Metamath Proof Explorer


Theorem latmmdir

Description: Lattice meet distributes over itself. ( inindir analog.) (Contributed by NM, 6-Jun-2012)

Ref Expression
Hypotheses olmass.b 𝐵 = ( Base ‘ 𝐾 )
olmass.m = ( meet ‘ 𝐾 )
Assertion latmmdir ( ( 𝐾 ∈ OL ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( ( 𝑋 𝑌 ) 𝑍 ) = ( ( 𝑋 𝑍 ) ( 𝑌 𝑍 ) ) )

Proof

Step Hyp Ref Expression
1 olmass.b 𝐵 = ( Base ‘ 𝐾 )
2 olmass.m = ( meet ‘ 𝐾 )
3 ollat ( 𝐾 ∈ OL → 𝐾 ∈ Lat )
4 3 adantr ( ( 𝐾 ∈ OL ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → 𝐾 ∈ Lat )
5 simpr3 ( ( 𝐾 ∈ OL ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → 𝑍𝐵 )
6 1 2 latmidm ( ( 𝐾 ∈ Lat ∧ 𝑍𝐵 ) → ( 𝑍 𝑍 ) = 𝑍 )
7 4 5 6 syl2anc ( ( 𝐾 ∈ OL ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( 𝑍 𝑍 ) = 𝑍 )
8 7 oveq2d ( ( 𝐾 ∈ OL ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( ( 𝑋 𝑌 ) ( 𝑍 𝑍 ) ) = ( ( 𝑋 𝑌 ) 𝑍 ) )
9 simpl ( ( 𝐾 ∈ OL ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → 𝐾 ∈ OL )
10 simpr1 ( ( 𝐾 ∈ OL ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → 𝑋𝐵 )
11 simpr2 ( ( 𝐾 ∈ OL ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → 𝑌𝐵 )
12 1 2 latm4 ( ( 𝐾 ∈ OL ∧ ( 𝑋𝐵𝑌𝐵 ) ∧ ( 𝑍𝐵𝑍𝐵 ) ) → ( ( 𝑋 𝑌 ) ( 𝑍 𝑍 ) ) = ( ( 𝑋 𝑍 ) ( 𝑌 𝑍 ) ) )
13 9 10 11 5 5 12 syl122anc ( ( 𝐾 ∈ OL ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( ( 𝑋 𝑌 ) ( 𝑍 𝑍 ) ) = ( ( 𝑋 𝑍 ) ( 𝑌 𝑍 ) ) )
14 8 13 eqtr3d ( ( 𝐾 ∈ OL ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( ( 𝑋 𝑌 ) 𝑍 ) = ( ( 𝑋 𝑍 ) ( 𝑌 𝑍 ) ) )