Nuprl Lemma : stable-union-metric-subspace
∀[X:Type]. ∀[d:metric(X)]. ∀[T:Type]. ∀[P:T ⟶ X ⟶ ℙ].
metric-subspace(X;d;stable-union(X;T;i,x.P[i;x])) supposing ∀i:T. ∀x,y:X. (P[i;x] ⇒ y ≡ x ⇒ P[i;y])
Proof
Definitions occuring in Statement :
metric-subspace: metric-subspace(X;d;A),
meq: x ≡ y,
metric: metric(X),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
guard: {T},
false: False,
not: ¬A,
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_apply: x[s],
so_apply: x[s1;s2],
exists: ∃x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
stable-union: Error :stable-union,
and: P ∧ Q,
metric-subspace: metric-subspace(X;d;A),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
istype-void,
istype-universe,
metric_wf,
subtype_rel_self,
strong-subtype_witness,
Error :stable-union_wf,
meq_wf,
strong-subtype-self,
not_wf,
strong-subtype-set3
Rules used in proof :
voidElimination,
productIsType,
dependent_pairFormation_alt,
promote_hyp,
dependent_set_memberEquality_alt,
isectIsTypeImplies,
isect_memberEquality_alt,
universeEquality,
instantiate,
functionIsType,
functionIsTypeImplies,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
dependent_functionElimination,
independent_functionElimination,
independent_pairEquality,
productElimination,
inhabitedIsType,
rename,
setElimination,
lambdaFormation_alt,
independent_isectElimination,
universeIsType,
hypothesis,
applyEquality,
productEquality,
lambdaEquality_alt,
sqequalRule,
because_Cache,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
independent_pairFormation,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} X {}\mrightarrow{} \mBbbP{}].
metric-subspace(X;d;stable-union(X;T;i,x.P[i;x]))
supposing \mforall{}i:T. \mforall{}x,y:X. (P[i;x] {}\mRightarrow{} y \mequiv{} x {}\mRightarrow{} P[i;y])
Date html generated:
2019_10_30-AM-06_31_28
Last ObjectModification:
2019_10_24-AM-10_23_16
Theory : reals
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