Nuprl Lemma : ifun-alt
∀I:Interval. ∀[f:I ⟶ℝ]. (ifun(f;I)) supposing ((∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ ((f x) = (f y)))) and icompact(I))
Proof
Definitions occuring in Statement :
ifun: ifun(f;I),
icompact: icompact(I),
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
req: x = y,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a
Definitions unfolded in proof :
guard: {T},
cand: A c∧ B,
top: Top,
subinterval: I ⊆ J ,
and: P ∧ Q,
icompact: icompact(I),
squash: ↓T,
sq_stable: SqStable(P),
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
real-fun: real-fun(f;a;b),
ifun: ifun(f;I),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
r-ap: f(x)
Lemmas referenced :
sq_stable__req,
sq_stable__rleq,
member_rccint_lemma,
trivial-subinterval,
sq_stable__icompact,
interval_wf,
rfun_wf,
icompact_wf,
sq_stable__i-member,
r-ap_wf,
all_wf,
right-endpoint_wf,
left-endpoint_wf,
rccint_wf,
i-member_wf,
real_wf,
set_wf,
req_wf
Rules used in proof :
dependent_set_memberEquality,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
productElimination,
imageElimination,
baseClosed,
imageMemberEquality,
independent_functionElimination,
dependent_functionElimination,
functionEquality,
setEquality,
independent_isectElimination,
lambdaEquality,
because_Cache,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
isect_memberFormation,
lambdaFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}I:Interval
\mforall{}[f:I {}\mrightarrow{}\mBbbR{}]
(ifun(f;I)) supposing ((\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . ((x = y) {}\mRightarrow{} ((f x) = (f y)))) and icompact(I))
Date html generated:
2018_07_29-AM-09_40_30
Last ObjectModification:
2018_07_02-PM-00_31_17
Theory : reals
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