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Metamath Proof Explorer

Theorem latj13

Description: Swap 1st and 3rd members of lattice join. (Contributed by NM, 4-Jun-2012)

Ref Expression
Hypotheses latjass.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
latjass.j ⊢ ∨ = ( join ‘ 𝐾 )
Assertion latj13 ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ∨ ( 𝑌 ∨ 𝑍 ) ) = ( 𝑍 ∨ ( 𝑌 ∨ 𝑋 ) ) )

Proof

Step Hyp Ref Expression
1 latjass.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 latjass.j ⊢ ∨ = ( join ‘ 𝐾 )
3 simpl ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝐾 ∈ Lat )
4 simpr2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
5 simpr3 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑍 ∈ 𝐵 )
6 simpr1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
7 1 2 latj32 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ) → ( ( 𝑌 ∨ 𝑍 ) ∨ 𝑋 ) = ( ( 𝑌 ∨ 𝑋 ) ∨ 𝑍 ) )
8 3 4 5 6 7 syl13anc ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑌 ∨ 𝑍 ) ∨ 𝑋 ) = ( ( 𝑌 ∨ 𝑋 ) ∨ 𝑍 ) )
9 1 2 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑌 ∨ 𝑍 ) ∈ 𝐵 )
10 9 3adant3r1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑌 ∨ 𝑍 ) ∈ 𝐵 )
11 1 2 latjcom ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑌 ∨ 𝑍 ) ∈ 𝐵 ) → ( 𝑋 ∨ ( 𝑌 ∨ 𝑍 ) ) = ( ( 𝑌 ∨ 𝑍 ) ∨ 𝑋 ) )
12 3 6 10 11 syl3anc ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ∨ ( 𝑌 ∨ 𝑍 ) ) = ( ( 𝑌 ∨ 𝑍 ) ∨ 𝑋 ) )
13 1 2 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑌 ∨ 𝑋 ) ∈ 𝐵 )
14 3 4 6 13 syl3anc ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑌 ∨ 𝑋 ) ∈ 𝐵 )
15 1 2 latjcom ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑍 ∈ 𝐵 ∧ ( 𝑌 ∨ 𝑋 ) ∈ 𝐵 ) → ( 𝑍 ∨ ( 𝑌 ∨ 𝑋 ) ) = ( ( 𝑌 ∨ 𝑋 ) ∨ 𝑍 ) )
16 3 5 14 15 syl3anc ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑍 ∨ ( 𝑌 ∨ 𝑋 ) ) = ( ( 𝑌 ∨ 𝑋 ) ∨ 𝑍 ) )
17 8 12 16 3eqtr4d ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ∨ ( 𝑌 ∨ 𝑍 ) ) = ( 𝑍 ∨ ( 𝑌 ∨ 𝑋 ) ) )