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Metamath Proof Explorer

Theorem latmrot

Description: Rotate lattice meet of 3 classes. (Contributed by NM, 9-Oct-2012)

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

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 simpr2 ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
7 1 2 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 )
8 4 5 6 7 syl3anc ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 )
9 simpr3 ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑍 ∈ 𝐵 )
10 1 2 latmcom ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( ( 𝑋 ∧ 𝑌 ) ∧ 𝑍 ) = ( 𝑍 ∧ ( 𝑋 ∧ 𝑌 ) ) )
11 4 8 9 10 syl3anc ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ∧ 𝑌 ) ∧ 𝑍 ) = ( 𝑍 ∧ ( 𝑋 ∧ 𝑌 ) ) )
12 simpl ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝐾 ∈ OL )
13 1 2 latmassOLD ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑍 ∧ 𝑋 ) ∧ 𝑌 ) = ( 𝑍 ∧ ( 𝑋 ∧ 𝑌 ) ) )
14 12 9 5 6 13 syl13anc ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑍 ∧ 𝑋 ) ∧ 𝑌 ) = ( 𝑍 ∧ ( 𝑋 ∧ 𝑌 ) ) )
15 11 14 eqtr4d ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ∧ 𝑌 ) ∧ 𝑍 ) = ( ( 𝑍 ∧ 𝑋 ) ∧ 𝑌 ) )