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Metamath Proof Explorer

Theorem latjjdir

Description: Lattice join distributes over itself. (Contributed by NM, 2-Aug-2012)

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

Proof

Step Hyp Ref Expression
1 latjass.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 latjass.j ⊢ ∨ = ( join ‘ 𝐾 )
3 1 2 latjidm ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑍 ∈ 𝐵 ) → ( 𝑍 ∨ 𝑍 ) = 𝑍 )
4 3 3ad2antr3 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑍 ∨ 𝑍 ) = 𝑍 )
5 4 oveq2d ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ∨ 𝑌 ) ∨ ( 𝑍 ∨ 𝑍 ) ) = ( ( 𝑋 ∨ 𝑌 ) ∨ 𝑍 ) )
6 simpl ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝐾 ∈ Lat )
7 simpr1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
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 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ∨ 𝑌 ) ∨ ( 𝑍 ∨ 𝑍 ) ) = ( ( 𝑋 ∨ 𝑍 ) ∨ ( 𝑌 ∨ 𝑍 ) ) )
11 6 7 8 9 9 10 syl122anc ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ∨ 𝑌 ) ∨ ( 𝑍 ∨ 𝑍 ) ) = ( ( 𝑋 ∨ 𝑍 ) ∨ ( 𝑌 ∨ 𝑍 ) ) )
12 5 11 eqtr3d ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ∨ 𝑌 ) ∨ 𝑍 ) = ( ( 𝑋 ∨ 𝑍 ) ∨ ( 𝑌 ∨ 𝑍 ) ) )