Nuprl Lemma : implies-DeMorgan-algebra-axioms
∀[dma:DeMorganAlgebraStructure]
((∀x:Point(dma). (¬(¬(x)) = x ∈ Point(dma)))
⇒ (∀x,y:Point(dma). (¬(x ∨ y) = ¬(x) ∧ ¬(y) ∈ Point(dma)))
⇒ DeMorgan-algebra-axioms(dma))
Proof
Definitions occuring in Statement :
DeMorgan-algebra-axioms: DeMorgan-algebra-axioms(dma),
dma-neg: ¬(x),
DeMorgan-algebra-structure: DeMorganAlgebraStructure,
lattice-join: a ∨ b,
lattice-meet: a ∧ b,
lattice-point: Point(l),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
DeMorgan-algebra-axioms: DeMorgan-algebra-axioms(dma),
and: P ∧ Q,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
dma-neg_wf,
lattice-point_wf,
all_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
DeMorgan-algebra-structure_wf,
bounded-lattice-structure_wf,
lattice-structure_wf,
equal_wf,
lattice-join_wf,
lattice-meet_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
sqequalRule,
instantiate,
independent_isectElimination,
lambdaEquality,
productElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[dma:DeMorganAlgebraStructure]
((\mforall{}x:Point(dma). (\mneg{}(\mneg{}(x)) = x))
{}\mRightarrow{} (\mforall{}x,y:Point(dma). (\mneg{}(x \mvee{} y) = \mneg{}(x) \mwedge{} \mneg{}(y)))
{}\mRightarrow{} DeMorgan-algebra-axioms(dma))
Date html generated:
2020_05_20-AM-08_55_42
Last ObjectModification:
2017_07_28-AM-09_17_04
Theory : lattices
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