Nuprl Lemma : rset-member-rrange
∀I:Interval. ∀[f:I ⟶ℝ]. ∀r:ℝ. ((r ∈ I) ⇒ (f[r] ∈ f[x](x∈I)))
Proof
Definitions occuring in Statement :
rrange: f[x](x∈I),
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
rset-member: x ∈ A,
real: ℝ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
rrange: f[x](x∈I),
rset-member: x ∈ A,
exists: ∃x:A. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
so_apply: x[s],
rfun: I ⟶ℝ,
prop: ℙ,
uimplies: b supposing a
Lemmas referenced :
req_weakening,
i-member_wf,
req_wf,
real_wf,
rfun_wf,
interval_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
dependent_pairFormation,
hypothesisEquality,
cut,
hypothesis,
independent_pairFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
dependent_set_memberEquality,
because_Cache,
independent_isectElimination,
productEquality
Latex:
\mforall{}I:Interval. \mforall{}[f:I {}\mrightarrow{}\mBbbR{}]. \mforall{}r:\mBbbR{}. ((r \mmember{} I) {}\mRightarrow{} (f[r] \mmember{} f[x](x\mmember{}I)))
Date html generated:
2016_05_18-AM-09_08_27
Last ObjectModification:
2015_12_27-PM-11_31_00
Theory : reals
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