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Metamath Proof Explorer

Theorem latm32

Description: A rearrangement of lattice meet. ( in12 analog.) (Contributed by NM, 13-Nov-2012)

Ref Expression
Hypotheses olmass.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
olmass.m ⊢ ∧ = ( meet ‘ 𝐾 )
Assertion latm32 ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ∧ 𝑌 ) ∧ 𝑍 ) = ( ( 𝑋 ∧ 𝑍 ) ∧ 𝑌 ) )

Proof

Step Hyp Ref Expression
1 olmass.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 olmass.m ⊢ ∧ = ( meet ‘ 𝐾 )
3 ollat ⊢ ( 𝐾 ∈ OL → 𝐾 ∈ Lat )
4 1 2 latmcom ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑌 ∧ 𝑍 ) = ( 𝑍 ∧ 𝑌 ) )
5 3 4 syl3an1 ⊢ ( ( 𝐾 ∈ OL ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑌 ∧ 𝑍 ) = ( 𝑍 ∧ 𝑌 ) )
6 5 3adant3r1 ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑌 ∧ 𝑍 ) = ( 𝑍 ∧ 𝑌 ) )
7 6 oveq2d ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ∧ ( 𝑌 ∧ 𝑍 ) ) = ( 𝑋 ∧ ( 𝑍 ∧ 𝑌 ) ) )
8 1 2 latmassOLD ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ∧ 𝑌 ) ∧ 𝑍 ) = ( 𝑋 ∧ ( 𝑌 ∧ 𝑍 ) ) )
9 simpl ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝐾 ∈ OL )
10 simpr1 ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
11 simpr3 ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑍 ∈ 𝐵 )
12 simpr2 ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
13 1 2 latmassOLD ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 ∧ 𝑍 ) ∧ 𝑌 ) = ( 𝑋 ∧ ( 𝑍 ∧ 𝑌 ) ) )
14 9 10 11 12 13 syl13anc ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ∧ 𝑍 ) ∧ 𝑌 ) = ( 𝑋 ∧ ( 𝑍 ∧ 𝑌 ) ) )
15 7 8 14 3eqtr4d ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ∧ 𝑌 ) ∧ 𝑍 ) = ( ( 𝑋 ∧ 𝑍 ) ∧ 𝑌 ) )