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Step * of Lemma dM-neg-properties

∀[I:fset(ℕ)]
  ((∀[x,y:Point(dM(I))].  (dm-neg(names(I);NamesDeq;x ∧ y) = ¬(x) ∨ ¬(y) ∈ Point(dM(I))))
  ∧ (∀[x,y:Point(dM(I))].  (dm-neg(names(I);NamesDeq;x ∨ y) = ¬(x) ∧ ¬(y) ∈ Point(dM(I))))
  ∧ (∀[x:Point(dM(I))]. (dm-neg(names(I);NamesDeq;¬(x)) = x ∈ Point(dM(I)))))
BY
{ ((D 0 THENA Auto)
   THEN (InstLemma `dm-neg-properties` [⌜names(I)⌝;⌜NamesDeq⌝]⋅ THENA Auto)
   THEN ExRepD
   THEN SplitAndConcl) }

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))
⊢ ∀[x,y:Point(dM(I))].  (dm-neg(names(I);NamesDeq;x ∧ y) = ¬(x) ∨ ¬(y) ∈ Point(dM(I)))

2
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,y:Point(dM(I))].  (dm-neg(names(I);NamesDeq;x ∨ y) = ¬(x) ∧ ¬(y) ∈ Point(dM(I)))

3
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)))

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Latex:

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Latex:
\mforall{}[I:fset(\mBbbN{})]
    ((\mforall{}[x,y:Point(dM(I))].    (dm-neg(names(I);NamesDeq;x  \mwedge{}  y)  =  \mneg{}(x)  \mvee{}  \mneg{}(y)))
    \mwedge{}  (\mforall{}[x,y:Point(dM(I))].    (dm-neg(names(I);NamesDeq;x  \mvee{}  y)  =  \mneg{}(x)  \mwedge{}  \mneg{}(y)))
    \mwedge{}  (\mforall{}[x:Point(dM(I))].  (dm-neg(names(I);NamesDeq;\mneg{}(x))  =  x)))


By

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Latex:
((D  0  THENA  Auto)
  THEN  (InstLemma  `dm-neg-properties`  [\mkleeneopen{}names(I)\mkleeneclose{};\mkleeneopen{}NamesDeq\mkleeneclose{}]\mcdot{}  THENA  Auto)
  THEN  ExRepD
  THEN  SplitAndConcl)

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