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Metamath Proof Explorer

Theorem oldmj2

Description: De Morgan's law for join in an ortholattice. ( chdmj2 analog.) (Contributed by NM, 7-Nov-2011)

Ref Expression
Hypotheses oldmm1.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
oldmm1.j ⊢ ∨ = ( join ‘ 𝐾 )
oldmm1.m ⊢ ∧ = ( meet ‘ 𝐾 )
oldmm1.o ⊢ ⊥ = ( oc ‘ 𝐾 )
Assertion oldmj2 ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ( ⊥ ‘ 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 ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ 𝑋 ) ∈ 𝐵 )
7 5 6 sylan ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ 𝑋 ) ∈ 𝐵 )
8 7 3adant3 ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ 𝑋 ) ∈ 𝐵 )
9 1 2 3 4 oldmj1 ⊢ ( ( 𝐾 ∈ OL ∧ ( ⊥ ‘ 𝑋 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ( ⊥ ‘ 𝑋 ) ∨ 𝑌 ) ) = ( ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ∧ ( ⊥ ‘ 𝑌 ) ) )
10 8 9 syld3an2 ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ( ⊥ ‘ 𝑋 ) ∨ 𝑌 ) ) = ( ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ∧ ( ⊥ ‘ 𝑌 ) ) )
11 1 4 opococ ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 )
12 5 11 sylan ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 )
13 12 3adant3 ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 )
14 13 oveq1d ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ∧ ( ⊥ ‘ 𝑌 ) ) = ( 𝑋 ∧ ( ⊥ ‘ 𝑌 ) ) )
15 10 14 eqtrd ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ( ⊥ ‘ 𝑋 ) ∨ 𝑌 ) ) = ( 𝑋 ∧ ( ⊥ ‘ 𝑌 ) ) )