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Metamath Proof Explorer

Theorem oldmj3

Description: De Morgan's law for join in an ortholattice. ( chdmj3 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 oldmj3 ( ( 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 5 3ad2ant1 ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ OP )
7 simp3 ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ∈ 𝐵 )
8 1 4 opoccl ⊢ ( ( 𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ 𝑌 ) ∈ 𝐵 )
9 6 7 8 syl2anc ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ 𝑌 ) ∈ 𝐵 )
10 1 2 3 4 oldmj1 ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘ 𝑌 ) ∈ 𝐵 ) → ( ⊥ ‘ ( 𝑋 ∨ ( ⊥ ‘ 𝑌 ) ) ) = ( ( ⊥ ‘ 𝑋 ) ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) )
11 9 10 syld3an3 ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( 𝑋 ∨ ( ⊥ ‘ 𝑌 ) ) ) = ( ( ⊥ ‘ 𝑋 ) ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) )
12 1 4 opococ ⊢ ( ( 𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) = 𝑌 )
13 6 7 12 syl2anc ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) = 𝑌 )
14 13 oveq2d ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ⊥ ‘ 𝑋 ) ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) = ( ( ⊥ ‘ 𝑋 ) ∧ 𝑌 ) )
15 11 14 eqtrd ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( 𝑋 ∨ ( ⊥ ‘ 𝑌 ) ) ) = ( ( ⊥ ‘ 𝑋 ) ∧ 𝑌 ) )