---

Step * 3 of Lemma dM-neg-properties


1. I : fset(ℕ)
2. ∀[x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq))].
     (dm-neg(names(I);NamesDeq;x ∧ y) = ¬(x) ∨ ¬(y) ∈ Point(free-DeMorgan-lattice(names(I);NamesDeq)))
3. ∀[x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq))].
     (dm-neg(names(I);NamesDeq;x ∨ y) = ¬(x) ∧ ¬(y) ∈ Point(free-DeMorgan-lattice(names(I);NamesDeq)))
4. dm-neg(names(I);NamesDeq;0) = 1 ∈ Point(free-DeMorgan-lattice(names(I);NamesDeq))
5. dm-neg(names(I);NamesDeq;1) = 0 ∈ Point(free-DeMorgan-lattice(names(I);NamesDeq))
⊢ ∀[x:Point(dM(I))]. (dm-neg(names(I);NamesDeq;¬(x)) = x ∈ Point(dM(I)))
BY
{ (InstLemma `dm-neg-neg` [⌜names(I)⌝;⌜NamesDeq⌝]⋅ THENA Auto) }

1
1. I : fset(ℕ)
2. ∀[x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq))].
     (dm-neg(names(I);NamesDeq;x ∧ y) = ¬(x) ∨ ¬(y) ∈ Point(free-DeMorgan-lattice(names(I);NamesDeq)))
3. ∀[x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq))].
     (dm-neg(names(I);NamesDeq;x ∨ y) = ¬(x) ∧ ¬(y) ∈ Point(free-DeMorgan-lattice(names(I);NamesDeq)))
4. dm-neg(names(I);NamesDeq;0) = 1 ∈ Point(free-DeMorgan-lattice(names(I);NamesDeq))
5. dm-neg(names(I);NamesDeq;1) = 0 ∈ Point(free-DeMorgan-lattice(names(I);NamesDeq))
6. ∀[x:Point(free-DeMorgan-lattice(names(I);NamesDeq))]
     (dm-neg(names(I);NamesDeq;¬(x)) = x ∈ Point(free-DeMorgan-lattice(names(I);NamesDeq)))
⊢ ∀[x:Point(dM(I))]. (dm-neg(names(I);NamesDeq;¬(x)) = x ∈ Point(dM(I)))

---

Latex:

---

Latex:

1.  I  :  fset(\mBbbN{})
2.  \mforall{}[x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq))].
          (dm-neg(names(I);NamesDeq;x  \mwedge{}  y)  =  \mneg{}(x)  \mvee{}  \mneg{}(y))
3.  \mforall{}[x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq))].
          (dm-neg(names(I);NamesDeq;x  \mvee{}  y)  =  \mneg{}(x)  \mwedge{}  \mneg{}(y))
4.  dm-neg(names(I);NamesDeq;0)  =  1
5.  dm-neg(names(I);NamesDeq;1)  =  0
\mvdash{}  \mforall{}[x:Point(dM(I))].  (dm-neg(names(I);NamesDeq;\mneg{}(x))  =  x)


By

---

Latex:
(InstLemma  `dm-neg-neg`  [\mkleeneopen{}names(I)\mkleeneclose{};\mkleeneopen{}NamesDeq\mkleeneclose{}]\mcdot{}  THENA  Auto)

---

Home Index