Nuprl Lemma : lattice-hom-meet
∀[l1,l2:BoundedLattice]. ∀[f:Hom(l1;l2)]. ∀[a,b:Point(l1)]. ((f a ∧ b) = f a ∧ f b ∈ Point(l2))
Proof
Definitions occuring in Statement :
bounded-lattice-hom: Hom(l1;l2),
bdd-lattice: BoundedLattice,
lattice-meet: a ∧ b,
lattice-point: Point(l),
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
and: P ∧ Q,
subtype_rel: A ⊆r B,
bdd-lattice: BoundedLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
lattice-point_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
bounded-lattice-hom_wf,
bdd-lattice_wf,
lattice-meet_wf,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
hypothesis,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
sqequalRule,
instantiate,
lambdaEquality,
productEquality,
cumulativity,
independent_isectElimination,
isect_memberEquality,
axiomEquality,
because_Cache,
functionExtensionality,
natural_numberEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[l1,l2:BoundedLattice]. \mforall{}[f:Hom(l1;l2)]. \mforall{}[a,b:Point(l1)]. ((f a \mwedge{} b) = f a \mwedge{} f b)
Date html generated:
2020_05_20-AM-08_44_36
Last ObjectModification:
2017_07_28-AM-09_14_12
Theory : lattices
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