Nuprl Lemma : order-preserving-map-is-lattice-hom
∀[l1,l2:Lattice]. ∀[f:Point(l1) ⟶ Point(l2)].
f ∈ Hom(l1;l2)
supposing (∀x,y:Point(l1). (x ≤ y ⇒ f x ≤ f y))
∧ (∀a,b:Point(l1). f a ∧ f b ≤ f a ∧ b)
∧ (∀a,b:Point(l1). f a ∨ b ≤ f a ∨ f b)
Proof
Definitions occuring in Statement :
lattice-hom: Hom(l1;l2),
lattice-le: a ≤ b,
lattice: Lattice,
lattice-join: a ∨ b,
lattice-meet: a ∧ b,
lattice-point: Point(l),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
all: ∀x:A. B[x],
order: Order(T;x,y.R[x; y]),
cand: A c∧ B,
lattice: Lattice,
lattice-hom: Hom(l1;l2),
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
guard: {T},
anti_sym: AntiSym(T;x,y.R[x; y])
Lemmas referenced :
lattice-le-order,
order-preserving-map-lattice-lemma,
lattice-point_wf,
uall_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
all_wf,
lattice-le_wf,
lattice_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
isectElimination,
because_Cache,
independent_isectElimination,
hypothesis,
independent_pairFormation,
sqequalRule,
independent_pairEquality,
axiomEquality,
setElimination,
rename,
isect_memberEquality,
dependent_set_memberEquality,
lambdaEquality,
productEquality,
applyEquality,
functionExtensionality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
independent_functionElimination
Latex:
\mforall{}[l1,l2:Lattice]. \mforall{}[f:Point(l1) {}\mrightarrow{} Point(l2)].
f \mmember{} Hom(l1;l2)
supposing (\mforall{}x,y:Point(l1). (x \mleq{} y {}\mRightarrow{} f x \mleq{} f y))
\mwedge{} (\mforall{}a,b:Point(l1). f a \mwedge{} f b \mleq{} f a \mwedge{} b)
\mwedge{} (\mforall{}a,b:Point(l1). f a \mvee{} b \mleq{} f a \mvee{} f b)
Date html generated:
2020_05_20-AM-08_26_21
Last ObjectModification:
2017_07_28-AM-09_13_14
Theory : lattices
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