Nuprl Lemma : real-fun-implies-sfun
∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[f:[a, b] ⟶ℝ]. real-sfun(f;a;b) supposing real-fun(f;a;b)
Proof
Definitions occuring in Statement :
real-sfun: real-sfun(f;a;b),
real-fun: real-fun(f;a;b),
rfun: I ⟶ℝ,
rccint: [l, u],
rleq: x ≤ y,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
real-fun: real-fun(f;a;b),
all: ∀x:A. B[x],
implies: P ⇒ Q,
rfun: I ⟶ℝ,
prop: ℙ,
real-sfun: real-sfun(f;a;b),
so_lambda: λ2x.t[x],
so_apply: x[s],
or: P ∨ Q,
not: ¬A,
guard: {T},
top: Top,
iff: P ⇐⇒ Q,
and: P ∧ Q,
false: False,
squash: ↓T,
sq_stable: SqStable(P),
rev_implies: P ⇐ Q,
cand: A c∧ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
req_wf,
real_wf,
i-member_wf,
rccint_wf,
rneq_wf,
set_wf,
real-fun_wf,
rfun_wf,
rleq_wf,
real-weak-Markov,
rneq-cases,
not_wf,
member_rccint_lemma,
req_inversion,
rneq_functionality,
req_weakening,
rneq_irrefl,
rmin-rleq,
rleq-rmax,
sq_stable__rleq,
rccint-icompact,
rmin_ub,
rmax_wf,
rmin_wf,
req_functionality,
rmin_functionality,
rmax_functionality,
rmin-req,
rleq_functionality,
rmax-req
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
extract_by_obid,
isectElimination,
applyEquality,
because_Cache,
independent_functionElimination,
hypothesis,
setElimination,
rename,
setEquality,
lambdaFormation,
independent_isectElimination,
unionElimination,
inlFormation,
inrFormation,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
productEquality,
independent_pairFormation,
imageElimination,
baseClosed,
imageMemberEquality,
dependent_set_memberEquality
Latex:
\mforall{}[a:\mBbbR{}]. \mforall{}[b:\{b:\mBbbR{}| a \mleq{} b\} ]. \mforall{}[f:[a, b] {}\mrightarrow{}\mBbbR{}]. real-sfun(f;a;b) supposing real-fun(f;a;b)
Date html generated:
2019_10_30-AM-07_17_14
Last ObjectModification:
2018_08_23-AM-11_24_31
Theory : reals
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