Theorem oldmj4

Description: De Morgan's law for join in an ortholattice. ( chdmj4 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	oldmj4	⊢ ( ( 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 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ 𝑌 ) ∈ 𝐵 )
7	5 6	sylan	⊢ ( ( 𝐾 ∈ OL ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ 𝑌 ) ∈ 𝐵 )
8	7	3adant2	⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ 𝑌 ) ∈ 𝐵 )
9	1 2 3 4	oldmj2	⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘ 𝑌 ) ∈ 𝐵 ) → ( ⊥ ‘ ( ( ⊥ ‘ 𝑋 ) ∨ ( ⊥ ‘ 𝑌 ) ) ) = ( 𝑋 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) )
10	8 9	syld3an3	⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ( ⊥ ‘ 𝑋 ) ∨ ( ⊥ ‘ 𝑌 ) ) ) = ( 𝑋 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) )
11	1 4	opococ	⊢ ( ( 𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) = 𝑌 )
12	5 11	sylan	⊢ ( ( 𝐾 ∈ OL ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) = 𝑌 )
13	12	3adant2	⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) = 𝑌 )
14	13	oveq2d	⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) = ( 𝑋 ∧ 𝑌 ) )
15	10 14	eqtrd	⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ( ⊥ ‘ 𝑋 ) ∨ ( ⊥ ‘ 𝑌 ) ) ) = ( 𝑋 ∧ 𝑌 ) )