Nuprl Lemma : face_lattice-hom-fixes-sublattice
∀[I,J:fset(ℕ)].
∀[f:Hom(face_lattice(J);face_lattice(I))]. ∀[x:Point(face_lattice(J))].
(f x) = x ∈ Point(face_lattice(I))
supposing ∀i:names(J)
(((f (i=0)) = (i=0) ∈ Point(face_lattice(I))) ∧ ((f (i=1)) = (i=1) ∈ Point(face_lattice(I))))
supposing J ⊆ I
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),
f-subset: xs ⊆ ys,
fset: fset(T),
int-deq: IntDeq,
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],
uimplies: b supposing a,
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
so_apply: x[s],
implies: P ⇒ Q,
prop: ℙ,
and: P ∧ Q,
cand: A c∧ B,
guard: {T},
bdd-distributive-lattice: BoundedDistributiveLattice,
nat: ℕ,
lattice-0: 0,
record-select: r.x,
face_lattice: face_lattice(I),
face-lattice: face-lattice(T;eq),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
empty-fset: {},
nil: [],
it: ⋅,
lattice-1: 1,
fset-singleton: {x},
cons: [a / b],
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
top: Top
Lemmas referenced :
face_lattice-induction,
equal_wf,
lattice-point_wf,
face_lattice_wf,
face_lattice-point-subtype,
sq_stable__equal,
names_wf,
all_wf,
fl0_wf,
names-subtype,
fl1_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,
f-subset_wf,
int-deq_wf,
strong-subtype-deq-subtype,
nat_wf,
strong-subtype-set3,
le_wf,
strong-subtype-self,
fset_wf,
squash_wf,
true_wf,
iff_weakening_equal,
face_lattice-join-invariant,
face_lattice-meet-invariant
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
isectElimination,
sqequalRule,
lambdaEquality,
hypothesis,
applyEquality,
because_Cache,
setElimination,
rename,
independent_isectElimination,
independent_functionElimination,
lambdaFormation,
independent_pairFormation,
productEquality,
instantiate,
cumulativity,
universeEquality,
intEquality,
natural_numberEquality,
productElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
equalityUniverse,
levelHypothesis,
imageMemberEquality,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[I,J:fset(\mBbbN{})].
\mforall{}[f:Hom(face\_lattice(J);face\_lattice(I))]. \mforall{}[x:Point(face\_lattice(J))].
(f x) = x supposing \mforall{}i:names(J). (((f (i=0)) = (i=0)) \mwedge{} ((f (i=1)) = (i=1)))
supposing J \msubseteq{} I
Date html generated:
2017_10_05-AM-01_10_37
Last ObjectModification:
2017_03_02-PM-10_26_59
Theory : cubical!type!theory
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