Metamath Proof Explorer


Theorem oldmm2

Description: De Morgan's law for meet in an ortholattice. ( chdmm2 analog.) (Contributed by NM, 6-Nov-2011)

Ref Expression
Hypotheses oldmm1.b 𝐵 = ( Base ‘ 𝐾 )
oldmm1.j = ( join ‘ 𝐾 )
oldmm1.m = ( meet ‘ 𝐾 )
oldmm1.o = ( oc ‘ 𝐾 )
Assertion oldmm2 ( ( 𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵 ) → ( ‘ ( ( 𝑋 ) 𝑌 ) ) = ( 𝑋 ( 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 oldmm1.b 𝐵 = ( Base ‘ 𝐾 )
2 oldmm1.j = ( join ‘ 𝐾 )
3 oldmm1.m = ( meet ‘ 𝐾 )
4 oldmm1.o = ( oc ‘ 𝐾 )
5 olop ( 𝐾 ∈ OL → 𝐾 ∈ OP )
6 1 4 opoccl ( ( 𝐾 ∈ OP ∧ 𝑋𝐵 ) → ( 𝑋 ) ∈ 𝐵 )
7 5 6 sylan ( ( 𝐾 ∈ OL ∧ 𝑋𝐵 ) → ( 𝑋 ) ∈ 𝐵 )
8 7 3adant3 ( ( 𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑋 ) ∈ 𝐵 )
9 1 2 3 4 oldmm1 ( ( 𝐾 ∈ OL ∧ ( 𝑋 ) ∈ 𝐵𝑌𝐵 ) → ( ‘ ( ( 𝑋 ) 𝑌 ) ) = ( ( ‘ ( 𝑋 ) ) ( 𝑌 ) ) )
10 8 9 syld3an2 ( ( 𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵 ) → ( ‘ ( ( 𝑋 ) 𝑌 ) ) = ( ( ‘ ( 𝑋 ) ) ( 𝑌 ) ) )
11 1 4 opococ ( ( 𝐾 ∈ OP ∧ 𝑋𝐵 ) → ( ‘ ( 𝑋 ) ) = 𝑋 )
12 5 11 sylan ( ( 𝐾 ∈ OL ∧ 𝑋𝐵 ) → ( ‘ ( 𝑋 ) ) = 𝑋 )
13 12 3adant3 ( ( 𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵 ) → ( ‘ ( 𝑋 ) ) = 𝑋 )
14 13 oveq1d ( ( 𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵 ) → ( ( ‘ ( 𝑋 ) ) ( 𝑌 ) ) = ( 𝑋 ( 𝑌 ) ) )
15 10 14 eqtrd ( ( 𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵 ) → ( ‘ ( ( 𝑋 ) 𝑌 ) ) = ( 𝑋 ( 𝑌 ) ) )