Nuprl Lemma : bounded-lattice-hom-equal
∀[l1,l2:BoundedLatticeStructure]. ∀[f,g:Hom(l1;l2)].
f = g ∈ Hom(l1;l2) supposing ∀x:Point(l1). ((f x) = (g x) ∈ Point(l2))
Proof
Definitions occuring in Statement :
bounded-lattice-hom: Hom(l1;l2),
bounded-lattice-structure: BoundedLatticeStructure,
lattice-point: Point(l),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
bounded-lattice-hom: Hom(l1;l2),
and: P ∧ Q,
lattice-hom: Hom(l1;l2),
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
lattice-point_wf,
bounded-lattice-structure-subtype,
iff_weakening_equal,
uall_wf,
lattice-meet_wf,
lattice-join_wf,
lattice-0_wf,
lattice-1_wf,
all_wf,
bounded-lattice-hom_wf,
bounded-lattice-structure_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality,
productElimination,
functionExtensionality,
applyEquality,
lambdaEquality,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
sqequalRule,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
because_Cache,
productEquality,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[l1,l2:BoundedLatticeStructure]. \mforall{}[f,g:Hom(l1;l2)]. f = g supposing \mforall{}x:Point(l1). ((f x) = (g x))
Date html generated:
2020_05_20-AM-08_24_53
Last ObjectModification:
2017_07_28-AM-09_12_39
Theory : lattices
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