Nuprl Lemma : DeMorgan-algebra-laws
∀[dma:DeMorganAlgebra]
((∀x:Point(dma). (¬(¬(x)) = x ∈ Point(dma)))
∧ (∀x,y:Point(dma). (¬(x ∧ y) = ¬(x) ∨ ¬(y) ∈ Point(dma)))
∧ (∀x,y:Point(dma). (¬(x ∨ y) = ¬(x) ∧ ¬(y) ∈ Point(dma)))
∧ (¬(0) = 1 ∈ Point(dma))
∧ (¬(1) = 0 ∈ Point(dma)))
Proof
Definitions occuring in Statement :
DeMorgan-algebra: DeMorganAlgebra,
dma-neg: ¬(x),
lattice-0: 0,
lattice-1: 1,
lattice-join: a ∨ b,
lattice-meet: a ∧ b,
lattice-point: Point(l),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
DeMorgan-algebra: DeMorganAlgebra,
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
DeMorgan-algebra-axioms: DeMorgan-algebra-axioms(dma),
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
lattice-axioms: lattice-axioms(l)
Lemmas referenced :
lattice-point_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
DeMorgan-algebra-structure_wf,
bounded-lattice-structure_wf,
lattice-structure_wf,
subtype_rel_set,
lattice-axioms_wf,
bounded-lattice-axioms_wf,
uall_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
DeMorgan-algebra_wf,
lattice-0_wf,
and_wf,
dma-neg_wf,
squash_wf,
true_wf,
iff_weakening_equal,
lattice-1_wf,
bdd-distributive-lattice-subtype-bdd-lattice,
DeMorgan-algebra-subtype,
bdd-distributive-lattice_wf,
bdd-lattice_wf,
lattice-meet-0,
lattice-join-1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesisEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
lambdaFormation,
productElimination,
hypothesis,
extract_by_obid,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
independent_pairFormation,
because_Cache,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
productEquality,
cumulativity,
universeEquality,
hyp_replacement,
equalitySymmetry,
dependent_set_memberEquality,
equalityTransitivity,
applyLambdaEquality,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[dma:DeMorganAlgebra]
((\mforall{}x:Point(dma). (\mneg{}(\mneg{}(x)) = x))
\mwedge{} (\mforall{}x,y:Point(dma). (\mneg{}(x \mwedge{} y) = \mneg{}(x) \mvee{} \mneg{}(y)))
\mwedge{} (\mforall{}x,y:Point(dma). (\mneg{}(x \mvee{} y) = \mneg{}(x) \mwedge{} \mneg{}(y)))
\mwedge{} (\mneg{}(0) = 1)
\mwedge{} (\mneg{}(1) = 0))
Date html generated:
2017_10_05-AM-00_42_07
Last ObjectModification:
2017_07_28-AM-09_17_00
Theory : lattices
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