Nuprl Lemma : ifun_subtype_subinterval
∀[I,J:{J:Interval| icompact(J)} ]. {f:I ⟶ℝ| ifun(f;I)} ⊆r {f:J ⟶ℝ| ifun(f;J)} supposing J ⊆ I
Proof
Definitions occuring in Statement :
ifun: ifun(f;I),
subinterval: I ⊆ J ,
icompact: icompact(I),
rfun: I ⟶ℝ,
interval: Interval,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
subinterval: I ⊆ J ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rbetween: x≤y≤z,
icompact: icompact(I),
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
top: Top,
cand: A c∧ B,
guard: {T}
Lemmas referenced :
rfun_subtype,
ifun_wf,
sq_stable__icompact,
rfun_wf,
subinterval_wf,
set_wf,
interval_wf,
icompact_wf,
left-endpoint_wf,
i-member-compact,
rbetween_wf,
right-endpoint_wf,
rleq_weakening_equal,
icompact-endpoints-rleq,
member_rccint_lemma,
rleq_transitivity,
rleq_wf,
req_wf,
real_wf,
i-member_wf,
rccint_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
setElimination,
thin,
rename,
dependent_set_memberEquality,
hypothesisEquality,
applyEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
hypothesis,
independent_isectElimination,
sqequalRule,
dependent_functionElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
setEquality,
axiomEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
lambdaFormation,
productElimination,
promote_hyp,
voidElimination,
voidEquality,
productEquality
Latex:
\mforall{}[I,J:\{J:Interval| icompact(J)\} ]. \{f:I {}\mrightarrow{}\mBbbR{}| ifun(f;I)\} \msubseteq{}r \{f:J {}\mrightarrow{}\mBbbR{}| ifun(f;J)\} supposing J \msubseteq{} I
Date html generated:
2016_10_26-AM-09_48_58
Last ObjectModification:
2016_08_22-AM-11_38_04
Theory : reals
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