---

Step * of Lemma dM-to-FL-properties

∀[I:fset(ℕ)]
  ((∀x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq)).
      (dM-to-FL(I;x ∨ y) = dM-to-FL(I;x) ∨ dM-to-FL(I;y) ∈ Point(face_lattice(I))))
  ∧ (∀x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq)).
       (dM-to-FL(I;x ∧ y) = dM-to-FL(I;x) ∧ dM-to-FL(I;y) ∈ Point(face_lattice(I))))
  ∧ (dM-to-FL(I;0) = 0 ∈ Point(face_lattice(I)))
  ∧ (dM-to-FL(I;1) = 1 ∈ Point(face_lattice(I))))
BY
{ (InstLemma `dM-to-FL-is-hom` [] THEN ParallelLast THEN Auto) }

1
1. ∀[I:fset(ℕ)]. (λz.dM-to-FL(I;z) ∈ Hom(free-DeMorgan-lattice(names(I);NamesDeq);face_lattice(I)))
2. I : fset(ℕ)
3. λz.dM-to-FL(I;z) ∈ Hom(free-DeMorgan-lattice(names(I);NamesDeq);face_lattice(I))
4. x : Point(free-DeMorgan-lattice(names(I);NamesDeq))
5. y : Point(free-DeMorgan-lattice(names(I);NamesDeq))
⊢ dM-to-FL(I;x ∨ y) = dM-to-FL(I;x) ∨ dM-to-FL(I;y) ∈ Point(face_lattice(I))

2
1. ∀[I:fset(ℕ)]. (λz.dM-to-FL(I;z) ∈ Hom(free-DeMorgan-lattice(names(I);NamesDeq);face_lattice(I)))
2. I : fset(ℕ)
3. λz.dM-to-FL(I;z) ∈ Hom(free-DeMorgan-lattice(names(I);NamesDeq);face_lattice(I))
4. ∀x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq)).
     (dM-to-FL(I;x ∨ y) = dM-to-FL(I;x) ∨ dM-to-FL(I;y) ∈ Point(face_lattice(I)))
5. x : Point(free-DeMorgan-lattice(names(I);NamesDeq))
6. y : Point(free-DeMorgan-lattice(names(I);NamesDeq))
⊢ dM-to-FL(I;x ∧ y) = dM-to-FL(I;x) ∧ dM-to-FL(I;y) ∈ Point(face_lattice(I))

3
1. ∀[I:fset(ℕ)]. (λz.dM-to-FL(I;z) ∈ Hom(free-DeMorgan-lattice(names(I);NamesDeq);face_lattice(I)))
2. I : fset(ℕ)
3. λz.dM-to-FL(I;z) ∈ Hom(free-DeMorgan-lattice(names(I);NamesDeq);face_lattice(I))
4. ∀x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq)).
     (dM-to-FL(I;x ∨ y) = dM-to-FL(I;x) ∨ dM-to-FL(I;y) ∈ Point(face_lattice(I)))
5. ∀x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq)).
     (dM-to-FL(I;x ∧ y) = dM-to-FL(I;x) ∧ dM-to-FL(I;y) ∈ Point(face_lattice(I)))
6. dM-to-FL(I;0) = 0 ∈ Point(face_lattice(I))
⊢ dM-to-FL(I;1) = 1 ∈ Point(face_lattice(I))

---

Latex:

---

Latex:
\mforall{}[I:fset(\mBbbN{})]
    ((\mforall{}x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq)).
            (dM-to-FL(I;x  \mvee{}  y)  =  dM-to-FL(I;x)  \mvee{}  dM-to-FL(I;y)))
    \mwedge{}  (\mforall{}x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq)).
              (dM-to-FL(I;x  \mwedge{}  y)  =  dM-to-FL(I;x)  \mwedge{}  dM-to-FL(I;y)))
    \mwedge{}  (dM-to-FL(I;0)  =  0)
    \mwedge{}  (dM-to-FL(I;1)  =  1))


By

---

Latex:
(InstLemma  `dM-to-FL-is-hom`  []  THEN  ParallelLast  THEN  Auto)

---

Home Index