Nuprl Lemma : order-preserving-map-lattice-lemma
∀[l1,l2:Lattice]. ∀[f:Point(l1) ⟶ Point(l2)].
(∀a,b:Point(l1). f a ∧ b ≤ f a ∧ f b) ∧ (∀a,b:Point(l1). f a ∨ f b ≤ f a ∨ b)
supposing ∀x,y:Point(l1). (x ≤ y ⇒ f x ≤ f y)
Proof
Definitions occuring in Statement :
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,
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,
cand: A c∧ B,
all: ∀x:A. B[x],
lattice: Lattice,
lattice-le: a ≤ b,
prop: ℙ,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
so_apply: x[s],
greatest-lower-bound: greatest-lower-bound(T;x,y.R[x; y];a;b;c),
guard: {T},
least-upper-bound: least-upper-bound(T;x,y.R[x; y];a;b;c)
Lemmas referenced :
lattice-point_wf,
all_wf,
lattice-le_wf,
lattice_wf,
lattice-meet-is-glb,
lattice-meet_wf,
lattice-join-is-lub,
lattice-join_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
independent_pairFormation,
because_Cache,
sqequalRule,
productElimination,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
functionEquality,
applyEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination
Latex:
\mforall{}[l1,l2:Lattice]. \mforall{}[f:Point(l1) {}\mrightarrow{} Point(l2)].
(\mforall{}a,b:Point(l1). f a \mwedge{} b \mleq{} f a \mwedge{} f b) \mwedge{} (\mforall{}a,b:Point(l1). f a \mvee{} f b \mleq{} f a \mvee{} b)
supposing \mforall{}x,y:Point(l1). (x \mleq{} y {}\mRightarrow{} f x \mleq{} f y)
Date html generated:
2020_05_20-AM-08_26_19
Last ObjectModification:
2015_12_28-PM-02_01_55
Theory : lattices
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