Nuprl Lemma : face_lattice-hom-equal
∀[I,J:fset(ℕ)]. ∀[g,h:Hom(face_lattice(I);face_lattice(J))].
g = h ∈ Hom(face_lattice(I);face_lattice(J))
supposing (∀x:names(I). ((g (x=0)) = (h (x=0)) ∈ Point(face_lattice(J))))
∧ (∀x:names(I). ((g (x=1)) = (h (x=1)) ∈ Point(face_lattice(J))))
Proof
Definitions occuring in Statement :
fl1: (x=1),
fl0: (x=0),
face_lattice: face_lattice(I),
names: names(I),
bounded-lattice-hom: Hom(l1;l2),
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
face_lattice: face_lattice(I),
all: ∀x:A. B[x],
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
subtype_rel: A ⊆r B,
cand: A c∧ B,
fl1: (x=1),
fl0: (x=0),
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
bdd-distributive-lattice: BoundedDistributiveLattice,
so_apply: x[s]
Lemmas referenced :
face-lattice-hom-unique,
names_wf,
names-deq_wf,
face-lattice_wf,
face_lattice-deq_wf,
fl0_wf,
lattice-point_wf,
face_lattice_wf,
fl1_wf,
equal_wf,
squash_wf,
true_wf,
FL-meet-0-1,
iff_weakening_equal,
all_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
uall_wf,
lattice-meet_wf,
lattice-join_wf,
bounded-lattice-hom_wf,
bdd-distributive-lattice_wf,
fset_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
dependent_functionElimination,
isectElimination,
hypothesisEquality,
hypothesis,
lambdaEquality,
applyEquality,
setElimination,
rename,
sqequalRule,
because_Cache,
independent_isectElimination,
lambdaFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
independent_pairFormation,
productEquality,
instantiate,
cumulativity,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[I,J:fset(\mBbbN{})]. \mforall{}[g,h:Hom(face\_lattice(I);face\_lattice(J))].
g = h supposing (\mforall{}x:names(I). ((g (x=0)) = (h (x=0)))) \mwedge{} (\mforall{}x:names(I). ((g (x=1)) = (h (x=1))))
Date html generated:
2017_10_05-AM-01_11_05
Last ObjectModification:
2017_07_28-AM-09_30_03
Theory : cubical!type!theory
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