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Metamath Proof Explorer

Theorem latjrot

Description: Rotate lattice join of 3 classes. (Contributed by NM, 23-Jul-2012)

Ref Expression
Hypotheses latjass.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
latjass.j ⊢ ∨ = ( join ‘ 𝐾 )
Assertion latjrot ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ∨ 𝑌 ) ∨ 𝑍 ) = ( ( 𝑍 ∨ 𝑋 ) ∨ 𝑌 ) )

Proof

Step Hyp Ref Expression
1 latjass.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 latjass.j ⊢ ∨ = ( join ‘ 𝐾 )
3 1 2 latj31 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ∨ 𝑌 ) ∨ 𝑍 ) = ( ( 𝑍 ∨ 𝑌 ) ∨ 𝑋 ) )
4 simpl ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝐾 ∈ Lat )
5 simpr3 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑍 ∈ 𝐵 )
6 simpr2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
7 simpr1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
8 1 2 latj32 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ) → ( ( 𝑍 ∨ 𝑌 ) ∨ 𝑋 ) = ( ( 𝑍 ∨ 𝑋 ) ∨ 𝑌 ) )
9 4 5 6 7 8 syl13anc ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑍 ∨ 𝑌 ) ∨ 𝑋 ) = ( ( 𝑍 ∨ 𝑋 ) ∨ 𝑌 ) )
10 3 9 eqtrd ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ∨ 𝑌 ) ∨ 𝑍 ) = ( ( 𝑍 ∨ 𝑋 ) ∨ 𝑌 ) )