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Step * 2 2 2 1 2 of Lemma mk-DeMorgan-algebra_wf


1. L : BoundedLatticeStructure
2. lattice-axioms(L)
3. bounded-lattice-axioms(L)
4. ∀[a,b,c:Point(L)].  (a ∧ b ∨ c = a ∧ b ∨ a ∧ c ∈ Point(L))
5. n : Point(L) ⟶ Point(L)
6. ∀x:Point(L). ((n (n x)) = x ∈ Point(L))
7. ∀x,y:Point(L).  ((n x ∨ y) = n x ∧ n y ∈ Point(L))
8. L["neg" := n] ∈ BoundedLatticeStructure
9. L["neg" := n] ∈ DeMorganAlgebraStructure
10. L["neg" := n] = L ∈ BoundedDistributiveLattice
11. L ∈ BoundedDistributiveLattice
12. lattice-axioms(L["neg" := n])
13. bounded-lattice-axioms(L["neg" := n])
14. ∀[a,b,c:Point(L["neg" := n])].  (a ∧ b ∨ c = a ∧ b ∨ a ∧ c ∈ Point(L["neg" := n]))
⊢ DeMorgan-algebra-axioms(L["neg" := n])
BY
{ (BLemma `implies-DeMorgan-algebra-axioms`
   THEN Try (Trivial)
   THEN RepUR ``lattice-meet lattice-join lattice-point`` 0
   THEN Folds ``lattice-meet lattice-join lattice-point`` 0
   THEN RepUR ``dma-neg`` 0
   THEN Auto) }

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Latex:

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Latex:

1.  L  :  BoundedLatticeStructure
2.  lattice-axioms(L)
3.  bounded-lattice-axioms(L)
4.  \mforall{}[a,b,c:Point(L)].    (a  \mwedge{}  b  \mvee{}  c  =  a  \mwedge{}  b  \mvee{}  a  \mwedge{}  c)
5.  n  :  Point(L)  {}\mrightarrow{}  Point(L)
6.  \mforall{}x:Point(L).  ((n  (n  x))  =  x)
7.  \mforall{}x,y:Point(L).    ((n  x  \mvee{}  y)  =  n  x  \mwedge{}  n  y)
8.  L["neg"  :=  n]  \mmember{}  BoundedLatticeStructure
9.  L["neg"  :=  n]  \mmember{}  DeMorganAlgebraStructure
10.  L["neg"  :=  n]  =  L
11.  L  \mmember{}  BoundedDistributiveLattice
12.  lattice-axioms(L["neg"  :=  n])
13.  bounded-lattice-axioms(L["neg"  :=  n])
14.  \mforall{}[a,b,c:Point(L["neg"  :=  n])].    (a  \mwedge{}  b  \mvee{}  c  =  a  \mwedge{}  b  \mvee{}  a  \mwedge{}  c)
\mvdash{}  DeMorgan-algebra-axioms(L["neg"  :=  n])


By

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Latex:
(BLemma  `implies-DeMorgan-algebra-axioms`
  THEN  Try  (Trivial)
  THEN  RepUR  ``lattice-meet  lattice-join  lattice-point``  0
  THEN  Folds  ``lattice-meet  lattice-join  lattice-point``  0
  THEN  RepUR  ``dma-neg``  0
  THEN  Auto)

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