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Metamath Proof Explorer

Theorem latmmdiN

Description: Lattice meet distributes over itself. ( inindi analog.) (Contributed by NM, 8-Nov-2011) (New usage is discouraged.)

Ref Expression
Hypotheses olmass.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
olmass.m ⊢ ∧ = ( meet ‘ 𝐾 )
Assertion latmmdiN ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ∧ ( 𝑌 ∧ 𝑍 ) ) = ( ( 𝑋 ∧ 𝑌 ) ∧ ( 𝑋 ∧ 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 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝐾 ∈ Lat )
5 simpr1 ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
6 1 2 latmidm ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑋 ) = 𝑋 )
7 4 5 6 syl2anc ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ∧ 𝑋 ) = 𝑋 )
8 7 oveq1d ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ∧ 𝑋 ) ∧ ( 𝑌 ∧ 𝑍 ) ) = ( 𝑋 ∧ ( 𝑌 ∧ 𝑍 ) ) )
9 simpl ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝐾 ∈ OL )
10 simpr2 ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
11 simpr3 ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑍 ∈ 𝐵 )
12 1 2 latm4 ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ∧ 𝑋 ) ∧ ( 𝑌 ∧ 𝑍 ) ) = ( ( 𝑋 ∧ 𝑌 ) ∧ ( 𝑋 ∧ 𝑍 ) ) )
13 9 5 5 10 11 12 syl122anc ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ∧ 𝑋 ) ∧ ( 𝑌 ∧ 𝑍 ) ) = ( ( 𝑋 ∧ 𝑌 ) ∧ ( 𝑋 ∧ 𝑍 ) ) )
14 8 13 eqtr3d ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ∧ ( 𝑌 ∧ 𝑍 ) ) = ( ( 𝑋 ∧ 𝑌 ) ∧ ( 𝑋 ∧ 𝑍 ) ) )