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Metamath Proof Explorer

Theorem latjjdi

Description: Lattice join distributes over itself. (Contributed by NM, 30-Jul-2012)

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

Proof

Step Hyp Ref Expression
1 latjass.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 latjass.j ⊢ ∨ = ( join ‘ 𝐾 )
3 simpr1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
4 1 2 latjidm ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑋 ) = 𝑋 )
5 3 4 syldan ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ∨ 𝑋 ) = 𝑋 )
6 5 oveq1d ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ∨ 𝑋 ) ∨ ( 𝑌 ∨ 𝑍 ) ) = ( 𝑋 ∨ ( 𝑌 ∨ 𝑍 ) ) )
7 simpl ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝐾 ∈ Lat )
8 simpr2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
9 simpr3 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑍 ∈ 𝐵 )
10 1 2 latj4 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ∨ 𝑋 ) ∨ ( 𝑌 ∨ 𝑍 ) ) = ( ( 𝑋 ∨ 𝑌 ) ∨ ( 𝑋 ∨ 𝑍 ) ) )
11 7 3 3 8 9 10 syl122anc ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ∨ 𝑋 ) ∨ ( 𝑌 ∨ 𝑍 ) ) = ( ( 𝑋 ∨ 𝑌 ) ∨ ( 𝑋 ∨ 𝑍 ) ) )
12 6 11 eqtr3d ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ∨ ( 𝑌 ∨ 𝑍 ) ) = ( ( 𝑋 ∨ 𝑌 ) ∨ ( 𝑋 ∨ 𝑍 ) ) )