 ∧ 𝑋 ) ≤ ( 𝑍 ∧ 𝑌 ) ) )