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 ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( ( 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 ∧ 𝑌𝐵𝑍𝐵 ) → ( 𝑌 𝑍 ) = ( 𝑍 𝑌 ) )
5 3 4 syl3an1 ( ( 𝐾 ∈ OL ∧ 𝑌𝐵𝑍𝐵 ) → ( 𝑌 𝑍 ) = ( 𝑍 𝑌 ) )
6 5 3adant3r1 ( ( 𝐾 ∈ OL ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( 𝑌 𝑍 ) = ( 𝑍 𝑌 ) )
7 6 oveq2d ( ( 𝐾 ∈ OL ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( 𝑋 ( 𝑌 𝑍 ) ) = ( 𝑋 ( 𝑍 𝑌 ) ) )
8 1 2 latmassOLD ( ( 𝐾 ∈ OL ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( ( 𝑋 𝑌 ) 𝑍 ) = ( 𝑋 ( 𝑌 𝑍 ) ) )
9 simpl ( ( 𝐾 ∈ OL ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → 𝐾 ∈ OL )
10 simpr1 ( ( 𝐾 ∈ OL ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → 𝑋𝐵 )
11 simpr3 ( ( 𝐾 ∈ OL ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → 𝑍𝐵 )
12 simpr2 ( ( 𝐾 ∈ OL ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → 𝑌𝐵 )
13 1 2 latmassOLD ( ( 𝐾 ∈ OL ∧ ( 𝑋𝐵𝑍𝐵𝑌𝐵 ) ) → ( ( 𝑋 𝑍 ) 𝑌 ) = ( 𝑋 ( 𝑍 𝑌 ) ) )
14 9 10 11 12 13 syl13anc ( ( 𝐾 ∈ OL ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( ( 𝑋 𝑍 ) 𝑌 ) = ( 𝑋 ( 𝑍 𝑌 ) ) )
15 7 8 14 3eqtr4d ( ( 𝐾 ∈ OL ∧ ( 𝑋𝐵𝑌𝐵𝑍𝐵 ) ) → ( ( 𝑋 𝑌 ) 𝑍 ) = ( ( 𝑋 𝑍 ) 𝑌 ) )