Nuprl Lemma : order-preserving-map-is-bounded-lattice-hom
∀[l1,l2:BoundedLattice]. ∀[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))
∧ ((f 0) = 0 ∈ Point(l2))
∧ ((f 1) = 1 ∈ Point(l2))
Proof
Definitions occuring in Statement :
bounded-lattice-hom: Hom(l1;l2),
bdd-lattice: BoundedLattice,
lattice-0: 0,
lattice-1: 1,
lattice-le: a ≤ b,
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],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
bounded-lattice-hom: Hom(l1;l2),
cand: A c∧ B,
subtype_rel: A ⊆r B,
bdd-lattice: BoundedLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
lattice-hom: Hom(l1;l2),
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
equal_wf,
lattice-point_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure-subtype,
lattice-0_wf,
lattice-1_wf,
all_wf,
lattice-le_wf,
lattice-meet_wf,
lattice-join_wf,
bdd-lattice_wf,
order-preserving-map-is-lattice-hom,
bdd-lattice-subtype-lattice
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
dependent_set_memberEquality,
hypothesis,
independent_pairFormation,
productEquality,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
sqequalRule,
instantiate,
lambdaEquality,
cumulativity,
because_Cache,
independent_isectElimination,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
functionEquality,
functionExtensionality,
isect_memberEquality
Latex:
\mforall{}[l1,l2:BoundedLattice]. \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))
\mwedge{} ((f 0) = 0)
\mwedge{} ((f 1) = 1)
Date html generated:
2020_05_20-AM-08_26_23
Last ObjectModification:
2017_07_28-AM-09_13_15
Theory : lattices
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