Nuprl Lemma : ifun_subtype_3
∀[a,b,c,d:ℝ]. ((a ≤ c) ⇒ (c ≤ d) ⇒ (d ≤ b) ⇒ ({f:[a, b] ⟶ℝ| ifun(f;[a, b])} ⊆r {f:[c, d] ⟶ℝ| ifun(f;[c, d])} ))
Proof
Definitions occuring in Statement :
ifun: ifun(f;I),
rfun: I ⟶ℝ,
rccint: [l, u],
rleq: x ≤ y,
real: ℝ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
prop: ℙ,
guard: {T},
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B
Lemmas referenced :
rfun_subtype_3,
ifun_wf,
rccint_wf,
rccint-icompact,
rfun_wf,
rleq_transitivity,
rleq_wf,
real_wf,
member_rccint_lemma,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
subtype_rel_sets,
set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
lambdaEquality,
setElimination,
thin,
rename,
dependent_set_memberEquality,
hypothesisEquality,
applyEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
independent_functionElimination,
hypothesis,
sqequalRule,
independent_isectElimination,
dependent_functionElimination,
productElimination,
setEquality,
because_Cache,
axiomEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productEquality,
independent_pairFormation
Latex:
\mforall{}[a,b,c,d:\mBbbR{}].
((a \mleq{} c)
{}\mRightarrow{} (c \mleq{} d)
{}\mRightarrow{} (d \mleq{} b)
{}\mRightarrow{} (\{f:[a, b] {}\mrightarrow{}\mBbbR{}| ifun(f;[a, b])\} \msubseteq{}r \{f:[c, d] {}\mrightarrow{}\mBbbR{}| ifun(f;[c, d])\} ))
Date html generated:
2016_10_26-AM-09_48_45
Last ObjectModification:
2016_08_20-PM-07_30_31
Theory : reals
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