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Metamath Proof Explorer

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 ) ) )