Metamath Proof Explorer

Theorem oldmj1

Description: De Morgan's law for join in an ortholattice. ( chdmj1 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 oldmj1 ( ( 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 1 2 3 4 oldmm4 ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ( ⊥ ‘ 𝑋 ) ∧ ( ⊥ ‘ 𝑌 ) ) ) = ( 𝑋 ∨ 𝑌 ) )
6 5 fveq2d ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ ( ( ⊥ ‘ 𝑋 ) ∧ ( ⊥ ‘ 𝑌 ) ) ) ) = ( ⊥ ‘ ( 𝑋 ∨ 𝑌 ) ) )
7 olop ⊢ ( 𝐾 ∈ OL → 𝐾 ∈ OP )
8 7 3ad2ant1 ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ OP )
9 ollat ⊢ ( 𝐾 ∈ OL → 𝐾 ∈ Lat )
10 9 3ad2ant1 ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ Lat )
11 1 4 opoccl ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ 𝑋 ) ∈ 𝐵 )
12 7 11 sylan ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ 𝑋 ) ∈ 𝐵 )
13 12 3adant3 ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ 𝑋 ) ∈ 𝐵 )
14 1 4 opoccl ⊢ ( ( 𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ 𝑌 ) ∈ 𝐵 )
15 7 14 sylan ⊢ ( ( 𝐾 ∈ OL ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ 𝑌 ) ∈ 𝐵 )
16 15 3adant2 ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ 𝑌 ) ∈ 𝐵 )
17 1 3 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ ( ⊥ ‘ 𝑋 ) ∈ 𝐵 ∧ ( ⊥ ‘ 𝑌 ) ∈ 𝐵 ) → ( ( ⊥ ‘ 𝑋 ) ∧ ( ⊥ ‘ 𝑌 ) ) ∈ 𝐵 )
18 10 13 16 17 syl3anc ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ⊥ ‘ 𝑋 ) ∧ ( ⊥ ‘ 𝑌 ) ) ∈ 𝐵 )
19 1 4 opococ ⊢ ( ( 𝐾 ∈ OP ∧ ( ( ⊥ ‘ 𝑋 ) ∧ ( ⊥ ‘ 𝑌 ) ) ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ ( ( ⊥ ‘ 𝑋 ) ∧ ( ⊥ ‘ 𝑌 ) ) ) ) = ( ( ⊥ ‘ 𝑋 ) ∧ ( ⊥ ‘ 𝑌 ) ) )
20 8 18 19 syl2anc ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ ( ( ⊥ ‘ 𝑋 ) ∧ ( ⊥ ‘ 𝑌 ) ) ) ) = ( ( ⊥ ‘ 𝑋 ) ∧ ( ⊥ ‘ 𝑌 ) ) )
21 6 20 eqtr3d ⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( 𝑋 ∨ 𝑌 ) ) = ( ( ⊥ ‘ 𝑋 ) ∧ ( ⊥ ‘ 𝑌 ) ) )